long-term room and pillar mine production planning based...

Research ArticleLong-Term Room and Pillar Mine Production Planning Basedon Fuzzy 0-1 Linear Programing and Multicriteria ClusteringAlgorithm with Uncertainty

Miloš V GligoriT ZoranM GligoriT Hedomir R BeljiTSuzana M Lutovac and Vesna M DamnjanoviT

Faculty of Mining and Geology University of Belgrade Belgrade 11000 Serbia

Correspondence should be addressed to Milos V Gligoric milosgligoricrgfbgacrs

Received 6 December 2018 Accepted 13 June 2019 Published 27 June 2019

Academic Editor Alessandro Formisano

Copyright copy 2019 Milos VGligoric et alThis is an open access article distributed under theCreativeCommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Production planning in an underground mine plays a key activity in the mining company business It is supported by the fact thatmineral industry is unique and volatile environment There are two uncertain parameters that cannot be managed by plannersmetal price and operating costs Having ability to quantify and incorporate them in the process of planning can help companies todo their business in much easier way We quantify these uncertainties by the simulation of mean reverting process and Ito-Doobstochastic differential equation respectively Mineral deposit is represented as a set of mineable blocks and room and pillar miningmethod is selected as a way of mining Multicriteria clustering algorithm is used to create areas inside of mineral deposit that havetechnological characteristics required by the plannersWe also developed a way to forecast the volatility of economic values of theseareas through the planning period Fuzzy 0-1 linear programming model is used to define the sequence of mining of these areas bymaximization of the expected value of the fuzzy future cash flow Model was tested on small hypothetical lead-zinc mineral depositand results showed that our approach was able to solve such complex problem

1 Introduction

Certainly the investment environment associated with themining industry is unique when comparedwith environmentencountered by typical manufacturing industries Some ofcharacteristics which are often proclaimed as being uniqueare as follows Virtually every knowledgeable observer wouldagree that mining ventures are extremely capital intensiveEven extremely small high grade precious metal minesemploying only a handful of miners can rarely be developedfor operation less than a million dollars The amount of timerequired to develop a mining property for production canvary significantly Once the occurrence of an ore deposithas been well established it takes a number of years ofintensive effort before the property is brought on streamand ore is produced on a continuous basis In addition tothe obvious risks associated with capital intensity and longlead times there are a number of other risks associated withmining ventures In general these risks may be placed under

the general headings of geological risks engineering riskseconomic risks and political risks Perhaps the most uniqueaspect of theminerals industry is the fact that it deals with theextraction of a nonrenewable resource [1]

If we consider all these characteristics of the miningindustry we can conclude that production planning is essen-tial activity which helps mining companies to do businessin such environment Planning is an optimization problemwhere the searching for the global optimum solution isvery difficult and time consumption task There are manyapproaches that try to solve this optimization problem

OrsquoSullivan et al explained very well why undergroundmine production planning is very difficult task and provideddirections related to the application of heuristic techniquesthat rely on spatial andor temporal aggregation to producesolutions [2] Carlyle et al used mixed-integer programmingmodel which considers some planning constraints Valida-tion of thismodel was tested in one sector of the undergroundplatinum mine [3]

HindawiMathematical Problems in EngineeringVolume 2019 Article ID 3078234 26 pageshttpsdoiorg10115520193078234

2 Mathematical Problems in Engineering

Anani applied discrete event simulation to determinethe optimal width of coal room and pillars panels underspecific mining conditions She also tested the hypothesisthat heuristic preprocessing can be used to increase thecomputational efficiency of branch and cut solutions to thebinary integer linear programming problem of room andpillar mine sequencing The findings of her research includepanel width optimization a deterministic modelling frame-work that incorporates multiple mining risk in room andpillar production sequencing and accounting for changingduty cycles in continuous miner-shuttle car matching [4]Bakhtavar et al used 0-1 integer programming to createmodel which optimizes the way of transition from open pitmine to the underground mine [5] Nehring et al developeda new mathematical programming model for optimizationof production scheduling of a sublevel stopping operationwhich significantly reduces solution times without alteringresults while maintaining all constraints They representedall stope production phases by single binary variable andincreased efficiency of mixed-integer programming in theprocess of optimization [6] Bai et al developed algorithmfor stope design optimization at sublevel mining methodOptimization problemwas treated asmaximumflow over theadequate graph [7] A general capacitated multicommoditynetwork flow model has been used for long-term mineplanning by Epstein et al [8] Grieco et al applied a prob-abilistic mixed-integer programming approach to optimizestope in an underground mine ie to define location sizeand number of active stopes with uncertainty related toore grade and acceptable level of risk [9] Nehring et alintegrated short andmedium-termproduction plans by com-bining the short-termobjective ofminimizing deviation fromtargeted mill feed grade with the medium-term objectiveof maximizing net present value into a single mathematicaloptimization model [10] Terblanche and Bley reduced theresolution of underground mine scheduling problem andapplied mixed-integer programming to improve profitabilitythrough selective mining [11] Kuchta et al used mixed-integer programming to schedule Kirunarsquos operations specif-ically which production blocks to mine and when to minethem to minimize deviations from monthly planned pro-duction quantities while adhering to operational restrictions[12] Topal developed an early start and late start algorithmthat defines the precedence restrictions for each miningunit in their mixed-integer linear programming model ofthe underground Kiruna Mine [13] Hirschi developed adynamic programming algorithm to supplant that trial anderror practice of determining and evaluating room and pillarmining sequences Dynamic programming has been used inmining to optimize multistage processes where productionparameters are stage-specific [14] Gligoric et al developed aproduction planningmodel whichminimizes deviation fromthe acceptable rate of return using multivariable weightedFrobenius distance function thatmeasures the deviation fromestablished targets [15]

All developed mine production planning models werebased on 0-1 linear programming and different methods havebeen used to find the extreme value of the linear objectivefunction for example simplex method simulated annealing

Branch and Bound algorithm ant colony optimization neu-ral networks etc We applied fuzzy 0-1 linear programmingto incorporate the uncertainties in the objective function andmake the problem of production planning more realisticBy this way we increase the precision of the obtainedresults If we take into consideration that mine productionplanning belongs to the decision making field then fuzzymodel really helps us to make final decision in more efficientway

The main aim of this paper is to provide efficiencysupport to decision making on production planning inundergroundmines that use room and pillar mining methodas a way of mining Model is based on the maximizationof fuzzy objective function which represents the presentor discounted value of the future cash flow of productionplan with respect to the set of constraints We consider thisproblem as zero-one linear programing problem in whichonly coefficients in the objective function are triangular fuzzynumbers Coefficients represent the discounted economicvalue of the technological mining cut (TMC) which is a partof mineral deposit characterized with respect to the givenset of technological requirements such as annual capacity ofproduction compactness of the shape of TMC and standarddeviation of ore grade in the TMC Total number of techno-logical mining cuts is equal to the total number of years ofproduction

The first step in the production planning model is relatedto the creation of TMCs having the value of attributesclosely to the values of technological requirements In thepurpose of creation such TMCs (clusters) we developed fuzzymulticriteria clustering algorithm where uncertainties ofsome input data are quantified by triangular fuzzy numbersMining engineers uses a block model of the deposit thatrepresents the deposit as three-dimensional array of blocksAccordingly clustering algorithm is applied on the set ofthese blocks The second step concerns the calculation ofdiscounted economic value of TMCs It indicates that we arefacing dynamic problem burdened with some uncertaintiesThese uncertainties come primary from the metal price andoperating costs fluctuation through the time of planningTo estimate the future state of metal price we developedforecasting algorithm which represents the hybrid of thefuzzy C-mean clustering algorithm and stochastic diffu-sion process called mean reverting process This algorithmquantifies the future states of metal price by the fuzzyseries Operating costs are modelled by Ito-Doob stochasticdifferential equation Applying concurrently simulations ofthese two parameters we can estimate the expected fuzzyvalue of each TMC for every year of the planning time Afterthat we discount these values by fuzzy discount rate anddefine the values of coefficient of objective function Solutionof the fuzzy objective function gives the order of mining ofTMCs

The proposed model is a mathematical representationof mining business reality and allows mining companymanagement to run a dynamic optimization of the businesswith uncertainty It helps mining company to survive in veryrisky environment

Mathematical Problems in Engineering 3

2 Model of Production Planning

Production planning models based on the linear program-ming use block as a basic variable in the objective functionThese models also use the constant values of metal priceand operating costs through the planning time It meansthat these models are static from the point of view ofthese two parameters If we want to include fluctuation ofthese parameters in the objective function then number ofvariables significantly increases Suppose we have a mineraldeposit contains of 1 000 blocks and we want to mine themfor 10 years with a different metal price for every year thenthe number of variables is about 10 000 Our model reducesthe number of variables in the objective function by creationof TMCs It means that mentioned example would have only100 variables obtained as years of mining to the power of twoThis reduction becomesmore significant when dimensions ofthe block are small By decreasing the number of variables weenable uncertainties to be included in the model We believethat including of uncertainties is much more importantthan maximum value of the objective function obtained byusing blocks as variables with constant values of influencingparameters

Applying fuzzy set theory and simulation of differentstochastic processes we increased flexibility of the model andmade the problem more realistic The model was tested ona small hypothetical lead-zinc mineral deposit and resultsshowed that model can be used for solving the problem ofmine production planning

21 Basic Concepts of the Fuzzy Linear Programming Fuzzyset theory introduced by Zadeh deals with problems inwhich a source of vagueness is involved and has been utilizedfor incorporating imprecise data into decision framework[16 17]

The characteristic function 120583119860 of a crisp set 119860 sube 119883assigns a value either 0 or 1 to each member in X Thisfunction can be generalized to a function 120583119860 such that valueassigned to the element of the universal set X falls within aspecified range ie 120583119860 119883 997888rarr [0 1] The assigned valueindicates the membership grade of the element in the set AThe function 120583119860 is called the membership function and theset 119860 = (119909 120583119860(119909)) 119909 isin 119883 defined by 120583119860(119909) for each 119909 isin 119883is called a fuzzy set [18 19]

A fuzzy number 119860 = (119886 119887 119888) is said to be a triangularfuzzy number if its membership function is given by

120583119860 (119909) =

119909 minus 119886119887 minus 119886 119886 le 119909 le 119887119909 minus 119888119887 minus 119888 119887 le 119909 le 1198880 119900119905ℎ119890119903119908119894119904119890

(1)

For more details of arithmetic operations on triangular fuzzynumbers see [16 18]

The absolute value of the triangular fuzzy number 119860 =(119886 119887 119888) is denoted by |119860| and defined as follows [16]

10038161003816100381610038161003816119860 (119909)10038161003816100381610038161003816 = max 119860 (119909) minus119860 (119909) 119894119891 119909 ge 00 119894119891 119909 lt 0 (2)

A ranking function is a functionR 119865(119877) 997888rarr 119877 where F(R)is a set of fuzzy numbers definedon set of real numbers whichmaps each fuzzy number into real line where a natural orderexists Let119860 = (119886 119887 119888) be a triangular fuzzy number then [18]

R (119860) = 119886 + 2119887 + 1198884 (3)

Linear programming is one of the most frequently appliedoperations research techniques In the conventional approachvalue of the parameters of linear programming modelsmust be well defined and precise However in real worldenvironment this is not realistic assumption In the real-lifeproblems there may exist uncertainty about the parametersIn such a situation the parameters of linear programmingproblems may be represented as fuzzy numbers [18]

In this paper we consider zero-one linear programingproblem in which only coefficients in the objective functionare triangular fuzzy numbers Such problem is first convertedinto an adequate crisp model and after that being solved byone of the existing methods

Suppose we have a linear programming problem withfuzzy coefficients as follows

max119899sum119895=1

(119886119895 119887119895 119888119895) 119909119895 (4)

subject to119899sum119895=1

119902119894119895119909119895 le 119901119894 1 le 119894 le 119898 1 le 119895 le 119899 (5)

119909119895 isin [0 1] (6)

Since variables xj and coefficients qij are crisp values it isnecessary only to convert fuzzy objective function into crispfunction The process of conversion is based on the waydeveloped by Kumar et al [18] Fuzzy objective function maybe expressed as follows

maxR( 119899sum119895=1

(119886119895 119887119895 119888119895) 119909119895) (7)

Example 1 Let us consider the following fuzzy objectivefunction and convert it by the proposed method

max ((1 6 9) 1199091 + (2 3 8) 1199092 + (4 6 7) 1199093) (8)

The fuzzy objective function may be written as follows

maxR (11199091 + 21199092 + 41199093 61199091 + 31199092 + 61199093 91199091 + 81199092+ 71199093) (9)

Using arithmetic operations on triangular fuzzy numbersand (3) the fuzzy objective function is converted into thefollowing crisp objective function

max (14 (221199091 + 161199092 + 231199093)) (10)


Suppose the solution of this objective function with respectto a given set of constraints is x1=x3=1 and x2=0 Fuzzyoptimal value of our objective function is obtained by puttingx1 x2 and x3 in (8) The value of the given objective functionis

((1 6 9) ∙ 1 + (2 3 8) ∙ 0 + (4 6 7) ∙ 1) = (5 12 16) (11)

An important concept related to the applications of fuzzynumbers is defuzzification which converts a fuzzy numberinto a crisp value Such a transformation is not uniquebecause different methods are possible The most commonlyused defuzzification method is the centroid defuzzificationmethod which is also known as center of gravity or centerof area defuzzification The centroid defuzzification methodcan be expressed as follows (Yager 1981) [20]

1199090 (119860) = int119888119886119909120583119860 (119909) 119889119909int119888119886120583119860 (119909) 119889119909 (12)

where 1199090(119860) is the defuzzified value The defuzzificationformula of triangular fuzzy number 119860 = (119886 119887 119888) is

1199090 (119860) = 119886 + 119887 + 1198883 (13)

This formula will be used in this paper Defuzzified (crisp)value of our objective function is 1199090(5 12 16) = 1122TheModel In general terms any economic evaluation ofa mine production plan is defined by its financial outcomesProduction planning model from the economic point ofview is defined by the following objective function and setof constraints

119865 ( 119883) = 119873sum119894=1

119879sum119905=1

V119889119894119905119909119894119905 = 119873sum119894=1

119879sum119905=1

V119894119905(1 + 119889)119905 119909119894119905 997888rarr 119898119886119909 (14)

subject to

119879sum119905=1

119909119894119905 = 1 forall119894 isin 119873 (15)

119873sum119894=1

119909119894119905 = 1 forall119905 isin 119879 (16)

119909119894119905 + Ssums=1119909s119905+1 le 1 forall119894 isin 119873 forall119905 isin 119879 s isin S (17)

119909119894119905 isin [0 1] forall119894 isin 119873 forall119905 isin 119879 (18)

119865 minus 119868 ge 0 (19)

where

V119889119894119905 is fuzzy present value of the technological miningcut discounted valueV119894119905 is fuzzy economic value of the technologicalmining cut in time t

benching ofthicker partsof ore body

copy 2007 Encyclopaedia Britannica Inc

front benching

drill jumbo

connecting dri

verticalbenching

pillar

pillar

Source H Hamrin Guide to UndergroundMining Methods and Applications(Stockholm Atlas Copco 1980)

Figure 1 Room and pillar mining method [22]

119909119894119905 is binary variable which equals 1 if and only if thetechnological mining cut i is mined in time tN is number of technological mining cuts consideredfor planning It is equal to the number ofmining years(T)S is set of technological mining cuts whose arenot accessible from xit ie this set is composed ofnonneighbouring mining cuts119889 is fuzzy discount rateT is the planning time119868 is fuzzy capital investment (capital costs)

Objective function represents the present value of the futurecash flow of production plan Equations (15) and (16) ensurethat each technological mining cut can be mined only onceover the planning time Equation (17) defines temporal-spatial nonconnectivity between cuts from t to t+1 andensures the concentration of production Equation (19) repre-sents the decision making constraint A positive value of (19)indicates that the obtained production plan is profitable andshould be accepted

Model is to maximize the present value of the productionplan which is generated by mining the technological miningcuts over the planning time

23 Creation of Technological Mining Cuts Room and pillarminingmethod is designed for flat bedded deposits of limitedthickness This method is used to recover resources in openstopes The method leaves pillars to support the hangingwall to recover the maximum amount of ore miners aimto leave the smallest possible pillars Rooms and pillars arenormally arranged in regular patterns Pillars can be designedwith circular or square cross sections see Figure 1 Mineralscontained in pillars are nonrecoverable and therefore are notincluded in the ore reserves of the mine [21]

Technological mining cut (TMC) is a part of mineraldeposit characterized with respect to the given set of tech-nological requirements (criteria) It means that TMC is a


multiattribute object Suppose the mineral deposit is dividedinto finite number of mineable blocks The first step in theproduction planning is related to the creation ofTMCs havingthe value of attributes closely to the values of technologicalrequirements Hence the first step concerns partition ofthe deposit in adequate number of TMCs To create theprocess of production planning more realistic we applythe concept of fuzzy set theory for some input data Bythis approach uncertainties of input data are decreasedand planning becomes much more flexible To create suchTMCs (clusters) we developed fuzzy multicriteria clusteringalgorithm which is based on Technique for order preferenceby similarity to ideal solution [23] and constrained polygonalspatial clustering algorithm [24ndash26]

Mining engineers uses a block model of the deposit thatrepresents the deposit as three-dimensional array of blocksSuch model is created by applying geostatistical methodson data obtained by exploration drilling In the process ofproduction planning block is defined as a basic object

Mineral deposit can be represented as a set of mineableblocks 119861 = 119887ℎℎ=12119867 and each block is characterized bythe block attribute vector 119861119860119881ℎ = 119886ℎ119895 119895=12119860forallℎisin119867 where His the total number of blocks andA total number of attributessuch as block tonnage ore grade etc

A set 119879119872119862119894 = 119887119894119897 119897lt119867forall119894isin119873 is defined as a subset of B andcalled technological mining cut

Each TMC is characterized by the mining cut attributevector 119872119862119860119881119894 = 119886119894119888119888=12119870forall119894isin119873 where K is the totalnumber of attributes and it is equal to the total numberof technological criteria Vector of technological criteria isdefined by themine production planer (decisionmaker)119862 =119862119895119895=12119870

At last creation of technological mining cuts can bemathematically formulated as a multiobjective partitionproblem wherein the TMCs must meet technological perfor-mance criteria subject to the given criteria constraints

max⏟⏟⏟⏟⏟⏟⏟1198861119888isin1198791198721198621

119891 (1198861119888) = [1198911 (11988611) 1198912 (11988612) 119891119870 (1198861119870)]max⏟⏟⏟⏟⏟⏟⏟

1198862119888isin1198791198721198622

119891 (1198862119888) = [1198911 (11988621) 1198912 (11988622) 119891119870 (1198862119870)]

max⏟⏟⏟⏟⏟⏟⏟119886119894119888isin119879119872119862119894

119891 (119886119894119888) = [1198911 (1198861198941) 1198912 (1198861198942) 119891119870 (119886119873119870)](20)

subject to

119886119897119888 le 119886119888 le 119886119906119888 119888 = 1 2 119870 (21)

where

119891(119886119894119888) is the ultimate relative closeness (URC) of the ithtechnological mining cut to the positive ideal techno-logical solution taking into account all technologicalcriteria119886119897119888 is the lower bound of value of the cth technologicalcriterion

119886119906119888 is the upper bound of value of the cth technologicalcriterion

Note some of criteria can be excluded from the set of criteriaconstraints it depends on the nature of the criterion Solutionof this problem is given as follows

119861 = 119873⋃119894=1

119879119872119862119894= 1198791198721198621 cup 1198791198721198622 cup cup 119879119872119862119894119879119872119862120572 cap 119879119872119862120573 = 0 120572 = 120573

(22)

Creation of the set119879119872119862119894 = 119887119894119897 119897lt119867forall119894isin119873 is the two-stage fuzzymulticriteria clustering process which can be treated as two-stage multicriteria decision making process At first stage wemake decision on which cluster (TMC) is to grow while atsecond which block should be added to the selected clusterThese two stages represent the one iteration and the processof clustering is iteratively repeated until no free mineableblocks in the deposit

Maximization of the ultimate similarity between vector119872119862119860119881119894 = 119886119894119888119888=12119870forall119894isin119873 and 119862119905119890ℎ = 119888119905119890ℎ119895 119895=12119870where 119862119905119890ℎ represents the required technological vector isessential to clustering mineable blocks Measure of similarityis expressed by the relative closeness coefficient It is calcu-lated by technique for order preference by similarity to idealsolution (TOPSIS) For detailed description of the methodsee [23 27ndash30] and we have given brief description of itsapplication in the context of clustering

Suppose that we defined number (N) of technologicalmining cuts The first stage problem that considers whichTMC is to grow can be concisely expressed by the followingdecision making matrix

119863 = [119909119894119895]119873times119870 =[[[[[[[[[[[

119879119872119862119862 1198621 1198622 sdot sdot sdot 1198621198951198791198721198621 11990911 11990912 sdot sdot sdot 11990911198951198791198721198622 11990921 11990922 sdot sdot sdot 1199092119895 d119879119872119862119894 1199091198941 1199091198942 sdot sdot sdot 119909119894119895

]]]]]]]]]]](23)

where 119909119894119895 is the estimated value of technological mining cutTMCi with respect to the technological criteria Cj Note thatthere is a difference between required technological vectorand vector of technological criteria 119862119905119890ℎ = 119862 and it will beexplained latter

For simplicity of notation we expressed all values astriangular fuzzy numbers but some of them can be expressedas crisp value The weighted normalized decision makingmatrix is computed by multiplying normalized value of 119909119894119895with weights (119908119895) of technological criteria = [119901119894119895]119873times119870 = [119903119894119895 ∙ 119908119895]119873times119870= [ 119909119894119895sum119873119894=1 119909119894119895 ∙ 119908119895]119873times119870 forall119895 isin [1 119870] 119870sum

119895=1

119908119895 = 1 (24)


To avoid decision makerrsquos subjectivity about weights ofcriteria we applied concept of the entropy method [31 32]Entropy value of each criterion can be calculated as follows

119890119895 = minus 1ln (119873)

119873sum119894=1

119903119894119895 ∙ ln (119903119894119895) forall119895 isin [1 119870] 0 le 119890119895 le 1

(25)

The objective weight for each criterion is given by thefollowing equation

119908119895 = 1 minus 119890119895sum119870119895=1 1 minus 119890119895 forall119895 isin [1 119870] 0 le 119908119895 le 1 (26)

The fuzzy positive ideal solution 119860+ and negative one 119860minus isdefined as

119860+ = ( max⏟⏟⏟⏟⏟⏟⏟119894=12119873

119901119894119895 | 119895 isin 119869) ( min⏟⏟⏟⏟⏟⏟⏟119894=12119873

119901119894119895 | 119895 isin 1198691015840)= 119901+1 119901+2 119901+119870

(27)

119860minus = ( min⏟⏟⏟⏟⏟⏟⏟119894=12119873

119901119894119895 | 119895 isin 119869) ( max⏟⏟⏟⏟⏟⏟⏟119894=12119873

119901119894119895 | 119895 isin 1198691015840)= 119901minus1 119901minus2 119901minus119870

(28)

max⏟⏟⏟⏟⏟⏟⏟119894=12119873

119901119894119895 = max⏟⏟⏟⏟⏟⏟⏟119894=12119873

(119886119894119895 119887119894119895 119888119894119895) = max⏟⏟⏟⏟⏟⏟⏟119894=12119873

(119887119894119895) forall119895 isin [1 119870] (29)

min⏟⏟⏟⏟⏟⏟⏟119894=12119873

119901119894119895 = min⏟⏟⏟⏟⏟⏟⏟119894=12119873

(119886119894119895 119887119894119895 119888119894119895) = min⏟⏟⏟⏟⏟⏟⏟119894=12119873

(119887119894119895) forall119895 isin [1 119870] (30)

where

J = 119895 = 12K | j associated with criteria that shouldbe maximized1198691015840 = 119895 = 12K | j associated with criteria that shouldbe minimized

The distance from each TMC to 119860+ and 119860minus is calculatedaccording to the following equations

119889+119894 = 119870sum119895=1

119889 (119901119894119895 119901+119895 ) forall119894 isin [1119873] (31)

119889minus119894 = 119870sum119895=1

119889 (119901119894119895 119901minus119895 ) forall119894 isin [1119873] (32)

where 119889(119901119894119895 119901119895) is the distance measurement between twofuzzy triangular numbers calculated as follows

119889 (119901119894119895 119901119895)= radic13 [(119886119894119895 minus 119886119895)2 + (119887119894119895 minus 119887119895)2 + (119888119894119895 minus 119888119895)2]

(33)

The relative closeness coefficient of each TMC is calculated as

119877119862119862119894 = 119889minus119894119889minus119894 + 119889+119894 forall119894 isin [1119873] (34)

Decision on which TMC is to grow is making according tothe following selection rule

119879119872119862119892119903119900119908 = max (1198771198621198621 1198771198621198622 119877119862119862119873) (35)

The second stage problem that considers which block shouldbe added to the selected cluster (TMCgrow) is defined by thefollowing form

119887119886119889119889 = max (1198771198621198621198861198891198891 + 1198741198861198891198891 1198771198621198621198861198891198892 + 1198741198861198891198892 + sdot sdot sdot+ 119877119862119862119886119889119889119872 + 119874119886119889119889119872 ) (36)

where

119877119862119862119886119889119889119872 is the new relative closeness coefficient of thenewTMC obtained after adding themth neighbouringblock to the TMCgrow and neighbouring block isthe block that has at least one common edge withTMCgrow119874119886119889119889119872 is penalty or cost functionM is number of neighbouring mineable blocks

The new relative closeness coefficients are calculated byapplying TOPSIS on the following decision making matrix

119863119887119897119900119888119896 = [120593119898119895]119872times119870

=[[[[[[[[[[[[

(119879119872119862119892119903119900119908 cup 119887)119862 1198621 1198622 sdot sdot sdot 119862119895119879119872119862119892119903119900119908 cup 1198871 12059311 12059312 sdot sdot sdot 1205931119895119879119872119862119892119903119900119908 cup 1198872 12059321 12059322 sdot sdot sdot 1205932119895 d119879119872119862119892119903119900119908 cup 119887119898 1205931198981 1205931198982 sdot sdot sdot 120593119898119895

]]]]]]]]]]]]

(37)

where

120593119898119895 is the new estimated value of the 119879119872119862119892119903119900119908 cup119887119898 119898 = 1 2 119872 with respect to the technologicalcriteria Cj

The value 120593119898119895 is estimated after making the union of theattribute vector of the 119879119872119862119892119903119900119908 and the attribute vector ofthe neighbouring block

120593119898119895 = 119872119862119860119881119892119903119900119908 cup 119861119860119881119898= 119886119892119903119900119908119888 119888=12119870 cup 119886119898119895 119895=12119860forall119898isin119872 119870 = 119860 (38)

With the use of the cost function our objective is to select acluster to be grown that will preserve the maximum degreeof flexibility for the other clusters to grow In order to select acost function that measures the reduction in flexibility on the


growth of the clusters we observe the effect of the growth ofone cluster on the ability of growth of the other clusters Thiscost function is as follows [24]

119874119886119889119889119872 = 119861119887119890119891 minus 119861119886119891119905119898119861119887119890119891 = 119861119888119900119898119898 + 119861119899119892ℎ119898 + 119861119908119886119904119905119890119898 minus 3119861119887119890119891 forall119898 isin [1119872] (39)

where

119861119887119890119891 is number of mineable blocks surrounding theTMCgrow before adding the newmth block119861119886119891119905119898 is number of mineable blocks surrounding theTMCgrow after adding the newmth block119861119888119900119898119898 is number of common edges between TMCgrow

and block to be added119861119899119892ℎ119898 is number of common edges between block to beadded and remaining TMCs119861119908119886119904119905119890119898 is number of common edges between block tobe added and waste blocks Waste block is block thathas not grade

When we define the set 119861119887119890119891 and 119861119886119891119905119898 it is necessary to meetthe following two spatial constraints

(i) only blocks having at least one common edge withTMCgrow can be added to the TMCgrow

(ii) 119879119872119862119894 forall119894 isin [1119873] must not be divided in two ormore parts ie technological mining cut must bealways homogeneous

The second constraint means that any mineable block thatviolates the spatial homogeneity of any TMC cannot beelement of 119861119887119890119891 and 119861119886119891119905119898 respectively Suppose the TMC1 isselected to grow see Figure 2

According to spatial constraints only blocks 20 32 and33 can be elements of the set 119861119887119890119891 because block 26 violatesspatial homogeneity of the TMC2 If we add block 33 to theTMC1 than only blocks 20 32 and 34 can be elements of theset 119861119886119891119905119898 because block 46 violates spatial homogeneity of theTMC3 Hence for 1198791198721198621 cup 11988733 value of the penalty functionis equal to 0 Note mutual takeover of blocks by TMCs isallowed but spatial homogeneity of TMCs must be alwayspreserved

In our model the block attribute vector 119861119860119881ℎ =119886ℎ119895 119895=12119860forallℎisin119867 is composed of the following components

119861119860119881ℎ = 119886ℎ1 119886ℎ21 119886ℎ22 119886ℎ2120574119895=1+120574forallℎisin119867 (40)

where

119886ℎ1 is ore tonnage in block h (t) expressed as fuzzytriangular number119886ℎ2120574 is grade of the hth block with respect to the 120574thmetal ()120574 is total number of metal concentrates beneficiatedfrom the ore For polymetallic ore 120574 gt 1

21 27 35 48

20 26 34 47

19 25 33 46

18 24 32 45 Waste

TMC1

TMC2

TMC3

Figure 2 Spatial plan of TMCs

Themining cut attribute vector119872119862119860119881119894 = 119886119894119888119888=12119870forall119894isin119873 iscomposed of the following components

119872119862119860119881119894 = 119886119894119888119888=12119870forall119894isin119873= 1198861198941 1198861198942 11988611989431 11988611989432 1198861198943120574119888=2+120574forall119894isin119873 (41)

where

1198861198941 is ore tonnage in the TMC (t) expressed as fuzzytriangular number1198861198942 is compactness of the TMC1198861198943120574 is standard deviation of the grade in the TMCwith respect to the 120574th metal ()

Ore tonnage in theTMC represents a total sumof ore tonnagein blocks contained within TMC

1198861198941 = 119897119894sumℎ=1

119886ℎ1 forall119894 isin [1119873] 1198971 + 1198972 + sdot sdot sdot + 119897119873 = 119867 (42)

Compactness of the TMC is defined as the following ratio ofthe square of the perimeter and the area of the TMC

1198861198942 = 1198752119894119860 119894 = 1198752119894119897119894 ∙ (1198872119890 ) forall119894 isin [1119873] (43)

where

119875119894 is perimeter of the ith TMC

119860 119894 is area of the ith TMC

119897119894 is total number of blocks in the ith TMC119887119890 is length of the block edge (m)

Standard deviation of the grade in the TMC with respect tothe 120574th metal is calculated as follows

1198861198943120574 = radic 1119897119894119897119894sumℎ=1

(1198862120574 minus 119886ℎ2120574)2119897119894 lt 119867 forall119894 isin [1119873] forall120574 isin [1Y]

(44)


The required technological vector 119862119905119890ℎ = 119888119905119890ℎ119895 119895=12119870includes the following components

119862119905119890ℎ = 119888119905119890ℎ119895 119895=12119870

= 119888119905119890ℎ1 119888119905119890ℎ2 119888119905119890ℎ31 119888119905119890ℎ32 119888119905119890ℎ3120574 119895=2+120574 (45)

where

119888119905119890ℎ1 is annual capacity of production (tyear)expressed as fuzzy triangular number119888119905119890ℎ2 is desired or target value of compactness of TMCand it is set up to 16119888119905119890ℎ3120574 is standard deviation of the grade with respect tothe 120574th metal

Annual capacity of production represents the quantity of orethat should be mined for one year It is calculated as a totalsum of ore tonnage in blocks divided by the total number ofplanning periods (number of technological mining cuts)

119888119905119890ℎ1 = 1119873119867sumℎ=1

119886ℎ1 (46)

Target value of compactness of the TMC is expressed by theSchwartzbergs index of the simple square geometric shape[33]

119888119905119890ℎ2 = (4119890)21198902 = 16 (47)

where e is the edge of the square or mineable block Standarddeviation of the grade with respect to the 120574th metal corre-sponds to the standard deviation of the grade in the TMCwith respect to the 120574th metal It is calculated as follows andthere is no target value for this component

119888119905119890ℎ3120574 = 1198861198943120574 (48)

The main aim of the vector of technological criteria is toenable creation of TMCs so that the ultimate similaritybetween vector 119872119862119860119881119894 = 119886119894119888119888=12119870forall119894isin119873 and 119862119905119890ℎ =119888119905119890ℎ119895 119895=12119870 is maximized It means that each TMC mustmeet technological requirements as maximum as possibleVector of technological criteria 119862 = 119862119895119895=12119870 is composedof the following components

119862 = 119862119895119895=12119870 = 1198621 1198622 11986231 11986232 1198623120574119895=2+120574 (49)

where

1198621 is absolute distance between annual capacity ofproduction and ore tonnage in the TMC (t)1198622 is absolute distance between target value of com-pactness and compactness of TMC1198623120574 is standard deviation of the grade in the TMCwith respect to the 120574th metal ()

Value of technological mining cut TMCi with respect to thetechnological criteria1198621 is calculated as follows and it shouldbe minimized

1199091198941 = 1198860 (1198861198941)1198880 (119888119905119890ℎ1 ) times 100 forall119894 isin [1119873] (50)


1199091198942 = 100381610038161003816100381616 minus 11988611989421003816100381610038161003816 forall119894 isin [1119873] (51)

Value of technological mining cut TMCi with respect to thetechnological criteria 1198623120574 is as follows and it should beminimized

1199091198943120574 = 1198861198943120574 forall119894 isin [1119873] forall120574 isin [1Y] (52)

The same calculations are applied when we define values ofthe decision making matrix119863119887119897119900119888119896 = [120593119898119895]119872times119870

Set of the technological criteria constraints 119886119897119888 le 119886119888 le119886119906119888 119888 = 1 2 119870 is composed of only constraint related tothe criterion 1198621minus10 le 1198861 le +10minus10 le (1 minus 1198860 (1198861198941)1198880 (119888119905119890ℎ1 )) times 100 le +10

forall119894 isin [1119873](53)

where 1198860(1198861198941) is defuzzified value of the ore tonnage in theTMC and 1198880(119888119905119890ℎ1 ) is defuzzified value of the annual capacityof production (see (13))

According to the graph theory in this paper the mineraldeposit (MD) composed of allmineable blocks is conceived asa graphMD=(BE) where119861 = 119887ℎℎ=12119867 is a set ofmineableblocks and 119864 = 119890119906V | 119906 V = 1 2 119867 119906 = V is a setof edges with euv representing the common edge betweenblock u and block v [34 35] The problem of creation ofTMCs is solved by the multicriteria partitioning of graphMD This approach attempts to address situation in whichthe creation of TMCs should simultaneously maximize theultimate relative closeness (URC) of the each TMC to thetechnological requirements with respect to technologicalcriteria and constraints Given a set119861 = 119887ℎℎ=12119867 and totalnumber ofTMCs119873 ge 2 then themodel of creation of TMCscan be formulated as follows

119885 = max⏟⏟⏟⏟⏟⏟⏟forall119894isin[1119873]

119880119877119862119894= max⏟⏟⏟⏟⏟⏟⏟forall119894isin[1119873]

119889minus119894 (⋃119897119894lt119867ℎ=1 119887ℎ119894 )119889minus119894 (⋃119897119894lt119867ℎ=1 119887ℎ119894 ) + 119889+119894 (⋃119897119894lt119867ℎ=1 119887ℎ119894 )

(54)


1 Set iteration to 120585=12 Create a set 119879119872119862119894 = 1198791198721198621 1198791198721198622 1198791198721198621198733 Create a set 119862 = 119862119895119895=12119870 = 1198621 1198622 11986231 11986232 1198623120574119895=2+1205744 Select the best technological mining cut (BTMC) to grow119861119879119872119862 = 119879119872119862119892119903119900119908 = max(1198771198621198621 1198771198621198622 119877119862119862119873)5 Create the list of neighbouring mineable blocks (NMB) as candidates for the grow ofBTMC such that

51 block has at least one common edge with TMCgrow

52 none of the homogeneous of the remaining TMCs are violated53 119861119872119861 notin 119873119872119861 if

for 120585+1 119861119872119861 isin 119879119872119862119892119903119900119908120572

for 120585+2 119861119872119861 isin 119879119872119862119892119903119900119908120573

for 120585+3 119887ℎ isin 119879119872119862119892119903119900119908120572

This is to avoid algorithm gets stuck6 Select the best mineable blocks (BMB) from NMB to add to BTMC119861119872119861 = max(1198771198621198621198861198891198891 + 1198741198861198891198891 1198771198621198621198861198891198892 + 1198741198861198891198892 + sdot sdot sdot + 119877119862119862119886119889119889119872 + 119874119886119889119889119872 )7 Add BMB to BTMC and create new technological mining cut119873119879119872119862 = 119861119879119872119862 cup 1198611198721198618 Update the state of the mining cut attribute vector of the NTMC9 If

91 no free mineable blocks

92 minus10 le (1 minus 1198860 (1198861198941)1198880 (119888119905119890ℎ1 )) times 100 le +10 forall119894 isin [1119873]93 1198861198943120574 forall119894 isin [1119873] forall120574 isin [1Y] 997888rarr converge

than Stop10 Else go to Step 1 and set 120585+1

Algorithm 1 Algorithm of the TMC creation

subject to

119873sum119894=1

119887ℎ119894 = 1 forallℎ isin [1119867] (55)

1198861198971 le (1 minus 119889119890119891119906119911119911119894119891119894119890119889 (sum119867ℎ=1 119886ℎ1119894119887ℎ119894 )119889119890119891119906119911119911119894119891119894119890119889 (119888119905119890ℎ1 ) ) times 100le 1198861199061 forall119894 isin [1119873]

(56)

119887ℎ119894 isin [0 1] forallℎ isin 119867 forall119894 isin 119873 (57)

where (⋃119897119894lt119867ℎ=1

119887ℎ119894 ) represents the union of mineable blockscontained in the ultimate ith TMC Constraint (55) forces thata mineable block can belong to one and only one TMC Itmeans no intersection between TMCs 119879119872119862120572 cap 119879119872119862120573 =0 120572 = 120573 Constraint (56) is already explained and it doesnot allow any TMC to be terminated 119879119872119862119894 = 0 forall119894 isin [1119873]Constraint (57) is related to binary variable 119887ℎ119894 which equals1 if and only if the mineable block h belongs to the ith TMC

Here we present the algorithm of partitioning the graphMD and it is based on the constrained polygonal spatialclustering algorithm [24ndash26] Code of the algorithm is rep-resented by the Algorithm 1

Our two-stage fuzzy multicriteria clustering algorithm isused to partition themineral deposit into adequate number ofparts such that each part satisfies technological requirementsgiven by the mine production planner These parts are called

technological mining cuts Algorithm represents the iterativeprocess which starts from the initial state with the aim ofapproaching the desired goal Initialization stage means theselection of N technological mining cuts from the set ofmineable blocks having at least one common edge withwaste These blocks are located along the perimeter of themineral deposit (see Figure 2 blocks 18 19 24 32 and 45are candidates to be selected as initial TMC) Obviouslythe initial TMC is composed of one and only one mineableblock From the set of peripheral blocks we select the first Nblocks according to the decreasing order of values of relativecloseness coefficient Once the initial TMCs are selected theybegin to be alive and the process of TMCs growing canrun Each TMC is grown by adding neighbouring blocks tohim one by one until the desired state of TMC is achievedGrowing of TMCs is an iterative process in nature At thebeginning of the each iteration we first select which TMCis to be grown Selection is based on the measuring themultidimensional distance between current and desired stateof the each TMC It is expressed by the relative closenesscoefficient and TMC having the largest value of RCC isthe best to be grown Upon the selection we proceed topursue which block is the best to be added From the setof neighbouring blocks we select the best one by the sameapproach we have done in the selection of the best TMCbut with adding the penalty function Mutual takeover ofthe block between two neighbouring TMCc is allowed buthomogeneity of each TMCmust always be preserved Duringthe process of growing the infinity mutual takeover can


arise If the block has mutually been acquired between twoTMCs in the three successive iterations we can say the blockoscillates between them In that case the algorithm gets stuckand the local optimum is achieved To enable algorithm toget global optimum we exclude oscillating block from theset of neighbouring blocks and algorithm can goes on Afterthat excluded block is coming back into process Once theblock was added the mining cut attribute vector 119872119862119860119881119894 =119886119894119888119888=12119870forall119894isin119873 must be updated

Stopping conditions 91 and 92 for the developed algo-rithm represent the point when algorithm can be executedwhile conditions 93 allows algorithm to continue fine tuningThe process of growing goes on until no free mineable blocksand desired technological state of the each mining cut isachieved with respect to given errors

24 Economic Value of the Technological Mining Cut Gener-ally the economic estimation of theTMC is based on the threefollowing main components metal price costs and discountrate

One of the most influencing variables on the economicvalue of TMC is the metal price This variable belongs tothe set of external variables and cannot be managed bythe planners It is primarily governed by the metal marketbehaviour Ability to define the law of variable behaviourtrough the time can help planners to find out much moreefficient and realistic solutions By this way we also increasethe flexibility in the process of decision making For thatpurpose we developed forecasting algorithm which is basedon the combination of fuzzy C-mean clustering and meanreverting process

Consider a metal price and denote it as variable X If thevalue of that variable is governed by the laws of probabilitythen variable X can be treated as a stochastic variableSuppose that we monitored values of X at equal interval120591119894 = 120591119894 minus 120591119894minus1 = 119888119900119899119904119905 119894 = 2 3 T In this paper intervalof monitoring is one month and we use symbols 120591 and T tomake distinction between interval of monitoring and intervalof planning (one year) Such a sequence of monitored values119883120591 = 1199091 1199092 119909T is called stochastic time series If weassign some underlying probabilistic distribution to the timeseries then it becomes stochastic process

Model of forecast does not provide the exact pointestimation (crisp value) of variable but rather the fuzzy statethat the variable will be at the next point ie model generatesthe future sequence of fuzzy states The general concept isas follows the monitored time series of metal price is trans-formed into fuzzy state series by applying the fuzzy C-meanclustering algorithm while the future states are forecasted bystochastic diffusion process called mean reverting processThe goal of the forecastingmodel is to estimate the fuzzy statethat the metal price will fall within one of the a priori definedstatesThismodel is able to account for the dynamics ofmetalprice process and distinguish increasing from a decreasingperiod Therefore efficiency of the model directly dependson the use of relevant monitored information pertainingto this process Usage of the monitored information isprimary related to the calibration of the model ie to the

defining of the parameters that will govern the forecastingprocess

Fuzzification of monitored metal price time series is per-formedby fuzzyC-mean clustering algorithmThis algorithmbelongs to the partitioning methods that consist of dividingN objects into a specified number of M disjoint groups thatare also called classes or clusters Fuzzy C-mean algorithm isbased on minimization of the following least-squared errorsfunction

119865 = (119880119883 119862) = Tsum120591=1

119872sum119898=1

119906120596120591119898 (119909120591 minus 119888119898)2 (58)

subject to

119872sum119898=1

119906120591119898 = 1 120591 = 1 2 T (59)

0 le 119906120591119898 le 1 119898 = 1 2 119872 120591 = 1 2 T (60)

0 lt Tsum120591=1

119906119894119898 lt 119898 119898 = 1 2 119872 (61)

where

X is the vector of monitored metal prices 119883 =1199091 1199092 119909T ⫅ 119877TC is the vector of class centers 119862 = 1198881 1198882 119888119872 ⫅119877119872U is the fuzzy partition matrix 119880 = |119906120591119898|Ttimes119872120596 is the coefficient of fuzzification and we take valueequal to 2

The objective function F is iteratively minimized The iter-ation process stops until 119865(119895+1) minus 119865(119895) lt 120575 where j is thenumber of iteration and 120575 represents the minimum amountof improvement For more details see [36ndash39] Suppose thatpartitioning F(XC) has been done and the sequence ofobtained centers is sorted in an ascending order 1198881 lt 1198882 ltsdot sdot sdot lt 119888119872

To describe the value (level) of metal price we uselinguistic variables A linguistic variable is variable whosevalues are words or sentences in natural artificial language[40] Following seven linguistic variables are used for thatpurpose very very low (VVL) very low (VL) low (L)medium (M) high (H) very high (VH) and very very high(VVH) Accordingly number of clusters equals also seven(M=12 7) Range boundaries of each linguistic variableare defined by the transformation of linguistic variable intoadequate triangular fuzzy number and corresponding rangecode Range code is expressed as crisp number

Triangular fuzzy number is defined as a triplet 119860 =(119886 119887 119888) where parameters a b and c respectively indicatethe smallest possible value the most promising value andthe largest possible value This formulation is interpreted asmembership function and holds the following conditions(1) 119886 le 119887 le 119888 (2) membership function is increasing inthe interval [119886 119887] and decreasing in the interval [119887 119888] Range


Table 1 The transformation of linguistic variables to triangular fuzzy numbers and range code

Linguistic Triangular fuzzy number Membership Range code valuevariable function 119910119898 119898 = 1 2 7VVL (119886 lt 1198881 119887 = 1198881 1198881 lt 119888 le 1198881 + 11988822 ) (0 1 0) 1199101 = 1VL (1198881 + 11988822 lt 119886 119887 = 1198882 1198882 lt 119888 le 1198882 + 11988832 ) (0 1 0) 1199102 = 2L (1198882 + 11988832 lt 119886 119887 = 1198883 1198883 lt 119888 le 1198883 + 11988842 ) (0 1 0) 1199103 = 3M (1198883 + 11988842 lt 119886 119887 = 1198884 1198884 lt 119888 le 1198884 + 11988852 ) (0 1 0) 1199104 = 4H (1198884 + 11988852 lt 119886 119887 = 1198885 1198885 lt 119888 le 1198885 + 11988862 ) (0 1 0) 1199105 = 5VH (1198885 + 11988862 lt 119886 119887 = 1198886 1198886 lt 119888 le 1198886 + 11988872 ) (0 1 0) 1199106 = 6VVH (1198886 + 11988872 lt 119886 119887 = 1198887 1198887 lt 119888) (0 1 0) 1199107 = 7

boundaries and value of code of linguistic variables are shownin Table 1

Suppose a monthly monitored metal price time seriesis given 119875 = 119901120591120591=12T and a range code vector 119884 =119910119898119898=12119872 is defined on it Transformation of monitoredseries to the fuzzy state series (range code series) is performedaccording to the following tagging mechanism

119875 (119884)forall120591isin[1T]

=

1 119894119891 min 119901120591120591=12T le 119901120591 le 1198881 + 11988822119898 119894119891 119888119898minus1 + 1198881198982 lt 119901120591 le 119888119898 + 119888119898+12119872 119894119891 119888119898minus1 + 1198881198982 lt 119901120591 lt max 119901120591120591=12T 119898 = 2 3 119872 minus 1

(62)

At last the monitored series is expressed as a range code timeseries119875 = 119901120591120591=12T 997888rarr 119885 (120591119898)forall120591isin[1T] = 119911120591119898forall120591isin[1T]

119898 = 1 2 119872 (63)

For example119885 = 2 4 3 1 5 7 6 Uncertainties relatedto behaviour of future states of the metal price are modelledby a simulation of stochastic diffusion process called meanreverting process The main characteristic of this process isthe tendency of the commodity prices to revert to a long-termequilibrium level after significant short-term fluctuationsUnderlying process is described by the following equation[41]

119889119911119898 = 120572 (ln 119911119898 minus ln 119911119898) 119911119898119889120591 + 120590119911119898119889119882 (64)

where

119911119898 is long-term equilibrium fuzzy state expressed bythe corresponding range code value120572 is speed of mean reverting to the long-term equilib-rium range code value

120590 is rate of range code volatility119889119882 is an increment to a standard Brownian motion

The correct discrete-time format for the continuous-timeprocess of mean reverting is the stationary first orderautoregressive process [42] so the sample path simulationequation for 119911119898(120591) is performed by using exact discrete-timeexpression as follows

119911119898 (120591) = expln (119911119898(120591minus1)) 119890minus120572120591

+ [ln (119911119898) minus 12059022120572] (1 minus 119890minus120572120591)+ 119873 (0 1) 120590radic (1 minus 119890minus2120572120591)2120572

(65)

where

120591 is constant time interval from point 120591 to 120591 + 1 onemonth119873(0 1) is normally distributed random variable

Calibration of the mean reverting process is based on thefollowing regression

119911119898(120591+1) = 1205730 + 1205731119911119898(120591) + 120576 (66)

The speed of mean reverting (120572) equals the negative of theslope long-term equilibrium (119911119898) is the intercept divided bythe speed and the rate of range code volatility (120590) is standarddeviation from the regression

Let 119911119898 = 119911119898(120591) 120591 = 0 1 T denote an evolutionpath of range code values with spot values 119911119898(120591) where119911119898(120591) is calculated by (65) Simulation of (65) for every 120591 =1 2 T minus 1 is composed of the following two steps (a) docalculation of (65) and (b) for obtained result determine the


3

44

555

4

5

67

6

5

444

3333333333

444444

33

4

3

22

333

4

33

4444

333333333

4

5

67

6

0

1

2

3

4

5

6

7

0 5 10 15 20 25 30 35 40 45 50 55 60

Rang

e co

de v

alue

Month

Figure 3 One simulated evolution path of the range code values ona monthly span

range code value Suppose the 119911119904119894119898119898 (120591) is the value obtainedby (65) at point 120591 then corresponding range code value isassigned as follows

119911119898 (120591)forall120591isin[1T]

=

1 119894119891 0 le 119911119904119894119898119898 (120591) le 1 + 22119898 119894119891 119910119898minus1 + 1199101198982 lt 119911119904119894119898119898 (120591) le 119910119898 + 119910119898+12119872 119894119891 119911119904119894119898119898 (120591) gt 119910119898minus1 + 1199101198982

119898 = 2 3 119872 minus 1

(67)

Note the long-term equilibrium range code value obtainedby the regression should also be defined by (67)

Space of simulation of future metal price and way offuzzification is described as follows

|119885|119904119894119898119878timesT = 1003816100381610038161003816119911119904120591119898 1003816100381610038161003816119878timesT 119864119902 (67)997888997888997888997888997888rarr|119884|119904119894119898119878timesT = 1003816100381610038161003816119910119904120591119898 1003816100381610038161003816119878timesT Table 1997888997888997888997888997888rarr1003816100381610038161003816100381610038161003816100381610038161003816119904119894119898119878timesT = 1003816100381610038161003816119901119904120591119898 1003816100381610038161003816119878timesT

(68)

where S represents the total number of simulations and Ttime span of simulation Figure 3 represents an evolution pathof the range code values while Figure 4 distribution of rangecode values in the 20th month upon S simulations

When developing cost estimates it is important to dis-tinguish between the types of costs being estimated Theestimator is concerned with two primary types of costsfor project estimation capital costs and operating costs Inthe minerals industry capital costs or capital investmentgenerally means the amount of total capital dollars requiredto bring a mining property into production [1] These costsinclude underground mine construction and mineral pro-cessing plant construction costs and equipment purchaseUnderground mine construction and mineral processingplant construction are activities that can last a few years Inthat period it is reasonable to expect the fluctuation of capitalcosts In such environment the right-skewed triangular fuzzy

18

95

137125

71

3321

020406080

100120140160

1 2 3 4 5 6 7Range code value

Figure 4 Distribution of range code values obtained by 500simulations (the 20th month)

number might express uncertainties related to capital costs119868 = (119886 119887 119888) 119886 lt 119887 lt 119888 119887 minus 119886 lt 119888 minus 119887Operating costs are also very significant source of uncer-

tainties related to the economic value of TMC Operatingcosts are incurred directly in the production process Thesecosts include the ore and waste development of individualstopes the actual stopping activities the mine services thatprovides logistical support to the miners and the milling andprocessing of the ore at the plant [43] Some componentsof operating cost such as fuel electricity lubricants tyresreplacement parts and inputs that is used for mineralprocessing etc are usually purchased at market prices thatfluctuatemonthly Note whenwe talk about operating cost wemean unit operating cost Uncertainties related to these costsarise as a consequence that many suppliers that do businessinmarket conditions aswell are offering short-term contractsto mine as a way to protect their business activities In suchenvironment and if we add the fact that production will becarried out for many years then it is very important to predictthe future behaviour of operating costs For that purposewe apply a continuous-time process using the following Ito-Doob type stochastic differential equation [44]

119889119888120591 = 120583119888120591119889120591 + 120590119889119882120591 (69)

where 120583 and 120590 are the drift and volatility and 119882120591 is anormalized Brownian motion Solution of this equation is asfollows and it describes behaviour of operating costs

119888120591 = 119888120591minus1 exp(120583 minus 12059022 )120591 + 119873 (0 1) 120590radic120591 (70)

Applying simulation on (70)we can createmany artificial sce-narios of operating cost behaviour trough the time Space ofsimulation of operating costs is represented by the followingmatrix

|119862|119904119894119898119878timesT = 10038161003816100381610038161198881199041205911003816100381610038161003816119878timesT (71)

Bearing inmind that ourmodel is based on themaximizationof the expected present value of production plan then it isobvious that discount rate has significant influence Deter-mining with precision the appropriate discount rate to use for


financial analysis studies is a difficult much debate topic It isclear that the cost of acquiring investment funds must coveropportunity costs and transaction costs must compensatefor risk and must cover anticipated inflation Although thisplaces some bounds on the problem each of these items isalso difficult to quantify with precision [1] For example if weask experts about value of discount rate we can expect thefollowing answer rdquodiscount rate is about 10rdquoThis statementis burdened with vagueness rather than randomness It canbe treated as linguistic variable which can be quantified bytriangular fuzzy number 119889 = (119886 119887 119888) 119886 lt 119887 lt 119888

Upon discussion on all main components separately wehave to calculate the economic value of TMC according totheir joined influence

Metal concentrate is a final product of undergroundmining companies that have not smelting facilities Theyrealize revenues by selling it on metal concentrate marketValue of the concentrate directly depends on the metal pricemetal content of concentrate and metal recovery rate Ifwe take into consideration that metal price uncertainty isdescribed by the simulation and fuzzification then the unitevalue of concentrate is calculated as follows10038161003816100381610038161003816V11988811990011989910038161003816100381610038161003816119904119894119898119878timesT = 119898119888119900119899 ∙ 119898119898119903 ∙ 1003816100381610038161003816100381610038161003816100381610038161003816119904119894119898119878timesT

= 119898119888119900119899 ∙ 119898119898119903 ∙ 1003816100381610038161003816119901119904120591119898 1003816100381610038161003816119878timesT (72)

where 119901119904120591119898 is unit selling price of the metal in period 120591 andsimulation s expressed by the triangular fuzzy number($t)mcon is metal content of concentrate ()mmr is metal recovery rate ()

Economic value of mineable block is calculated as follows [1]

10038161003816100381610038161003816Vℎ10038161003816100381610038161003816119904119894119898119878timesT = 119886ℎ1 (Ysum120574=1

119886ℎ2120574 ∙ 119872120574119898119888119900119899120574 10038161003816100381610038161003816V11988811990011989910038161003816100381610038161003816119904119894119898119878timesT) minus |119862|119904119894119898119878timesT forallℎ isin [1119867]

(73)

whereY is the total number of metal concentrates benefici-ated from the ore119872120574 is mill recovery rate of the 120574th metal ()

Space of economic value ofmineable block can be representedby the following fuzzy matrix10038161003816100381610038161003816Vℎ10038161003816100381610038161003816119904119894119898119878timesT = 10038161003816100381610038161003816vℎ11990412059110038161003816100381610038161003816119878timesT

=1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(119886 119887 119888)ℎ11 (119886 119887 119888)ℎ12 (119886 119887 119888)ℎ1120591(119886 119887 119888)ℎ21 (119886 119887 119888)ℎ22 (119886 119887 119888)ℎ2120591 d

(119886 119887 119888)ℎ1199041 (119886 119887 119888)ℎ1199042 (119886 119887 119888)ℎ119878T

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

forallℎ isin [1119867]

(74)

Expected economic values of mineable block trough theplanning time are obtained by averaging each column of thematrix |Vℎ|119904119894119898119878timesT119864 (Vℎ)

forall120591isin[1T] =10038161003816100381610038161003816100381610038161003816100381610038161119878119878sum119904=1

(119886 119887 119888)ℎ1199041205911003816100381610038161003816100381610038161003816100381610038161003816forall120591isin[1T] forallℎ isin [1119867]

(75)

If we take into consideration that our planningmodel is basedon the annual time span then it is necessary to transformforecasted economic values of mineable block from monthlyto annual level

119864 (119864 (Vℎ120591)119905)119905=[1119879] =100381610038161003816100381610038161003816100381610038161003816100381611212sum120591=1

119864 (Vℎ120591)1199051003816100381610038161003816100381610038161003816100381610038161003816119905=[1119879]

forallℎ isin [1119867](76)

Let Vℎ119905 forallℎ isin [1119867] 119905 = [1 119879] be the expected economicvalue of mineable block through the planning time then itsdiscounted value for every year is defined as follows

1003816100381610038161003816100381610038161003816(Vℎ119905 )1198891003816100381610038161003816100381610038161003816 =10038161003816100381610038161003816100381610038161003816100381610038161003816

Vℎ119905(1 + 119889)1Vℎ119905(1 + 119889)2 sdot sdot sdot Vℎ119905(1 + 119889)119905

10038161003816100381610038161003816100381610038161003816100381610038161003816 forallℎ isin [1119867] 119905 = [1 119879]

(77)

Finally the coefficient V119889119894119905 which represents the fuzzy presentvalue of the technological mining cut (discounted value) iscalculated by the following equation

V119889119894119905 = 119897119894sumℎ=1

(Vℎ119894119905)119889 forall119894 isin [1119873] 119905 = [1 119879] 1198971 + 1198972 + sdot sdot sdot + 119897119873 = 119867

(78)

It means that discounted economic value of the TMC is asum of discounted economic values of the mineable blockscontained in TMC

3 Numerical Example

To test developed production planning model we borroweddata of small hypothetical lead-zinc ore body from Gligoricet al [15] Room and pillar mining method was selectedas way of mining the mineral deposit The management ofmining company is looking for the best production plan withuncertainties related to the metal price and operating costsMineable ore reserves were estimated about 862 245 t Tostart up the production it is necessary to do developmentof ore body and buildup mineral processing plant All theseactivities are defined as follows

(i) construction of development system based on theramp and horizontal drive that enables accessing theore body It will be used for purpose of ore haulageand fresh air intake


Table 2 Input parameters

Parameter ValueNumber of blocks 115Block dimension 25times25times5 mPillar dimension 8times8times5 mPlanning time 5 years 60 monthsNumber of technological mining cuts 5Metal content of concentrateLead 70Zinc 50Metal recovery rateLead 95

Zinc

(119898119888119900119899 minus 8) ∙ 100119898119888119900119899 le 85 119898119888119900119899 ∙ 119898119903 997888rarr 119898119888119900119899 minus 8(119898119888119900119899 minus 8) ∙ 100119898119888119900119899 gt 85 119898119888119900119899 ∙ 119898119903 997888rarr 85

42 Mill recovery rateLead 92 Zinc 80 Capital costs ($) (12 000 000 13 000 000 15 000 000)Capacity of production (tyear)-triangular fuzzy number (155 204 172 449 189 694)Capacity constraint (tyear) plusmn 10Fuzzy C-mean clustering-coefficient of fuzzification 2

Metal price linguistic variablesM=7 very very low (VVL) very low (VL) low (L) medium (M) high (H) very high(VH) very very high (VVH)

Discount rate-triangular fuzzy number () (10 11 12)

(ii) construction of horizontal drive and decline thatenables exhaust of foul air after having ventilatedworking places of the mine

(iii) purchase of new mining equipment(iv) buildup of the mineral processing plant(v) purchase of mining and processing equipment

For the purpose of planning the ore body is represented by115 blocks each 25times25times5 m The input parameters used fortesting the developed model are given in Figures 5 and 6 andTables 2 and 3

Regular pattern of square pillars means that pillars are leftunmined and distance between them is equal

Applying the clustering algorithm on the set of peripheralblocks we selected five blocks according to the decreasingorder of values of relative closeness coefficient If we takeinto consideration that all peripheral blocks have the samecompactness and standard deviation of the zinc and leadgrade are equal to zero then selection is based only on theabsolute distance between annual capacity of production andore tonnage in the peripheral block By this way initial TMCs(seeds) were defined see Figure 7

Having selected initial TMCs algorithm proceeded todo iterations and characteristically phases of transition arepresented by Figure 8

Figure 5 Ore body with regular pattern of square pillars


Table 3 Reserves of mineable blocks

Block Reserves (t) Block Reserves (t)triangular fuzzy number triangular fuzzy number

1 5301 5891 6480 59 6458 7176 78932 6844 7604 8365 60 6954 7727 84993 5081 5646 6210 61 6293 6992 76914 7064 7849 8634 62 7119 7910 87015 7615 8461 9307 63 6128 6809 74896 6954 7727 8499 64 7284 8094 89037 7725 8583 9442 65 5962 6625 72878 5715 6350 6984 66 8977 9974 109729 6747 7497 8247 67 6128 6809 748910 5508 6120 6732 68 7119 7910 870111 7367 8186 9004 69 6293 6992 769112 6871 7635 8398 70 6954 7727 849913 7491 8323 9156 71 6458 7176 789314 6747 7497 8247 72 6789 7543 829715 7615 8461 9307 73 6623 7359 809516 6623 7359 8095 74 6623 7359 809517 7615 8461 9307 75 6789 7543 829718 5907 6564 7220 76 6458 7176 789319 6871 7635 8398 77 6954 7727 849920 5715 6350 6984 78 6293 6992 769121 7064 7849 8634 79 7119 7910 870122 5522 6135 6749 80 6128 6809 748923 6816 7574 8331 81 7284 8094 890324 6678 7421 8163 82 9130 10144 1115825 7367 8186 9004 83 6334 7038 774226 6541 7268 7994 84 7243 8048 885327 7505 8339 9172 85 6486 7206 792728 6458 7176 7893 86 7092 7880 866729 6637 7375 8112 87 6637 7375 811230 6279 6977 7674 88 6940 7711 848231 6816 7574 8331 89 6789 7543 829732 6100 6778 7456 90 6789 7543 829733 6995 7772 8550 91 6940 7711 848234 5921 6579 7237 92 6637 7375 811235 7174 7971 8768 93 7092 7880 866736 6486 7206 7927 94 6486 7206 792737 7092 7880 8667 95 7243 8048 885338 6637 7375 8112 96 8895 9883 1087239 6940 7711 8482 97 5921 6579 723740 6789 7543 8297 98 6995 7772 855041 6789 7543 8297 99 6100 6778 745642 6940 7711 8482 100 6816 7574 833143 6637 7375 8112 101 6279 6977 767444 7092 7880 8667 102 6637 7375 811245 6486 7206 7927 103 6458 7176 789346 7243 8048 8853 104 6458 7176 789347 6334 7038 7742 105 6637 7375 811248 7394 8216 9038 106 6279 6977 767449 6183 6870 7557 107 6816 7574 833150 6128 6809 7489 108 6100 6778 745651 7119 7910 8701 109 6678 7421 816352 6293 6992 7691 110 7229 8033 8836


Table 3 Continued


53 6954 7727 8499 111 6816 7574 833154 6458 7176 7893 112 6954 7727 849955 6789 7543 8297 113 6954 7727 849956 6623 7359 8095 114 7092 7880 866757 6623 7359 8095 115 6816 7574 833158 6789 7543 8297lowastBold italic font indicates peripheral blocks

Table 4 Ore tonnage and compactness of TMC

TMC Ore tonnage (t) Compactness Production capacity error ()

1152640

3913 -165169601186561

2153217

3072 -128170241187266

3154940

2939 -017172156189371

4155573

6278 +024172859190145

5159649

4816 +286177388195127

lowast(abc) triangular fuzzy number is expressed as column

(15-19](19-23](23-27](27-31](31-35]

(151-276](276-401](401-526](526-651](651-776]

Figure 6 Pb and Zn grade distribution ()


17

3 16 22 49 65

2 7 15 21 27 35 48 64 81

1 6 10 14 20 26 34 47 63 80 TMC1

5 9 13 19 25 33 46 62 79 95 TMC2

4 8 12 18 24 32 45 61 78 94 108 TMC3

11 31 44 60 77 93 107 TMC4

23 30 43 59 76 92 106 115 TMC5

29 42 58 75 91 105 114

28 41 57 74 90 104 113

40 56 73 89 103 112

39 55 72 88 102

38 54 71 87 101 111

37 53 70 86 100 110

36 52 69 85 99 109

51 68 84 98

50 67 83 97

66 82 96

Figure 7 Initial TMCs (seeds)

Table 5 Ore grade characteristics of TMC

TMC Mean value of the grade () Standard deviation of the grade () Coefficient of the grade variation ()Lead Zinc Lead Zinc Lead Zinc

1 2263 3364 0325 1198 14392 356072 1801 3832 0292 0418 16222 109313 2170 3937 0381 0820 17565 208234 1950 2573 0169 0928 8710 360795 2136 4636 0267 1262 12531 27226

Finally algorithm partitioned the original mineraldeposit into five ultimate technological mining cuts withrespect to the error of annual capacity of production andconvergence of standard deviation of lead and zinc in theTMC (see Figure 9)

Stopping condition 92 related to the standard deviationof lead and zinc in the TMC is represented by Figure 10

Realized technological mining cut attribute values thatconsider ore tonnage and compactness of theTMC are shownin Table 4

Mean absolute percentage error of the production capac-ity is

(165 + 128 + 017 + 024 + 286)5 = 124 (79)

and it indicates that efficiency of the algorithm is very high Ifwe take into consideration the fact that mineral deposit is ofvery complex shape then compactnesses ofTMCs achieved byalgorithm are suitable Algorithm succeeded to create threeTMCs that have the similar compactness and it is 60 of theoriginal deposit Summary characteristics that consider thegrade in the TMC are shown in Table 5 These characteristicsare very important for the mineral processing planners

Grade distribution is related to ore body genesis processesand cannot be influenced by planners Coefficient of lead andzinc grade variation for entire mineral deposit is 172 and

327 respectively Algorithm also succeeded to divide orebody in parts with suitable coefficients of grade variationWe can see that only in TMC1 and TMC4 coefficient of zincgrade variation is greater than 327 but not so significantlyRealized distribution of lead and zinc grade in TMCs enablesmineral processing planners to create much more flexibleplans because they are informed in advance about expectedgrade distribution

Since the all attributes of TMCs are defined we can nowproceed to define their economic values Two monitoredtime series represented by Figure 11 have been used to defineclasses of lead and zincmetal prices Period ofmonitoringwas125 months Fuzzy C-mean clustering algorithm producedthe following two vectors of class centers see Table 6

Minimum and maximum values of the lead price are963 $t and 3080 $t while for zinc they are 1101 $t and3533 $t Transformation of linguistic variables into adequatetriangular fuzzy numbers and corresponding range codesaccording to obtained class centers is represented by Table 7and Figure 12

Transformation of monitored series to the fuzzy stateseries (range code series) is performed according to thetagging mechanism (see (62)) Monitored value was fuzzifiedwith respect to where the maximum membership degreeoccurred Results of tagging mechanism are represented byFigure 13


Table 6 Class centers of metal price

Metal price Class center (m=12 7)c1 c2 c3 c4 c5 c6 c7

Lead ($t) 1176 1689 1863 2064 2205 2381 2665Zinc ($t) 1236 1816 2014 2176 2344 2695 3244

173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80

5 9 13 19 25 33 46 62 79 954 8 12 18 24 32 45 61 78 94 108

11 31 44 60 77 93 10723 30 43 59 76 92 106 115

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96

(a)

173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80

5 9 13 19 25 33 46 62 79 954 8 12 18 24 32 45 61 78 94 108

11 31 44 60 77 93 10723 30 43 59 76 92 106 115

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96

(b)

173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80

5 9 13 19 25 33 46 62 79 954 8 12 18 24 32 45 61 78 94 108

11 31 44 60 77 93 10723 30 43 59 76 92 106 115

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96

(c)

173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80

5 9 13 19 25 33 46 62 79 954 8 12 18 24 32 45 61 78 94 108

11 31 44 60 77 93 10723 30 43 59 76 92 106 115

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96

(d)

Figure 8 Results of the algorithm after (a) 50 iterations (b) 100 iterations (c) 150 iterations (d) 200 iterations


173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80 TMC1

5 9 13 19 25 33 46 62 79 95 TMC2

4 8 12 18 24 32 45 61 78 94 108 TMC3

11 31 44 60 77 93 107 TMC4

23 30 43 59 76 92 106 115 TMC5

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96

Figure 9 Final partition of the ore body

Table 7 The transformation of linguistic variables

Linguistic variable Triangular fuzzy number of metal price ($t) Membership function Range code value119910119898 119898 = 1 2 7Lead ZincVVL (963 1176 1433) (1101 1236 1526) (0 1 0) 1199101 = 1VL (1433 1689 1776) (1526 1816 1915) (0 1 0) 1199102 = 2L (1776 1863 1963) (1915 2014 2095) (0 1 0) 1199103 = 3M (1963 2064 2134) (2095 2176 2260) (0 1 0) 1199104 = 4H (2134 2205 2293) (2260 2344 2519) (0 1 0) 1199105 = 5VH (2293 2381 2523) (2519 2695 2970) (0 1 0) 1199106 = 6VVH (2523 2665 3080) (2970 3244 3533) (0 1 0) 1199107 = 7

Table 8 Parameters of simulation

Process Value

Lead metal pricemean reverting process-monthly span

spot value (120591=0)2361 $t997888rarrrange code 6997888rarr(2293 2381 2523) $t

equilibriumrange code 4997888rarr(1963 2064 2134) $tspeed of mean reversion 011251

volatility 018491

Zinc metal pricemean reverting process-monthly span

spot value (120591=0)3060 $t997888rarrrange code 7997888rarr(2970 3244 3533)

equilibriumrange code 4997888rarr(2095 2176 2260)speed of mean reversion 006133

volatility 016416

Operating costsIto-Doob stochastic process - monthly span

spot value (120591=0) 50 $tdrift 00025

volatility 00125Time span of simulation (120591=12 60) T=60 monthsNumber of simulations 500


0

01

02

03

04

05

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC3 StdDev Pb

0

02

04

06

08

1

12

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC3 StdDev Zn

0

01

02

03

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC4 StdDev Pb

0

02

04

06

08

1

12

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC4 StedDev Zn

0

01

02

03

04

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

Number of blocks

TMC5 StdDev Pb

002040608

112141618

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

Number of blocks

TMC5 StdDev Zn

0

01

02

03

04

05

06

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC1 StdDev Pb

002040608

112141618

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC1 StdDev Zn

0

01

02

03

04

05

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Number of blocks

TMC2 StdDev Pb

0

02

04

06

08

1

12

14

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Number of blocks

TMC2 StdDev Zn

Figure 10 Convergence of standard deviation of lead and zinc metal in the TMC

To calibrate the mean reverting process which is usedto forecast the future states of lead and zinc metal price wehave run first order linear regression on range code lead priceseries and range code zinc price series separately Calibrationof Ito-Doob stochastic process which is used to forecastthe future states of unit operating costs is based on theexpert knowledge Parameters needed for simulation of thesediffusion processes are shown in the Table 8

One simulation of lead and zinc metal price unit operat-ing cost and resulting economic value of mineable block 1 isrepresented in Figure 14

After 500 simulations we obtained sample which is usedto defined expected economic values of mineable block 1

through the planning time onmonthly span For everymonthwe average the sample and expected economic values ofmineable block 1 are represented by Figure 15

Transformation of forecasted economic values of block 1from monthly to annual level is represented by Figure 16

Discounted or present economic values ofmineable block1 are represented by Figure 17

Applying the process of simulation on all blocks simul-taneously we obtain their present economic values and wecan calculate the present economic values of TMCs Accord-ing to the results of mineral deposit partition followingpresent economic values of every TMC are shown inTable 9


0

500

1000

1500

2000

2500

3000

3500

4000

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125

US$

t

Month

(a)

0

500

1000

1500

2000

2500

3000

3500

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125

US$

t

Month

(b)

Figure 11 Monitored price on monthly span (a) zinc (b) lead

0010203040506070809

1

500 1000 1500 2000 2500 3000 3500$t

VVLVLLM

HVHVVH

(a)

0010203040506070809

1

500 1000 1500 2000 2500 3000 3500 4000$t

VVLVLLM

HVHVVH

(b)

Figure 12 Linguistic variables expressed by triangular fuzzy numbers (a) lead (b) zinc

Table 9 Present economic values of TMC

TMC Value ($)Year 1 Year 2 Year 3 Year 4 Year 5

15104734 3944737 3044364 2443462 19244336591226 5356427 4305812 3576349 29263948228167 6904516 5704581 4871250 4096627

25583555 4336349 3366286 2691067 21178617190095 5863841 4730052 3916424 32015418951887 7536746 6246798 5320340 4472001

36213929 4891545 3857917 3139633 25243907923489 6519182 5317593 4456306 36948059796030 8298845 6935676 5958908 5056129

42976598 2108873 1450337 1038819 6860214063918 3133980 2351322 1834430 13747555275030 4267887 3358689 2754948 2200817

56198802 4899506 3882193 3166488 25554757882521 6503582 5321728 4466255 37120859723915 8256167 6917828 5949400 5056656



0

500

1000

1500

2000

2500

3000

3500

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Lead price

Lower boundCenterUpper bound

(a)

0

1

2

3

4

5

6

7

8

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Lead price

Range code

(b)

0

500

1000

1500

2000

2500

3000

3500

4000

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Zinc price


(c)

0

1

2

3

4

5

6

7

8

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Zinc price

Range code

(d)

Figure 13 Monitored series expressed by fuzzy states and range codes (a) lead price as a fuzzy state (b) lead price as a range code (c) zincprice as a fuzzy state and (d) zinc price as a range code

By this way every coefficient V119889119894119905 119894 = 1 2 5 119905 =1 2 5 of the objective function (see (14)) is definedSolution of the fuzzy objective function is x14=1 x22=1 x31=1x45=1 x53=1 while the remains are equal to zero It means thatoptimal plan of production is TMC3 (year 1) 997888rarr TMC2 (year2) 997888rarr TMC5 (year 3) 997888rarr TMC1 (year 4) 997888rarr TMC4 (year 5)(see Figure 18)

Value of the objective function is 119865 =(17561955 24060165 31322673) $ Value of (19) is

119865 minus 119868 = (17561955 24060165 31322673)minus (12000000 13000000 15000000)

= (2561955 11060166 19322673) $(80)

This difference represents the net present value of the realizedproduction plan Since it is a positive production plan isprofitable and should be accepted

4 Conclusion

Having ability to optimize the production with uncertaintyis recognized as critical to compete and even survive ofmining company The presence of the uncertainty makesthat the model works in a dynamic environment producingthe optimal results and plans Metal price operating costsand annual capacity of production are identified as mainsources of uncertainties and they are described by fuzzynumbers Each mining project depends on these uncertainparameters which have a significant impact on the finalfinancial results By this way we incorporate risks into mineproduction planning

In this paper we have presented fuzzy 0-1 linear pro-gramming model for production planning of room andpillar underground mining method Model is based on themaximization of fuzzy objective function which representsthe present or discounted value of the future cash flow ofproduction plan with respect to the set of constraints

Our approach of the mine production planning con-tains two main parts First part is referred on partition-ing of the mineral deposit creating the TMCs (clusters)of the ore body Two-stage fuzzy multicriteria clustering


LEAD ZINC

UNIT OPERATING COST

RESULTING ECONOMIC VALUE OF BLOCK

01234567

0 5 10 15 20 25 30 35 40 45 50 55 60

Rang

e cod

e val

ue

Month

01234567

0 5 10 15 20 25 30 35 40 45 50 55 60

Rang

e cod

e val

ue

Month

0500

100015002000250030003500

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

0500

1000150020002500300035004000

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

4648505254565860

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

0100000200000300000400000500000600000700000800000

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

ℎ

sim

Stimes４= aℎ

1

(９sum

=1

aℎ2 M

mcon

con

sim

Stimes４) minus |C|simStimes４

forallℎ isin [1 H]V V

Figure 14 One possible artificial scenario of economic value of block 1

algorithm as iterative process is applied for creation ofTMCs where each cluster meets the following technologicalrequirements annual capacity of production compactness ofthe shape of TMC and standard deviation of ore grade inthe TMC

The second part of the mine production planning modelis related to calculation process of the economic value ofmineable blocks using the simulation of the mean reverting

process and Ito-Doob stochastic differential equation Eco-nomic value of each mineable block is defined by metal priceand operating costs with uncertainty Accordingly we devel-oped concurrently simulations of these twomain economicalsources whereby each mineable block has the different eco-nomic value through the planning time Based on this we canestimate the present economic values (expected fuzzy values)of each TMCs for every year of the planning time


0

100000

200000

300000

400000

500000

600000

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

Figure 15 Expected economic values of block 1

0

100000

200000

300000

400000

500000

600000

1 2 3 4 5

$t

Year

Figure 16 Forecasted economic values of block 1-annual span

Coefficients of the fuzzy linear objective function areobtained by discounting these expected values by fuzzydiscount rate Solution of the fuzzy objective function givesthe order of mining of TMCs ie the optimal productionplan with respect to operational constraints Bearing in mindthat the value of the objective function is positive ourdeveloped algorithm is absolutely realized and accepted forthe solving more complex problems

Themodel is not closed and can be extended by includingthe uncertainty related to the ore grade This uncertaintyis very important and significantly affects the productionplan Fuzzy numbers can also be used for this purpose butcalculation of standard deviation of fuzzy numbers is a verycomplex task Future exploration will be focused on theinclusion of fuzzy numbers as a way to describe ore gradeuncertainty

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

0

100000

200000

300000

400000

500000

600000

1 2 3 4 5

$t

Year

Figure 17 Present economic values of block 1

TMC 1year 4

TMC 4year 5

TMC 5year 3

TMC 3year 1

TMC 2year 2

Figure 18 Optimal plan of production

References

[1] DW Gentry and T J OrsquoNeilMine Investment Analysis Societyof Mining Engineers 1984

[2] D OrsquoSullivan A Brickey and A Newman ldquoIs open pit pro-duction scheduling rsquoeasierrsquo than its underground counterpartrdquoMining Engineering vol 67 no 4 pp 68ndash73 2015

[3] W M Carlyle and B C Eaves ldquoUnderground planning atstillwater mining companyrdquo Interfaces vol 31 no 4 pp 50ndash602001

[4] A Konanu ldquoApplications of simulation and optimizationtechniques in optimizing room and pillar mining systemsrdquoDissertations pp 1ndash2467 2016 httpscholarsminemstedudoctoral dissertations2467

[5] E Bakhtavar K Shahriar and A Mirhassani ldquoOptimizationof the transition from open-pit to underground operation incombined mining using (0-1) integer programmingrdquo Journal of


the Southern African Institute of Mining andMetallurgy vol 112no 12 pp 1059ndash1064 2012

[6] M Nehring E Topal and J Little ldquoA new mathematicalprogramming model for production schedule optimizationin underground mining operationsrdquo Journal of the SouthernAfrican Institute of Mining and Metallurgy vol 110 no 8 pp437ndash446 2010

[7] X Bai D Marcotte and R Simon ldquoUnderground stope opti-mizationwith network flowmethodrdquoComputers ampGeosciencesvol 52 pp 361ndash371 2013

[8] R EpsteinM Goic AWeintraub et al ldquoOptimizing long-termproduction plans in underground and open-pit copper minesrdquoOperations Research vol 60 no 1 pp 4ndash17 2012

[9] N Grieco and R Dimitrakopoulos ldquoManaging grade riskin stope design optimisation Probabilistic mathematical pro-gramming model and application in sublevel stopingrdquo Trans-actions of the Institutions of Mining and Metallurgy Section AMining Technology vol 116 no 2 pp 49ndash57 2007

[10] M Nehring E Topal M Kizil and P Knights ldquoInte-grated short- andmedium-term undergroundmine productionschedulingrdquo Journal of the Southern African Institute of Miningand Metallurgy vol 112 no 5 pp 365ndash378 2012

[11] S E Terblanche and A Bley ldquoAn improved formulation ofthe underground mine scheduling optimisation problem whenconsidering selective miningrdquo ORiON vol 31 no 1 pp 1ndash162015

[12] M Kuchta A Newman and E Topal ldquoImplementing a produc-tion schedule at LKABrsquos kiruna minerdquo Interfaces vol 34 no 2pp 124ndash134 2004

[13] E Topal ldquoEarly start and late start algorithms to improve thesolution time for long-term underground mine productionschedulingrdquo Journal of the Southern African Institute of Miningand Metallurgy vol 108 no 2 pp 99ndash107 2008

[14] J C Hirschi ldquoA dynamic programming approach to identifyingoptimal mining sequences for continuous miner coal produc-tion systemsrdquo Dissertations pp 1ndash542 2012

[15] Z Gligoric M Gligoric B Dimitrijevic et al ldquoModel ofroom and pillar production planning in small scale under-ground mines with metal price and operating cost uncertaintyrdquoResources Policy 2018

[16] R M Alguliyev R M Aliguliyev and R S Mahmudova ldquoMul-ticriteria personnel selection by the modified fuzzy VIKORmethodrdquo The Scientific World Journal vol 2015 Article ID612767 16 pages 2015

[17] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965

[18] A Kumar J Kaur and P Singh ldquoA newmethod for solving fullyfuzzy linear programming problemsrdquo Applied MathematicalModelling vol 35 no 2 pp 817ndash823 2011

[19] A Kaufmann and M M Gupta Introduction to Fuzzy Arith-metic Theory and Applications Van Nostrand Reinhold NewYork NY USA 1985

[20] R R Yager ldquoA procedure for ordering fuzzy subsets of the unitintervalrdquo Information Sciences vol 24 no 2 pp 143ndash161 1981

[21] httpwwwctufrgsbrlapromUnderground20Mining20Methodspdf

[22] H Hamrin Guide to Underground Mining Methods and Appli-cations Atlas Copco Stockholm Sweden 1980

[23] C L Hwang and K Yoon Multiple Attribute Decision Making-Methods and Applications A State of the Art Survey Springer-Verlag New York NY USA 1981

[24] D Joshi ldquoPolygonal spatial clusteringrdquo Dissertation LincolnNebraska 2011

[25] D Joshi L Soh and A Samal ldquoRedistricting using constrainedpolygonal clusteringrdquo IEEE Transactions on Knowledge andData Engineering vol 24 no 11 pp 2065ndash2079 2012

[26] D Joshi L-K Soh andA Samal ldquoRedistricting using heuristic-based polygonal clusteringrdquo in Proceedings of the 9th IEEEInternational Conference on Data Mining (ICDM rsquo09) pp 830ndash835 December 2009

[27] Y Wang and T M S Elhag ldquoFuzzy TOPSIS method based onalpha level sets with an application to bridge risk assessmentrdquoExpert Systems with Applications vol 31 no 2 pp 309ndash3192006

[28] C T Chen ldquoExtensions of the TOPSIS for group decision-making under fuzzy environmentrdquo Fuzzy Sets and Systems vol114 no 1 pp 1ndash9 2000

[29] T C Chu ldquoSelecting plant location via a fuzzy TOPSISapproachrdquo The International Journal of Advanced Manufactur-ing Technology vol 20 pp 859ndash864 2002

[30] T Yang and C C Hung ldquoMultiple-attribute decision mak-ing methods for plant layout design problemrdquo Robotics andComputer-Integrated Manufacturing vol 23 no 1 pp 126ndash1372007

[31] C E Shannon and V Weaver The Mathematical Theory ofCommunication The University of Illinois Press Urbana IllUSA 1947

[32] M Zeleny Multiple Criteria Decision Making McGraw HillNew York NY USA 1982

[33] J Schwartzberg ldquoReapportionment gerrymanders and thenotion of compactnessrdquo Minnesota Law Review vol 50 pp443ndash452 1996

[34] G Karypis and V Kumar ldquoMultilevel k-way partitioningscheme for irregular graphsrdquo Journal of Parallel and DistributedComputing vol 48 no 1 pp 96ndash129 1998

[35] G Von Laszewski Intelligent Structural Operators for the k-wayGraph Partitioning Problem Northeast Parallel ArchitectureCenter 1997 paper 30 httpsurfacesyredunpac30

[36] J C Bezdek R Enrlich andW Full ldquoFCMThe fuzzy c-meansclustering algorithmrdquo Computers amp Geosciences vol 10 no 2-3pp 191ndash203 1984

[37] Y Lu T Ma C Yin X Xie W Tian and S M ZhongldquoImplementation of the fuzzy c-means clustering algorithm inmeteorological datardquo Journal of Database Theory and Applica-tion vol 6 no 6 pp 1ndash18 2013

[38] X Wang Y Wang and L Wang ldquoImproving fuzzy c-meansclustering based on feature-weight learningrdquo Pattern Recogni-tion Letters vol 25 no 10 pp 1123ndash1132 2004

[39] C-T Chang J Z C Lai and M-D Jeng ldquoA fuzzy k-meansclustering algorithm using cluster center displacementrdquo Journalof Information Science and Engineering vol 27 no 3 pp 995ndash1009 2011

[40] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoning Irdquo Information Sciencesvol 8 pp 199ndash249 1975

[41] E S Schwartz ldquoThe stochastic behavior of commodity pricesImplications for valuation and hedgingrdquo Journal of Finance vol52 no 3 pp 923ndash973 1997

[42] A K Dixit and R S Pindyck Investment under UncertaintyPrinceton University Press Princeton NJ USA 1994

[43] Z Gligoric C Beljic B Gluscevic and S Jovanovic ldquoHybridmodel of evaluation of underground leadmdashzinc mine capacity


expansion project using Monte Carlo simulation and fuzzynumbersrdquo Simulation vol 87 no 8 pp 726ndash742 2011

[44] G S Ladde and M Sambandham Stochastic Versus Determin-istic Systems of Differential Equations Marcel Dekker Inc NewYork NY USA 2004

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Submit your manuscripts atwwwhindawicom


Anani applied discrete event simulation to determinethe optimal width of coal room and pillars panels underspecific mining conditions She also tested the hypothesisthat heuristic preprocessing can be used to increase thecomputational efficiency of branch and cut solutions to thebinary integer linear programming problem of room andpillar mine sequencing The findings of her research includepanel width optimization a deterministic modelling frame-work that incorporates multiple mining risk in room andpillar production sequencing and accounting for changingduty cycles in continuous miner-shuttle car matching [4]Bakhtavar et al used 0-1 integer programming to createmodel which optimizes the way of transition from open pitmine to the underground mine [5] Nehring et al developeda new mathematical programming model for optimizationof production scheduling of a sublevel stopping operationwhich significantly reduces solution times without alteringresults while maintaining all constraints They representedall stope production phases by single binary variable andincreased efficiency of mixed-integer programming in theprocess of optimization [6] Bai et al developed algorithmfor stope design optimization at sublevel mining methodOptimization problemwas treated asmaximumflow over theadequate graph [7] A general capacitated multicommoditynetwork flow model has been used for long-term mineplanning by Epstein et al [8] Grieco et al applied a prob-abilistic mixed-integer programming approach to optimizestope in an underground mine ie to define location sizeand number of active stopes with uncertainty related toore grade and acceptable level of risk [9] Nehring et alintegrated short andmedium-termproduction plans by com-bining the short-termobjective ofminimizing deviation fromtargeted mill feed grade with the medium-term objectiveof maximizing net present value into a single mathematicaloptimization model [10] Terblanche and Bley reduced theresolution of underground mine scheduling problem andapplied mixed-integer programming to improve profitabilitythrough selective mining [11] Kuchta et al used mixed-integer programming to schedule Kirunarsquos operations specif-ically which production blocks to mine and when to minethem to minimize deviations from monthly planned pro-duction quantities while adhering to operational restrictions[12] Topal developed an early start and late start algorithmthat defines the precedence restrictions for each miningunit in their mixed-integer linear programming model ofthe underground Kiruna Mine [13] Hirschi developed adynamic programming algorithm to supplant that trial anderror practice of determining and evaluating room and pillarmining sequences Dynamic programming has been used inmining to optimize multistage processes where productionparameters are stage-specific [14] Gligoric et al developed aproduction planningmodel whichminimizes deviation fromthe acceptable rate of return using multivariable weightedFrobenius distance function thatmeasures the deviation fromestablished targets [15]

All developed mine production planning models werebased on 0-1 linear programming and different methods havebeen used to find the extreme value of the linear objectivefunction for example simplex method simulated annealing

Branch and Bound algorithm ant colony optimization neu-ral networks etc We applied fuzzy 0-1 linear programmingto incorporate the uncertainties in the objective function andmake the problem of production planning more realisticBy this way we increase the precision of the obtainedresults If we take into consideration that mine productionplanning belongs to the decision making field then fuzzymodel really helps us to make final decision in more efficientway

The main aim of this paper is to provide efficiencysupport to decision making on production planning inundergroundmines that use room and pillar mining methodas a way of mining Model is based on the maximizationof fuzzy objective function which represents the presentor discounted value of the future cash flow of productionplan with respect to the set of constraints We consider thisproblem as zero-one linear programing problem in whichonly coefficients in the objective function are triangular fuzzynumbers Coefficients represent the discounted economicvalue of the technological mining cut (TMC) which is a partof mineral deposit characterized with respect to the givenset of technological requirements such as annual capacity ofproduction compactness of the shape of TMC and standarddeviation of ore grade in the TMC Total number of techno-logical mining cuts is equal to the total number of years ofproduction

The first step in the production planning model is relatedto the creation of TMCs having the value of attributesclosely to the values of technological requirements In thepurpose of creation such TMCs (clusters) we developed fuzzymulticriteria clustering algorithm where uncertainties ofsome input data are quantified by triangular fuzzy numbersMining engineers uses a block model of the deposit thatrepresents the deposit as three-dimensional array of blocksAccordingly clustering algorithm is applied on the set ofthese blocks The second step concerns the calculation ofdiscounted economic value of TMCs It indicates that we arefacing dynamic problem burdened with some uncertaintiesThese uncertainties come primary from the metal price andoperating costs fluctuation through the time of planningTo estimate the future state of metal price we developedforecasting algorithm which represents the hybrid of thefuzzy C-mean clustering algorithm and stochastic diffu-sion process called mean reverting process This algorithmquantifies the future states of metal price by the fuzzyseries Operating costs are modelled by Ito-Doob stochasticdifferential equation Applying concurrently simulations ofthese two parameters we can estimate the expected fuzzyvalue of each TMC for every year of the planning time Afterthat we discount these values by fuzzy discount rate anddefine the values of coefficient of objective function Solutionof the fuzzy objective function gives the order of mining ofTMCs

The proposed model is a mathematical representationof mining business reality and allows mining companymanagement to run a dynamic optimization of the businesswith uncertainty It helps mining company to survive in veryrisky environment








120583119860 (119909) =


(1)



10038161003816100381610038161003816119860 (119909)10038161003816100381610038161003816 = max 119860 (119909) minus119860 (119909) 119894119891 119909 ge 00 119894119891 119909 lt 0 (2)


R (119860) = 119886 + 2119887 + 1198884 (3)




max119899sum119895=1

(119886119895 119887119895 119888119895) 119909119895 (4)


119902119894119895119909119895 le 119901119894 1 le 119894 le 119898 1 le 119895 le 119899 (5)

119909119895 isin [0 1] (6)


maxR( 119899sum119895=1

(119886119895 119887119895 119888119895) 119909119895) (7)


max ((1 6 9) 1199091 + (2 3 8) 1199092 + (4 6 7) 1199093) (8)


maxR (11199091 + 21199092 + 41199093 61199091 + 31199092 + 61199093 91199091 + 81199092+ 71199093) (9)


max (14 (221199091 + 161199092 + 231199093)) (10)



((1 6 9) ∙ 1 + (2 3 8) ∙ 0 + (4 6 7) ∙ 1) = (5 12 16) (11)


1199090 (119860) = int119888119886119909120583119860 (119909) 119889119909int119888119886120583119860 (119909) 119889119909 (12)


1199090 (119860) = 119886 + 119887 + 1198883 (13)


119865 ( 119883) = 119873sum119894=1

119879sum119905=1

V119889119894119905119909119894119905 = 119873sum119894=1

119879sum119905=1

V119894119905(1 + 119889)119905 119909119894119905 997888rarr 119898119886119909 (14)

subject to

119879sum119905=1

119909119894119905 = 1 forall119894 isin 119873 (15)

119873sum119894=1

119909119894119905 = 1 forall119905 isin 119879 (16)



119865 minus 119868 ge 0 (19)

where




front benching

drill jumbo

connecting dri

verticalbenching

pillar

pillar















max⏟⏟⏟⏟⏟⏟⏟1198861119888isin1198791198721198621

119891 (1198861119888) = [1198911 (11988611) 1198912 (11988612) 119891119870 (1198861119870)]max⏟⏟⏟⏟⏟⏟⏟

1198862119888isin1198791198721198622

119891 (1198862119888) = [1198911 (11988621) 1198912 (11988622) 119891119870 (1198862119870)]

max⏟⏟⏟⏟⏟⏟⏟119886119894119888isin119879119872119862119894

119891 (119886119894119888) = [1198911 (1198861198941) 1198912 (1198861198942) 119891119870 (119886119873119870)](20)

subject to

119886119897119888 le 119886119888 le 119886119906119888 119888 = 1 2 119870 (21)

where




119861 = 119873⋃119894=1

119879119872119862119894= 1198791198721198621 cup 1198791198721198622 cup cup 119879119872119862119894119879119872119862120572 cap 119879119872119862120573 = 0 120572 = 120573

(22)




119863 = [119909119894119895]119873times119870 =[[[[[[[[[[[


]]]]]]]]]]](23)



119895=1

119908119895 = 1 (24)



119890119895 = minus 1ln (119873)

119873sum119894=1


(25)




119860+ = ( max⏟⏟⏟⏟⏟⏟⏟119894=12119873

119901119894119895 | 119895 isin 119869) ( min⏟⏟⏟⏟⏟⏟⏟119894=12119873

119901119894119895 | 119895 isin 1198691015840)= 119901+1 119901+2 119901+119870

(27)

119860minus = ( min⏟⏟⏟⏟⏟⏟⏟119894=12119873

119901119894119895 | 119895 isin 119869) ( max⏟⏟⏟⏟⏟⏟⏟119894=12119873


(28)

max⏟⏟⏟⏟⏟⏟⏟119894=12119873

119901119894119895 = max⏟⏟⏟⏟⏟⏟⏟119894=12119873

(119886119894119895 119887119894119895 119888119894119895) = max⏟⏟⏟⏟⏟⏟⏟119894=12119873

(119887119894119895) forall119895 isin [1 119870] (29)

min⏟⏟⏟⏟⏟⏟⏟119894=12119873

119901119894119895 = min⏟⏟⏟⏟⏟⏟⏟119894=12119873

(119886119894119895 119887119894119895 119888119894119895) = min⏟⏟⏟⏟⏟⏟⏟119894=12119873

(119887119894119895) forall119895 isin [1 119870] (30)

where



119889+119894 = 119870sum119895=1

119889 (119901119894119895 119901+119895 ) forall119894 isin [1119873] (31)

119889minus119894 = 119870sum119895=1




(33)




119879119872119862119892119903119900119908 = max (1198771198621198621 1198771198621198622 119877119862119862119873) (35)



where



119863119887119897119900119888119896 = [120593119898119895]119872times119870

=[[[[[[[[[[[[


]]]]]]]]]]]]

(37)

where








where










where


21 27 35 48

20 26 34 47

19 25 33 46

18 24 32 45 Waste

TMC1

TMC2

TMC3




where



1198861198941 = 119897119894sumℎ=1



1198861198942 = 1198752119894119860 119894 = 1198752119894119897119894 ∙ (1198872119890 ) forall119894 isin [1119873] (43)

where





1198861198943120574 = radic 1119897119894119897119894sumℎ=1


(44)



119862119905119890ℎ = 119888119905119890ℎ119895 119895=12119870

= 119888119905119890ℎ1 119888119905119890ℎ2 119888119905119890ℎ31 119888119905119890ℎ32 119888119905119890ℎ3120574 119895=2+120574 (45)

where



119888119905119890ℎ1 = 1119873119867sumℎ=1

119886ℎ1 (46)


119888119905119890ℎ2 = (4119890)21198902 = 16 (47)


119888119905119890ℎ3120574 = 1198861198943120574 (48)


119862 = 119862119895119895=12119870 = 1198621 1198622 11986231 11986232 1198623120574119895=2+120574 (49)

where










forall119894 isin [1119873](53)



119885 = max⏟⏟⏟⏟⏟⏟⏟forall119894isin[1119873]

119880119877119862119894= max⏟⏟⏟⏟⏟⏟⏟forall119894isin[1119873]


(54)





for 120585+1 119861119872119861 isin 119879119872119862119892119903119900119908120572

for 120585+2 119861119872119861 isin 119879119872119862119892119903119900119908120573

for 120585+3 119887ℎ isin 119879119872119862119892119903119900119908120572






subject to

119873sum119894=1

119887ℎ119894 = 1 forallℎ isin [1119867] (55)


(56)


where (⋃119897119894lt119867ℎ=1














119865 = (119880119883 119862) = Tsum120591=1

119872sum119898=1

119906120596120591119898 (119909120591 minus 119888119898)2 (58)

subject to

119872sum119898=1

119906120591119898 = 1 120591 = 1 2 T (59)

0 le 119906120591119898 le 1 119898 = 1 2 119872 120591 = 1 2 T (60)

0 lt Tsum120591=1

119906119894119898 lt 119898 119898 = 1 2 119872 (61)

where










119875 (119884)forall120591isin[1T]

=


(62)


119898 = 1 2 119872 (63)


119889119911119898 = 120572 (ln 119911119898 minus ln 119911119898) 119911119898119889120591 + 120590119911119898119889119882 (64)

where






(65)

where



119911119898(120591+1) = 1205730 + 1205731119911119898(120591) + 120576 (66)




3

44

555

4

5

67

6

5

444

3333333333

444444

33

4

3

22

333

4

33

4444

333333333

4

5

67

6

0

1

2

3

4

5

6

7

0 5 10 15 20 25 30 35 40 45 50 55 60

Rang

e co

de v

alue

Month



119911119898 (120591)forall120591isin[1T]

=


119898 = 2 3 119872 minus 1

(67)




(68)



18

95

137125

71

3321

020406080

100120140160





119889119888120591 = 120583119888120591119889120591 + 120590119889119882120591 (69)




|119862|119904119894119898119878timesT = 10038161003816100381610038161198881199041205911003816100381610038161003816119878timesT (71)






= 119898119888119900119899 ∙ 119898119898119903 ∙ 1003816100381610038161003816119901119904120591119898 1003816100381610038161003816119878timesT (72)



10038161003816100381610038161003816Vℎ10038161003816100381610038161003816119904119894119898119878timesT = 119886ℎ1 (Ysum120574=1


(73)



=1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(119886 119887 119888)ℎ11 (119886 119887 119888)ℎ12 (119886 119887 119888)ℎ1120591(119886 119887 119888)ℎ21 (119886 119887 119888)ℎ22 (119886 119887 119888)ℎ2120591 d

(119886 119887 119888)ℎ1199041 (119886 119887 119888)ℎ1199042 (119886 119887 119888)ℎ119878T

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816


(74)


forall120591isin[1T] =10038161003816100381610038161003816100381610038161003816100381610038161119878119878sum119904=1


(75)


119864 (119864 (Vℎ120591)119905)119905=[1119879] =100381610038161003816100381610038161003816100381610038161003816100381611212sum120591=1

119864 (Vℎ120591)1199051003816100381610038161003816100381610038161003816100381610038161003816119905=[1119879]



1003816100381610038161003816100381610038161003816(Vℎ119905 )1198891003816100381610038161003816100381610038161003816 =10038161003816100381610038161003816100381610038161003816100381610038161003816


10038161003816100381610038161003816100381610038161003816100381610038161003816 forallℎ isin [1119867] 119905 = [1 119879]

(77)


V119889119894119905 = 119897119894sumℎ=1


(78)


3 Numerical Example






Zinc















1 5301 5891 6480 59 6458 7176 78932 6844 7604 8365 60 6954 7727 84993 5081 5646 6210 61 6293 6992 76914 7064 7849 8634 62 7119 7910 87015 7615 8461 9307 63 6128 6809 74896 6954 7727 8499 64 7284 8094 89037 7725 8583 9442 65 5962 6625 72878 5715 6350 6984 66 8977 9974 109729 6747 7497 8247 67 6128 6809 748910 5508 6120 6732 68 7119 7910 870111 7367 8186 9004 69 6293 6992 769112 6871 7635 8398 70 6954 7727 849913 7491 8323 9156 71 6458 7176 789314 6747 7497 8247 72 6789 7543 829715 7615 8461 9307 73 6623 7359 809516 6623 7359 8095 74 6623 7359 809517 7615 8461 9307 75 6789 7543 829718 5907 6564 7220 76 6458 7176 789319 6871 7635 8398 77 6954 7727 849920 5715 6350 6984 78 6293 6992 769121 7064 7849 8634 79 7119 7910 870122 5522 6135 6749 80 6128 6809 748923 6816 7574 8331 81 7284 8094 890324 6678 7421 8163 82 9130 10144 1115825 7367 8186 9004 83 6334 7038 774226 6541 7268 7994 84 7243 8048 885327 7505 8339 9172 85 6486 7206 792728 6458 7176 7893 86 7092 7880 866729 6637 7375 8112 87 6637 7375 811230 6279 6977 7674 88 6940 7711 848231 6816 7574 8331 89 6789 7543 829732 6100 6778 7456 90 6789 7543 829733 6995 7772 8550 91 6940 7711 848234 5921 6579 7237 92 6637 7375 811235 7174 7971 8768 93 7092 7880 866736 6486 7206 7927 94 6486 7206 792737 7092 7880 8667 95 7243 8048 885338 6637 7375 8112 96 8895 9883 1087239 6940 7711 8482 97 5921 6579 723740 6789 7543 8297 98 6995 7772 855041 6789 7543 8297 99 6100 6778 745642 6940 7711 8482 100 6816 7574 833143 6637 7375 8112 101 6279 6977 767444 7092 7880 8667 102 6637 7375 811245 6486 7206 7927 103 6458 7176 789346 7243 8048 8853 104 6458 7176 789347 6334 7038 7742 105 6637 7375 811248 7394 8216 9038 106 6279 6977 767449 6183 6870 7557 107 6816 7574 833150 6128 6809 7489 108 6100 6778 745651 7119 7910 8701 109 6678 7421 816352 6293 6992 7691 110 7229 8033 8836


Table 3 Continued





1152640

3913 -165169601186561

2153217

3072 -128170241187266

3154940

2939 -017172156189371

4155573

6278 +024172859190145

5159649

4816 +286177388195127


(15-19](19-23](23-27](27-31](31-35]

(151-276](276-401](401-526](526-651](651-776]



17

3 16 22 49 65

2 7 15 21 27 35 48 64 81

1 6 10 14 20 26 34 47 63 80 TMC1

5 9 13 19 25 33 46 62 79 95 TMC2

4 8 12 18 24 32 45 61 78 94 108 TMC3

11 31 44 60 77 93 107 TMC4

23 30 43 59 76 92 106 115 TMC5

29 42 58 75 91 105 114

28 41 57 74 90 104 113

40 56 73 89 103 112

39 55 72 88 102

38 54 71 87 101 111

37 53 70 86 100 110

36 52 69 85 99 109

51 68 84 98

50 67 83 97

66 82 96




1 2263 3364 0325 1198 14392 356072 1801 3832 0292 0418 16222 109313 2170 3937 0381 0820 17565 208234 1950 2573 0169 0928 8710 360795 2136 4636 0267 1262 12531 27226





(165 + 128 + 017 + 024 + 286)5 = 124 (79)










Lead ($t) 1176 1689 1863 2064 2205 2381 2665Zinc ($t) 1236 1816 2014 2176 2344 2695 3244

173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80

5 9 13 19 25 33 46 62 79 954 8 12 18 24 32 45 61 78 94 108

11 31 44 60 77 93 10723 30 43 59 76 92 106 115

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96

(a)

173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80

5 9 13 19 25 33 46 62 79 954 8 12 18 24 32 45 61 78 94 108

11 31 44 60 77 93 10723 30 43 59 76 92 106 115

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96

(b)

173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80

5 9 13 19 25 33 46 62 79 954 8 12 18 24 32 45 61 78 94 108

11 31 44 60 77 93 10723 30 43 59 76 92 106 115

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96

(c)

173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80

5 9 13 19 25 33 46 62 79 954 8 12 18 24 32 45 61 78 94 108

11 31 44 60 77 93 10723 30 43 59 76 92 106 115

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96

(d)



173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80 TMC1

5 9 13 19 25 33 46 62 79 95 TMC2

4 8 12 18 24 32 45 61 78 94 108 TMC3

11 31 44 60 77 93 107 TMC4

23 30 43 59 76 92 106 115 TMC5

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96





Process Value




volatility 018491




volatility 016416





0

01

02

03

04

05

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC3 StdDev Pb

0

02

04

06

08

1

12

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC3 StdDev Zn

0

01

02

03

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC4 StdDev Pb

0

02

04

06

08

1

12

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC4 StedDev Zn

0

01

02

03

04

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

Number of blocks

TMC5 StdDev Pb

002040608

112141618

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

Number of blocks

TMC5 StdDev Zn

0

01

02

03

04

05

06

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC1 StdDev Pb

002040608

112141618

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC1 StdDev Zn

0

01

02

03

04

05

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Number of blocks

TMC2 StdDev Pb

0

02

04

06

08

1

12

14

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Number of blocks

TMC2 StdDev Zn










0

500

1000

1500

2000

2500

3000

3500

4000

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125

US$

t

Month

(a)

0

500

1000

1500

2000

2500

3000

3500

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125

US$

t

Month

(b)


0010203040506070809

1

500 1000 1500 2000 2500 3000 3500$t

VVLVLLM

HVHVVH

(a)

0010203040506070809

1

500 1000 1500 2000 2500 3000 3500 4000$t

VVLVLLM

HVHVVH

(b)




15104734 3944737 3044364 2443462 19244336591226 5356427 4305812 3576349 29263948228167 6904516 5704581 4871250 4096627

25583555 4336349 3366286 2691067 21178617190095 5863841 4730052 3916424 32015418951887 7536746 6246798 5320340 4472001

36213929 4891545 3857917 3139633 25243907923489 6519182 5317593 4456306 36948059796030 8298845 6935676 5958908 5056129

42976598 2108873 1450337 1038819 6860214063918 3133980 2351322 1834430 13747555275030 4267887 3358689 2754948 2200817

56198802 4899506 3882193 3166488 25554757882521 6503582 5321728 4466255 37120859723915 8256167 6917828 5949400 5056656



0

500

1000

1500

2000

2500

3000

3500

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Lead price


(a)

0

1

2

3

4

5

6

7

8

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Lead price

Range code

(b)

0

500

1000

1500

2000

2500

3000

3500

4000

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Zinc price


(c)

0

1

2

3

4

5

6

7

8

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Zinc price

Range code

(d)




119865 minus 119868 = (17561955 24060165 31322673)minus (12000000 13000000 15000000)

= (2561955 11060166 19322673) $(80)


4 Conclusion





LEAD ZINC

UNIT OPERATING COST


01234567

0 5 10 15 20 25 30 35 40 45 50 55 60

Rang

e cod

e val

ue

Month

01234567

0 5 10 15 20 25 30 35 40 45 50 55 60

Rang

e cod

e val

ue

Month

0500

100015002000250030003500

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

0500

1000150020002500300035004000

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

4648505254565860

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

0100000200000300000400000500000600000700000800000

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

ℎ

sim

Stimes４= aℎ

1

(９sum

=1

aℎ2 M

mcon

con

sim








0

100000

200000

300000

400000

500000

600000

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month


0

100000

200000

300000

400000

500000

600000

1 2 3 4 5

$t

Year




Data Availability




0

100000

200000

300000

400000

500000

600000

1 2 3 4 5

$t

Year


TMC 1year 4

TMC 4year 5

TMC 5year 3

TMC 3year 1

TMC 2year 2


References
























































Journal of












Journal of







Volume 2018






Volume 2018















120583119860 (119909) =


(1)



10038161003816100381610038161003816119860 (119909)10038161003816100381610038161003816 = max 119860 (119909) minus119860 (119909) 119894119891 119909 ge 00 119894119891 119909 lt 0 (2)


R (119860) = 119886 + 2119887 + 1198884 (3)




max119899sum119895=1

(119886119895 119887119895 119888119895) 119909119895 (4)


119902119894119895119909119895 le 119901119894 1 le 119894 le 119898 1 le 119895 le 119899 (5)

119909119895 isin [0 1] (6)


maxR( 119899sum119895=1

(119886119895 119887119895 119888119895) 119909119895) (7)


max ((1 6 9) 1199091 + (2 3 8) 1199092 + (4 6 7) 1199093) (8)


maxR (11199091 + 21199092 + 41199093 61199091 + 31199092 + 61199093 91199091 + 81199092+ 71199093) (9)


max (14 (221199091 + 161199092 + 231199093)) (10)



((1 6 9) ∙ 1 + (2 3 8) ∙ 0 + (4 6 7) ∙ 1) = (5 12 16) (11)


1199090 (119860) = int119888119886119909120583119860 (119909) 119889119909int119888119886120583119860 (119909) 119889119909 (12)


1199090 (119860) = 119886 + 119887 + 1198883 (13)


119865 ( 119883) = 119873sum119894=1

119879sum119905=1

V119889119894119905119909119894119905 = 119873sum119894=1

119879sum119905=1

V119894119905(1 + 119889)119905 119909119894119905 997888rarr 119898119886119909 (14)

subject to

119879sum119905=1

119909119894119905 = 1 forall119894 isin 119873 (15)

119873sum119894=1

119909119894119905 = 1 forall119905 isin 119879 (16)



119865 minus 119868 ge 0 (19)

where




front benching

drill jumbo

connecting dri

verticalbenching

pillar

pillar















max⏟⏟⏟⏟⏟⏟⏟1198861119888isin1198791198721198621

119891 (1198861119888) = [1198911 (11988611) 1198912 (11988612) 119891119870 (1198861119870)]max⏟⏟⏟⏟⏟⏟⏟

1198862119888isin1198791198721198622

119891 (1198862119888) = [1198911 (11988621) 1198912 (11988622) 119891119870 (1198862119870)]

max⏟⏟⏟⏟⏟⏟⏟119886119894119888isin119879119872119862119894

119891 (119886119894119888) = [1198911 (1198861198941) 1198912 (1198861198942) 119891119870 (119886119873119870)](20)

subject to

119886119897119888 le 119886119888 le 119886119906119888 119888 = 1 2 119870 (21)

where




119861 = 119873⋃119894=1

119879119872119862119894= 1198791198721198621 cup 1198791198721198622 cup cup 119879119872119862119894119879119872119862120572 cap 119879119872119862120573 = 0 120572 = 120573

(22)




119863 = [119909119894119895]119873times119870 =[[[[[[[[[[[


]]]]]]]]]]](23)



119895=1

119908119895 = 1 (24)



119890119895 = minus 1ln (119873)

119873sum119894=1


(25)




119860+ = ( max⏟⏟⏟⏟⏟⏟⏟119894=12119873

119901119894119895 | 119895 isin 119869) ( min⏟⏟⏟⏟⏟⏟⏟119894=12119873

119901119894119895 | 119895 isin 1198691015840)= 119901+1 119901+2 119901+119870

(27)

119860minus = ( min⏟⏟⏟⏟⏟⏟⏟119894=12119873

119901119894119895 | 119895 isin 119869) ( max⏟⏟⏟⏟⏟⏟⏟119894=12119873


(28)

max⏟⏟⏟⏟⏟⏟⏟119894=12119873

119901119894119895 = max⏟⏟⏟⏟⏟⏟⏟119894=12119873

(119886119894119895 119887119894119895 119888119894119895) = max⏟⏟⏟⏟⏟⏟⏟119894=12119873

(119887119894119895) forall119895 isin [1 119870] (29)

min⏟⏟⏟⏟⏟⏟⏟119894=12119873

119901119894119895 = min⏟⏟⏟⏟⏟⏟⏟119894=12119873

(119886119894119895 119887119894119895 119888119894119895) = min⏟⏟⏟⏟⏟⏟⏟119894=12119873

(119887119894119895) forall119895 isin [1 119870] (30)

where



119889+119894 = 119870sum119895=1

119889 (119901119894119895 119901+119895 ) forall119894 isin [1119873] (31)

119889minus119894 = 119870sum119895=1




(33)




119879119872119862119892119903119900119908 = max (1198771198621198621 1198771198621198622 119877119862119862119873) (35)



where



119863119887119897119900119888119896 = [120593119898119895]119872times119870

=[[[[[[[[[[[[


]]]]]]]]]]]]

(37)

where








where










where


21 27 35 48

20 26 34 47

19 25 33 46

18 24 32 45 Waste

TMC1

TMC2

TMC3




where



1198861198941 = 119897119894sumℎ=1



1198861198942 = 1198752119894119860 119894 = 1198752119894119897119894 ∙ (1198872119890 ) forall119894 isin [1119873] (43)

where





1198861198943120574 = radic 1119897119894119897119894sumℎ=1


(44)



119862119905119890ℎ = 119888119905119890ℎ119895 119895=12119870

= 119888119905119890ℎ1 119888119905119890ℎ2 119888119905119890ℎ31 119888119905119890ℎ32 119888119905119890ℎ3120574 119895=2+120574 (45)

where



119888119905119890ℎ1 = 1119873119867sumℎ=1

119886ℎ1 (46)


119888119905119890ℎ2 = (4119890)21198902 = 16 (47)


119888119905119890ℎ3120574 = 1198861198943120574 (48)


119862 = 119862119895119895=12119870 = 1198621 1198622 11986231 11986232 1198623120574119895=2+120574 (49)

where










forall119894 isin [1119873](53)



119885 = max⏟⏟⏟⏟⏟⏟⏟forall119894isin[1119873]

119880119877119862119894= max⏟⏟⏟⏟⏟⏟⏟forall119894isin[1119873]


(54)





for 120585+1 119861119872119861 isin 119879119872119862119892119903119900119908120572

for 120585+2 119861119872119861 isin 119879119872119862119892119903119900119908120573

for 120585+3 119887ℎ isin 119879119872119862119892119903119900119908120572






subject to

119873sum119894=1

119887ℎ119894 = 1 forallℎ isin [1119867] (55)


(56)


where (⋃119897119894lt119867ℎ=1














119865 = (119880119883 119862) = Tsum120591=1

119872sum119898=1

119906120596120591119898 (119909120591 minus 119888119898)2 (58)

subject to

119872sum119898=1

119906120591119898 = 1 120591 = 1 2 T (59)

0 le 119906120591119898 le 1 119898 = 1 2 119872 120591 = 1 2 T (60)

0 lt Tsum120591=1

119906119894119898 lt 119898 119898 = 1 2 119872 (61)

where










119875 (119884)forall120591isin[1T]

=


(62)


119898 = 1 2 119872 (63)


119889119911119898 = 120572 (ln 119911119898 minus ln 119911119898) 119911119898119889120591 + 120590119911119898119889119882 (64)

where






(65)

where



119911119898(120591+1) = 1205730 + 1205731119911119898(120591) + 120576 (66)




3

44

555

4

5

67

6

5

444

3333333333

444444

33

4

3

22

333

4

33

4444

333333333

4

5

67

6

0

1

2

3

4

5

6

7

0 5 10 15 20 25 30 35 40 45 50 55 60

Rang

e co

de v

alue

Month



119911119898 (120591)forall120591isin[1T]

=


119898 = 2 3 119872 minus 1

(67)




(68)



18

95

137125

71

3321

020406080

100120140160





119889119888120591 = 120583119888120591119889120591 + 120590119889119882120591 (69)




|119862|119904119894119898119878timesT = 10038161003816100381610038161198881199041205911003816100381610038161003816119878timesT (71)






= 119898119888119900119899 ∙ 119898119898119903 ∙ 1003816100381610038161003816119901119904120591119898 1003816100381610038161003816119878timesT (72)



10038161003816100381610038161003816Vℎ10038161003816100381610038161003816119904119894119898119878timesT = 119886ℎ1 (Ysum120574=1


(73)



=1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(119886 119887 119888)ℎ11 (119886 119887 119888)ℎ12 (119886 119887 119888)ℎ1120591(119886 119887 119888)ℎ21 (119886 119887 119888)ℎ22 (119886 119887 119888)ℎ2120591 d

(119886 119887 119888)ℎ1199041 (119886 119887 119888)ℎ1199042 (119886 119887 119888)ℎ119878T

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816


(74)


forall120591isin[1T] =10038161003816100381610038161003816100381610038161003816100381610038161119878119878sum119904=1


(75)


119864 (119864 (Vℎ120591)119905)119905=[1119879] =100381610038161003816100381610038161003816100381610038161003816100381611212sum120591=1

119864 (Vℎ120591)1199051003816100381610038161003816100381610038161003816100381610038161003816119905=[1119879]



1003816100381610038161003816100381610038161003816(Vℎ119905 )1198891003816100381610038161003816100381610038161003816 =10038161003816100381610038161003816100381610038161003816100381610038161003816


10038161003816100381610038161003816100381610038161003816100381610038161003816 forallℎ isin [1119867] 119905 = [1 119879]

(77)


V119889119894119905 = 119897119894sumℎ=1


(78)


3 Numerical Example






Zinc















1 5301 5891 6480 59 6458 7176 78932 6844 7604 8365 60 6954 7727 84993 5081 5646 6210 61 6293 6992 76914 7064 7849 8634 62 7119 7910 87015 7615 8461 9307 63 6128 6809 74896 6954 7727 8499 64 7284 8094 89037 7725 8583 9442 65 5962 6625 72878 5715 6350 6984 66 8977 9974 109729 6747 7497 8247 67 6128 6809 748910 5508 6120 6732 68 7119 7910 870111 7367 8186 9004 69 6293 6992 769112 6871 7635 8398 70 6954 7727 849913 7491 8323 9156 71 6458 7176 789314 6747 7497 8247 72 6789 7543 829715 7615 8461 9307 73 6623 7359 809516 6623 7359 8095 74 6623 7359 809517 7615 8461 9307 75 6789 7543 829718 5907 6564 7220 76 6458 7176 789319 6871 7635 8398 77 6954 7727 849920 5715 6350 6984 78 6293 6992 769121 7064 7849 8634 79 7119 7910 870122 5522 6135 6749 80 6128 6809 748923 6816 7574 8331 81 7284 8094 890324 6678 7421 8163 82 9130 10144 1115825 7367 8186 9004 83 6334 7038 774226 6541 7268 7994 84 7243 8048 885327 7505 8339 9172 85 6486 7206 792728 6458 7176 7893 86 7092 7880 866729 6637 7375 8112 87 6637 7375 811230 6279 6977 7674 88 6940 7711 848231 6816 7574 8331 89 6789 7543 829732 6100 6778 7456 90 6789 7543 829733 6995 7772 8550 91 6940 7711 848234 5921 6579 7237 92 6637 7375 811235 7174 7971 8768 93 7092 7880 866736 6486 7206 7927 94 6486 7206 792737 7092 7880 8667 95 7243 8048 885338 6637 7375 8112 96 8895 9883 1087239 6940 7711 8482 97 5921 6579 723740 6789 7543 8297 98 6995 7772 855041 6789 7543 8297 99 6100 6778 745642 6940 7711 8482 100 6816 7574 833143 6637 7375 8112 101 6279 6977 767444 7092 7880 8667 102 6637 7375 811245 6486 7206 7927 103 6458 7176 789346 7243 8048 8853 104 6458 7176 789347 6334 7038 7742 105 6637 7375 811248 7394 8216 9038 106 6279 6977 767449 6183 6870 7557 107 6816 7574 833150 6128 6809 7489 108 6100 6778 745651 7119 7910 8701 109 6678 7421 816352 6293 6992 7691 110 7229 8033 8836


Table 3 Continued





1152640

3913 -165169601186561

2153217

3072 -128170241187266

3154940

2939 -017172156189371

4155573

6278 +024172859190145

5159649

4816 +286177388195127


(15-19](19-23](23-27](27-31](31-35]

(151-276](276-401](401-526](526-651](651-776]



17

3 16 22 49 65

2 7 15 21 27 35 48 64 81

1 6 10 14 20 26 34 47 63 80 TMC1

5 9 13 19 25 33 46 62 79 95 TMC2

4 8 12 18 24 32 45 61 78 94 108 TMC3

11 31 44 60 77 93 107 TMC4

23 30 43 59 76 92 106 115 TMC5

29 42 58 75 91 105 114

28 41 57 74 90 104 113

40 56 73 89 103 112

39 55 72 88 102

38 54 71 87 101 111

37 53 70 86 100 110

36 52 69 85 99 109

51 68 84 98

50 67 83 97

66 82 96




1 2263 3364 0325 1198 14392 356072 1801 3832 0292 0418 16222 109313 2170 3937 0381 0820 17565 208234 1950 2573 0169 0928 8710 360795 2136 4636 0267 1262 12531 27226





(165 + 128 + 017 + 024 + 286)5 = 124 (79)










Lead ($t) 1176 1689 1863 2064 2205 2381 2665Zinc ($t) 1236 1816 2014 2176 2344 2695 3244

173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80

5 9 13 19 25 33 46 62 79 954 8 12 18 24 32 45 61 78 94 108

11 31 44 60 77 93 10723 30 43 59 76 92 106 115

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96

(a)

173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80

5 9 13 19 25 33 46 62 79 954 8 12 18 24 32 45 61 78 94 108

11 31 44 60 77 93 10723 30 43 59 76 92 106 115

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96

(b)

173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80

5 9 13 19 25 33 46 62 79 954 8 12 18 24 32 45 61 78 94 108

11 31 44 60 77 93 10723 30 43 59 76 92 106 115

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96

(c)

173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80

5 9 13 19 25 33 46 62 79 954 8 12 18 24 32 45 61 78 94 108

11 31 44 60 77 93 10723 30 43 59 76 92 106 115

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96

(d)



173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80 TMC1

5 9 13 19 25 33 46 62 79 95 TMC2

4 8 12 18 24 32 45 61 78 94 108 TMC3

11 31 44 60 77 93 107 TMC4

23 30 43 59 76 92 106 115 TMC5

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96





Process Value




volatility 018491




volatility 016416





0

01

02

03

04

05

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC3 StdDev Pb

0

02

04

06

08

1

12

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC3 StdDev Zn

0

01

02

03

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC4 StdDev Pb

0

02

04

06

08

1

12

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC4 StedDev Zn

0

01

02

03

04

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

Number of blocks

TMC5 StdDev Pb

002040608

112141618

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

Number of blocks

TMC5 StdDev Zn

0

01

02

03

04

05

06

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC1 StdDev Pb

002040608

112141618

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC1 StdDev Zn

0

01

02

03

04

05

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Number of blocks

TMC2 StdDev Pb

0

02

04

06

08

1

12

14

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Number of blocks

TMC2 StdDev Zn










0

500

1000

1500

2000

2500

3000

3500

4000

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125

US$

t

Month

(a)

0

500

1000

1500

2000

2500

3000

3500

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125

US$

t

Month

(b)


0010203040506070809

1

500 1000 1500 2000 2500 3000 3500$t

VVLVLLM

HVHVVH

(a)

0010203040506070809

1

500 1000 1500 2000 2500 3000 3500 4000$t

VVLVLLM

HVHVVH

(b)




15104734 3944737 3044364 2443462 19244336591226 5356427 4305812 3576349 29263948228167 6904516 5704581 4871250 4096627

25583555 4336349 3366286 2691067 21178617190095 5863841 4730052 3916424 32015418951887 7536746 6246798 5320340 4472001

36213929 4891545 3857917 3139633 25243907923489 6519182 5317593 4456306 36948059796030 8298845 6935676 5958908 5056129

42976598 2108873 1450337 1038819 6860214063918 3133980 2351322 1834430 13747555275030 4267887 3358689 2754948 2200817

56198802 4899506 3882193 3166488 25554757882521 6503582 5321728 4466255 37120859723915 8256167 6917828 5949400 5056656



0

500

1000

1500

2000

2500

3000

3500

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Lead price


(a)

0

1

2

3

4

5

6

7

8

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Lead price

Range code

(b)

0

500

1000

1500

2000

2500

3000

3500

4000

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Zinc price


(c)

0

1

2

3

4

5

6

7

8

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Zinc price

Range code

(d)




119865 minus 119868 = (17561955 24060165 31322673)minus (12000000 13000000 15000000)

= (2561955 11060166 19322673) $(80)


4 Conclusion





LEAD ZINC

UNIT OPERATING COST


01234567

0 5 10 15 20 25 30 35 40 45 50 55 60

Rang

e cod

e val

ue

Month

01234567

0 5 10 15 20 25 30 35 40 45 50 55 60

Rang

e cod

e val

ue

Month

0500

100015002000250030003500

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

0500

1000150020002500300035004000

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

4648505254565860

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

0100000200000300000400000500000600000700000800000

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

ℎ

sim

Stimes４= aℎ

1

(９sum

=1

aℎ2 M

mcon

con

sim








0

100000

200000

300000

400000

500000

600000

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month


0

100000

200000

300000

400000

500000

600000

1 2 3 4 5

$t

Year




Data Availability




0

100000

200000

300000

400000

500000

600000

1 2 3 4 5

$t

Year


TMC 1year 4

TMC 4year 5

TMC 5year 3

TMC 3year 1

TMC 2year 2


References
























































Journal of












Journal of







Volume 2018






Volume 2018










((1 6 9) ∙ 1 + (2 3 8) ∙ 0 + (4 6 7) ∙ 1) = (5 12 16) (11)


1199090 (119860) = int119888119886119909120583119860 (119909) 119889119909int119888119886120583119860 (119909) 119889119909 (12)


1199090 (119860) = 119886 + 119887 + 1198883 (13)


119865 ( 119883) = 119873sum119894=1

119879sum119905=1

V119889119894119905119909119894119905 = 119873sum119894=1

119879sum119905=1

V119894119905(1 + 119889)119905 119909119894119905 997888rarr 119898119886119909 (14)

subject to

119879sum119905=1

119909119894119905 = 1 forall119894 isin 119873 (15)

119873sum119894=1

119909119894119905 = 1 forall119905 isin 119879 (16)



119865 minus 119868 ge 0 (19)

where




front benching

drill jumbo

connecting dri

verticalbenching

pillar

pillar















max⏟⏟⏟⏟⏟⏟⏟1198861119888isin1198791198721198621

119891 (1198861119888) = [1198911 (11988611) 1198912 (11988612) 119891119870 (1198861119870)]max⏟⏟⏟⏟⏟⏟⏟

1198862119888isin1198791198721198622

119891 (1198862119888) = [1198911 (11988621) 1198912 (11988622) 119891119870 (1198862119870)]

max⏟⏟⏟⏟⏟⏟⏟119886119894119888isin119879119872119862119894

119891 (119886119894119888) = [1198911 (1198861198941) 1198912 (1198861198942) 119891119870 (119886119873119870)](20)

subject to

119886119897119888 le 119886119888 le 119886119906119888 119888 = 1 2 119870 (21)

where




119861 = 119873⋃119894=1

119879119872119862119894= 1198791198721198621 cup 1198791198721198622 cup cup 119879119872119862119894119879119872119862120572 cap 119879119872119862120573 = 0 120572 = 120573

(22)




119863 = [119909119894119895]119873times119870 =[[[[[[[[[[[


]]]]]]]]]]](23)



119895=1

119908119895 = 1 (24)



119890119895 = minus 1ln (119873)

119873sum119894=1


(25)




119860+ = ( max⏟⏟⏟⏟⏟⏟⏟119894=12119873

119901119894119895 | 119895 isin 119869) ( min⏟⏟⏟⏟⏟⏟⏟119894=12119873

119901119894119895 | 119895 isin 1198691015840)= 119901+1 119901+2 119901+119870

(27)

119860minus = ( min⏟⏟⏟⏟⏟⏟⏟119894=12119873

119901119894119895 | 119895 isin 119869) ( max⏟⏟⏟⏟⏟⏟⏟119894=12119873


(28)

max⏟⏟⏟⏟⏟⏟⏟119894=12119873

119901119894119895 = max⏟⏟⏟⏟⏟⏟⏟119894=12119873

(119886119894119895 119887119894119895 119888119894119895) = max⏟⏟⏟⏟⏟⏟⏟119894=12119873

(119887119894119895) forall119895 isin [1 119870] (29)

min⏟⏟⏟⏟⏟⏟⏟119894=12119873

119901119894119895 = min⏟⏟⏟⏟⏟⏟⏟119894=12119873

(119886119894119895 119887119894119895 119888119894119895) = min⏟⏟⏟⏟⏟⏟⏟119894=12119873

(119887119894119895) forall119895 isin [1 119870] (30)

where



119889+119894 = 119870sum119895=1

119889 (119901119894119895 119901+119895 ) forall119894 isin [1119873] (31)

119889minus119894 = 119870sum119895=1




(33)




119879119872119862119892119903119900119908 = max (1198771198621198621 1198771198621198622 119877119862119862119873) (35)



where



119863119887119897119900119888119896 = [120593119898119895]119872times119870

=[[[[[[[[[[[[


]]]]]]]]]]]]

(37)

where








where










where


21 27 35 48

20 26 34 47

19 25 33 46

18 24 32 45 Waste

TMC1

TMC2

TMC3




where



1198861198941 = 119897119894sumℎ=1



1198861198942 = 1198752119894119860 119894 = 1198752119894119897119894 ∙ (1198872119890 ) forall119894 isin [1119873] (43)

where





1198861198943120574 = radic 1119897119894119897119894sumℎ=1


(44)



119862119905119890ℎ = 119888119905119890ℎ119895 119895=12119870

= 119888119905119890ℎ1 119888119905119890ℎ2 119888119905119890ℎ31 119888119905119890ℎ32 119888119905119890ℎ3120574 119895=2+120574 (45)

where



119888119905119890ℎ1 = 1119873119867sumℎ=1

119886ℎ1 (46)


119888119905119890ℎ2 = (4119890)21198902 = 16 (47)


119888119905119890ℎ3120574 = 1198861198943120574 (48)


119862 = 119862119895119895=12119870 = 1198621 1198622 11986231 11986232 1198623120574119895=2+120574 (49)

where










forall119894 isin [1119873](53)



119885 = max⏟⏟⏟⏟⏟⏟⏟forall119894isin[1119873]

119880119877119862119894= max⏟⏟⏟⏟⏟⏟⏟forall119894isin[1119873]


(54)





for 120585+1 119861119872119861 isin 119879119872119862119892119903119900119908120572

for 120585+2 119861119872119861 isin 119879119872119862119892119903119900119908120573

for 120585+3 119887ℎ isin 119879119872119862119892119903119900119908120572






subject to

119873sum119894=1

119887ℎ119894 = 1 forallℎ isin [1119867] (55)


(56)


where (⋃119897119894lt119867ℎ=1














119865 = (119880119883 119862) = Tsum120591=1

119872sum119898=1

119906120596120591119898 (119909120591 minus 119888119898)2 (58)

subject to

119872sum119898=1

119906120591119898 = 1 120591 = 1 2 T (59)

0 le 119906120591119898 le 1 119898 = 1 2 119872 120591 = 1 2 T (60)

0 lt Tsum120591=1

119906119894119898 lt 119898 119898 = 1 2 119872 (61)

where










119875 (119884)forall120591isin[1T]

=


(62)


119898 = 1 2 119872 (63)


119889119911119898 = 120572 (ln 119911119898 minus ln 119911119898) 119911119898119889120591 + 120590119911119898119889119882 (64)

where






(65)

where



119911119898(120591+1) = 1205730 + 1205731119911119898(120591) + 120576 (66)




3

44

555

4

5

67

6

5

444

3333333333

444444

33

4

3

22

333

4

33

4444

333333333

4

5

67

6

0

1

2

3

4

5

6

7

0 5 10 15 20 25 30 35 40 45 50 55 60

Rang

e co

de v

alue

Month



119911119898 (120591)forall120591isin[1T]

=


119898 = 2 3 119872 minus 1

(67)




(68)



18

95

137125

71

3321

020406080

100120140160





119889119888120591 = 120583119888120591119889120591 + 120590119889119882120591 (69)




|119862|119904119894119898119878timesT = 10038161003816100381610038161198881199041205911003816100381610038161003816119878timesT (71)






= 119898119888119900119899 ∙ 119898119898119903 ∙ 1003816100381610038161003816119901119904120591119898 1003816100381610038161003816119878timesT (72)



10038161003816100381610038161003816Vℎ10038161003816100381610038161003816119904119894119898119878timesT = 119886ℎ1 (Ysum120574=1


(73)



=1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(119886 119887 119888)ℎ11 (119886 119887 119888)ℎ12 (119886 119887 119888)ℎ1120591(119886 119887 119888)ℎ21 (119886 119887 119888)ℎ22 (119886 119887 119888)ℎ2120591 d

(119886 119887 119888)ℎ1199041 (119886 119887 119888)ℎ1199042 (119886 119887 119888)ℎ119878T

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816


(74)


forall120591isin[1T] =10038161003816100381610038161003816100381610038161003816100381610038161119878119878sum119904=1


(75)


119864 (119864 (Vℎ120591)119905)119905=[1119879] =100381610038161003816100381610038161003816100381610038161003816100381611212sum120591=1

119864 (Vℎ120591)1199051003816100381610038161003816100381610038161003816100381610038161003816119905=[1119879]



1003816100381610038161003816100381610038161003816(Vℎ119905 )1198891003816100381610038161003816100381610038161003816 =10038161003816100381610038161003816100381610038161003816100381610038161003816


10038161003816100381610038161003816100381610038161003816100381610038161003816 forallℎ isin [1119867] 119905 = [1 119879]

(77)


V119889119894119905 = 119897119894sumℎ=1


(78)


3 Numerical Example






Zinc















1 5301 5891 6480 59 6458 7176 78932 6844 7604 8365 60 6954 7727 84993 5081 5646 6210 61 6293 6992 76914 7064 7849 8634 62 7119 7910 87015 7615 8461 9307 63 6128 6809 74896 6954 7727 8499 64 7284 8094 89037 7725 8583 9442 65 5962 6625 72878 5715 6350 6984 66 8977 9974 109729 6747 7497 8247 67 6128 6809 748910 5508 6120 6732 68 7119 7910 870111 7367 8186 9004 69 6293 6992 769112 6871 7635 8398 70 6954 7727 849913 7491 8323 9156 71 6458 7176 789314 6747 7497 8247 72 6789 7543 829715 7615 8461 9307 73 6623 7359 809516 6623 7359 8095 74 6623 7359 809517 7615 8461 9307 75 6789 7543 829718 5907 6564 7220 76 6458 7176 789319 6871 7635 8398 77 6954 7727 849920 5715 6350 6984 78 6293 6992 769121 7064 7849 8634 79 7119 7910 870122 5522 6135 6749 80 6128 6809 748923 6816 7574 8331 81 7284 8094 890324 6678 7421 8163 82 9130 10144 1115825 7367 8186 9004 83 6334 7038 774226 6541 7268 7994 84 7243 8048 885327 7505 8339 9172 85 6486 7206 792728 6458 7176 7893 86 7092 7880 866729 6637 7375 8112 87 6637 7375 811230 6279 6977 7674 88 6940 7711 848231 6816 7574 8331 89 6789 7543 829732 6100 6778 7456 90 6789 7543 829733 6995 7772 8550 91 6940 7711 848234 5921 6579 7237 92 6637 7375 811235 7174 7971 8768 93 7092 7880 866736 6486 7206 7927 94 6486 7206 792737 7092 7880 8667 95 7243 8048 885338 6637 7375 8112 96 8895 9883 1087239 6940 7711 8482 97 5921 6579 723740 6789 7543 8297 98 6995 7772 855041 6789 7543 8297 99 6100 6778 745642 6940 7711 8482 100 6816 7574 833143 6637 7375 8112 101 6279 6977 767444 7092 7880 8667 102 6637 7375 811245 6486 7206 7927 103 6458 7176 789346 7243 8048 8853 104 6458 7176 789347 6334 7038 7742 105 6637 7375 811248 7394 8216 9038 106 6279 6977 767449 6183 6870 7557 107 6816 7574 833150 6128 6809 7489 108 6100 6778 745651 7119 7910 8701 109 6678 7421 816352 6293 6992 7691 110 7229 8033 8836


Table 3 Continued





1152640

3913 -165169601186561

2153217

3072 -128170241187266

3154940

2939 -017172156189371

4155573

6278 +024172859190145

5159649

4816 +286177388195127


(15-19](19-23](23-27](27-31](31-35]

(151-276](276-401](401-526](526-651](651-776]



17

3 16 22 49 65

2 7 15 21 27 35 48 64 81

1 6 10 14 20 26 34 47 63 80 TMC1

5 9 13 19 25 33 46 62 79 95 TMC2

4 8 12 18 24 32 45 61 78 94 108 TMC3

11 31 44 60 77 93 107 TMC4

23 30 43 59 76 92 106 115 TMC5

29 42 58 75 91 105 114

28 41 57 74 90 104 113

40 56 73 89 103 112

39 55 72 88 102

38 54 71 87 101 111

37 53 70 86 100 110

36 52 69 85 99 109

51 68 84 98

50 67 83 97

66 82 96




1 2263 3364 0325 1198 14392 356072 1801 3832 0292 0418 16222 109313 2170 3937 0381 0820 17565 208234 1950 2573 0169 0928 8710 360795 2136 4636 0267 1262 12531 27226





(165 + 128 + 017 + 024 + 286)5 = 124 (79)










Lead ($t) 1176 1689 1863 2064 2205 2381 2665Zinc ($t) 1236 1816 2014 2176 2344 2695 3244

173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80

5 9 13 19 25 33 46 62 79 954 8 12 18 24 32 45 61 78 94 108

11 31 44 60 77 93 10723 30 43 59 76 92 106 115

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96

(a)

173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80

5 9 13 19 25 33 46 62 79 954 8 12 18 24 32 45 61 78 94 108

11 31 44 60 77 93 10723 30 43 59 76 92 106 115

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96

(b)

173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80

5 9 13 19 25 33 46 62 79 954 8 12 18 24 32 45 61 78 94 108

11 31 44 60 77 93 10723 30 43 59 76 92 106 115

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96

(c)

173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80

5 9 13 19 25 33 46 62 79 954 8 12 18 24 32 45 61 78 94 108

11 31 44 60 77 93 10723 30 43 59 76 92 106 115

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96

(d)



173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80 TMC1

5 9 13 19 25 33 46 62 79 95 TMC2

4 8 12 18 24 32 45 61 78 94 108 TMC3

11 31 44 60 77 93 107 TMC4

23 30 43 59 76 92 106 115 TMC5

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96





Process Value




volatility 018491




volatility 016416





0

01

02

03

04

05

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC3 StdDev Pb

0

02

04

06

08

1

12

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC3 StdDev Zn

0

01

02

03

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC4 StdDev Pb

0

02

04

06

08

1

12

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC4 StedDev Zn

0

01

02

03

04

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

Number of blocks

TMC5 StdDev Pb

002040608

112141618

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

Number of blocks

TMC5 StdDev Zn

0

01

02

03

04

05

06

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC1 StdDev Pb

002040608

112141618

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC1 StdDev Zn

0

01

02

03

04

05

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Number of blocks

TMC2 StdDev Pb

0

02

04

06

08

1

12

14

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Number of blocks

TMC2 StdDev Zn










0

500

1000

1500

2000

2500

3000

3500

4000

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125

US$

t

Month

(a)

0

500

1000

1500

2000

2500

3000

3500

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125

US$

t

Month

(b)


0010203040506070809

1

500 1000 1500 2000 2500 3000 3500$t

VVLVLLM

HVHVVH

(a)

0010203040506070809

1

500 1000 1500 2000 2500 3000 3500 4000$t

VVLVLLM

HVHVVH

(b)




15104734 3944737 3044364 2443462 19244336591226 5356427 4305812 3576349 29263948228167 6904516 5704581 4871250 4096627

25583555 4336349 3366286 2691067 21178617190095 5863841 4730052 3916424 32015418951887 7536746 6246798 5320340 4472001

36213929 4891545 3857917 3139633 25243907923489 6519182 5317593 4456306 36948059796030 8298845 6935676 5958908 5056129

42976598 2108873 1450337 1038819 6860214063918 3133980 2351322 1834430 13747555275030 4267887 3358689 2754948 2200817

56198802 4899506 3882193 3166488 25554757882521 6503582 5321728 4466255 37120859723915 8256167 6917828 5949400 5056656



0

500

1000

1500

2000

2500

3000

3500

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Lead price


(a)

0

1

2

3

4

5

6

7

8

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Lead price

Range code

(b)

0

500

1000

1500

2000

2500

3000

3500

4000

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Zinc price


(c)

0

1

2

3

4

5

6

7

8

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Zinc price

Range code

(d)




119865 minus 119868 = (17561955 24060165 31322673)minus (12000000 13000000 15000000)

= (2561955 11060166 19322673) $(80)


4 Conclusion





LEAD ZINC

UNIT OPERATING COST


01234567

0 5 10 15 20 25 30 35 40 45 50 55 60

Rang

e cod

e val

ue

Month

01234567

0 5 10 15 20 25 30 35 40 45 50 55 60

Rang

e cod

e val

ue

Month

0500

100015002000250030003500

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

0500

1000150020002500300035004000

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

4648505254565860

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

0100000200000300000400000500000600000700000800000

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

ℎ

sim

Stimes４= aℎ

1

(９sum

=1

aℎ2 M

mcon

con

sim








0

100000

200000

300000

400000

500000

600000

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month


0

100000

200000

300000

400000

500000

600000

1 2 3 4 5

$t

Year




Data Availability




0

100000

200000

300000

400000

500000

600000

1 2 3 4 5

$t

Year


TMC 1year 4

TMC 4year 5

TMC 5year 3

TMC 3year 1

TMC 2year 2


References
























































Journal of












Journal of







Volume 2018






Volume 2018















max⏟⏟⏟⏟⏟⏟⏟1198861119888isin1198791198721198621

119891 (1198861119888) = [1198911 (11988611) 1198912 (11988612) 119891119870 (1198861119870)]max⏟⏟⏟⏟⏟⏟⏟

1198862119888isin1198791198721198622

119891 (1198862119888) = [1198911 (11988621) 1198912 (11988622) 119891119870 (1198862119870)]

max⏟⏟⏟⏟⏟⏟⏟119886119894119888isin119879119872119862119894

119891 (119886119894119888) = [1198911 (1198861198941) 1198912 (1198861198942) 119891119870 (119886119873119870)](20)

subject to

119886119897119888 le 119886119888 le 119886119906119888 119888 = 1 2 119870 (21)

where




119861 = 119873⋃119894=1

119879119872119862119894= 1198791198721198621 cup 1198791198721198622 cup cup 119879119872119862119894119879119872119862120572 cap 119879119872119862120573 = 0 120572 = 120573

(22)




119863 = [119909119894119895]119873times119870 =[[[[[[[[[[[


]]]]]]]]]]](23)



119895=1

119908119895 = 1 (24)



119890119895 = minus 1ln (119873)

119873sum119894=1


(25)




119860+ = ( max⏟⏟⏟⏟⏟⏟⏟119894=12119873

119901119894119895 | 119895 isin 119869) ( min⏟⏟⏟⏟⏟⏟⏟119894=12119873

119901119894119895 | 119895 isin 1198691015840)= 119901+1 119901+2 119901+119870

(27)

119860minus = ( min⏟⏟⏟⏟⏟⏟⏟119894=12119873

119901119894119895 | 119895 isin 119869) ( max⏟⏟⏟⏟⏟⏟⏟119894=12119873


(28)

max⏟⏟⏟⏟⏟⏟⏟119894=12119873

119901119894119895 = max⏟⏟⏟⏟⏟⏟⏟119894=12119873

(119886119894119895 119887119894119895 119888119894119895) = max⏟⏟⏟⏟⏟⏟⏟119894=12119873

(119887119894119895) forall119895 isin [1 119870] (29)

min⏟⏟⏟⏟⏟⏟⏟119894=12119873

119901119894119895 = min⏟⏟⏟⏟⏟⏟⏟119894=12119873

(119886119894119895 119887119894119895 119888119894119895) = min⏟⏟⏟⏟⏟⏟⏟119894=12119873

(119887119894119895) forall119895 isin [1 119870] (30)

where



119889+119894 = 119870sum119895=1

119889 (119901119894119895 119901+119895 ) forall119894 isin [1119873] (31)

119889minus119894 = 119870sum119895=1




(33)




119879119872119862119892119903119900119908 = max (1198771198621198621 1198771198621198622 119877119862119862119873) (35)



where



119863119887119897119900119888119896 = [120593119898119895]119872times119870

=[[[[[[[[[[[[


]]]]]]]]]]]]

(37)

where








where










where


21 27 35 48

20 26 34 47

19 25 33 46

18 24 32 45 Waste

TMC1

TMC2

TMC3




where



1198861198941 = 119897119894sumℎ=1



1198861198942 = 1198752119894119860 119894 = 1198752119894119897119894 ∙ (1198872119890 ) forall119894 isin [1119873] (43)

where





1198861198943120574 = radic 1119897119894119897119894sumℎ=1


(44)



119862119905119890ℎ = 119888119905119890ℎ119895 119895=12119870

= 119888119905119890ℎ1 119888119905119890ℎ2 119888119905119890ℎ31 119888119905119890ℎ32 119888119905119890ℎ3120574 119895=2+120574 (45)

where



119888119905119890ℎ1 = 1119873119867sumℎ=1

119886ℎ1 (46)


119888119905119890ℎ2 = (4119890)21198902 = 16 (47)


119888119905119890ℎ3120574 = 1198861198943120574 (48)


119862 = 119862119895119895=12119870 = 1198621 1198622 11986231 11986232 1198623120574119895=2+120574 (49)

where










forall119894 isin [1119873](53)



119885 = max⏟⏟⏟⏟⏟⏟⏟forall119894isin[1119873]

119880119877119862119894= max⏟⏟⏟⏟⏟⏟⏟forall119894isin[1119873]


(54)





for 120585+1 119861119872119861 isin 119879119872119862119892119903119900119908120572

for 120585+2 119861119872119861 isin 119879119872119862119892119903119900119908120573

for 120585+3 119887ℎ isin 119879119872119862119892119903119900119908120572






subject to

119873sum119894=1

119887ℎ119894 = 1 forallℎ isin [1119867] (55)


(56)


where (⋃119897119894lt119867ℎ=1














119865 = (119880119883 119862) = Tsum120591=1

119872sum119898=1

119906120596120591119898 (119909120591 minus 119888119898)2 (58)

subject to

119872sum119898=1

119906120591119898 = 1 120591 = 1 2 T (59)

0 le 119906120591119898 le 1 119898 = 1 2 119872 120591 = 1 2 T (60)

0 lt Tsum120591=1

119906119894119898 lt 119898 119898 = 1 2 119872 (61)

where










119875 (119884)forall120591isin[1T]

=


(62)


119898 = 1 2 119872 (63)


119889119911119898 = 120572 (ln 119911119898 minus ln 119911119898) 119911119898119889120591 + 120590119911119898119889119882 (64)

where






(65)

where



119911119898(120591+1) = 1205730 + 1205731119911119898(120591) + 120576 (66)




3

44

555

4

5

67

6

5

444

3333333333

444444

33

4

3

22

333

4

33

4444

333333333

4

5

67

6

0

1

2

3

4

5

6

7

0 5 10 15 20 25 30 35 40 45 50 55 60

Rang

e co

de v

alue

Month



119911119898 (120591)forall120591isin[1T]

=


119898 = 2 3 119872 minus 1

(67)




(68)



18

95

137125

71

3321

020406080

100120140160





119889119888120591 = 120583119888120591119889120591 + 120590119889119882120591 (69)




|119862|119904119894119898119878timesT = 10038161003816100381610038161198881199041205911003816100381610038161003816119878timesT (71)






= 119898119888119900119899 ∙ 119898119898119903 ∙ 1003816100381610038161003816119901119904120591119898 1003816100381610038161003816119878timesT (72)



10038161003816100381610038161003816Vℎ10038161003816100381610038161003816119904119894119898119878timesT = 119886ℎ1 (Ysum120574=1


(73)



=1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(119886 119887 119888)ℎ11 (119886 119887 119888)ℎ12 (119886 119887 119888)ℎ1120591(119886 119887 119888)ℎ21 (119886 119887 119888)ℎ22 (119886 119887 119888)ℎ2120591 d

(119886 119887 119888)ℎ1199041 (119886 119887 119888)ℎ1199042 (119886 119887 119888)ℎ119878T

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816


(74)


forall120591isin[1T] =10038161003816100381610038161003816100381610038161003816100381610038161119878119878sum119904=1


(75)


119864 (119864 (Vℎ120591)119905)119905=[1119879] =100381610038161003816100381610038161003816100381610038161003816100381611212sum120591=1

119864 (Vℎ120591)1199051003816100381610038161003816100381610038161003816100381610038161003816119905=[1119879]



1003816100381610038161003816100381610038161003816(Vℎ119905 )1198891003816100381610038161003816100381610038161003816 =10038161003816100381610038161003816100381610038161003816100381610038161003816


10038161003816100381610038161003816100381610038161003816100381610038161003816 forallℎ isin [1119867] 119905 = [1 119879]

(77)


V119889119894119905 = 119897119894sumℎ=1


(78)


3 Numerical Example






Zinc















1 5301 5891 6480 59 6458 7176 78932 6844 7604 8365 60 6954 7727 84993 5081 5646 6210 61 6293 6992 76914 7064 7849 8634 62 7119 7910 87015 7615 8461 9307 63 6128 6809 74896 6954 7727 8499 64 7284 8094 89037 7725 8583 9442 65 5962 6625 72878 5715 6350 6984 66 8977 9974 109729 6747 7497 8247 67 6128 6809 748910 5508 6120 6732 68 7119 7910 870111 7367 8186 9004 69 6293 6992 769112 6871 7635 8398 70 6954 7727 849913 7491 8323 9156 71 6458 7176 789314 6747 7497 8247 72 6789 7543 829715 7615 8461 9307 73 6623 7359 809516 6623 7359 8095 74 6623 7359 809517 7615 8461 9307 75 6789 7543 829718 5907 6564 7220 76 6458 7176 789319 6871 7635 8398 77 6954 7727 849920 5715 6350 6984 78 6293 6992 769121 7064 7849 8634 79 7119 7910 870122 5522 6135 6749 80 6128 6809 748923 6816 7574 8331 81 7284 8094 890324 6678 7421 8163 82 9130 10144 1115825 7367 8186 9004 83 6334 7038 774226 6541 7268 7994 84 7243 8048 885327 7505 8339 9172 85 6486 7206 792728 6458 7176 7893 86 7092 7880 866729 6637 7375 8112 87 6637 7375 811230 6279 6977 7674 88 6940 7711 848231 6816 7574 8331 89 6789 7543 829732 6100 6778 7456 90 6789 7543 829733 6995 7772 8550 91 6940 7711 848234 5921 6579 7237 92 6637 7375 811235 7174 7971 8768 93 7092 7880 866736 6486 7206 7927 94 6486 7206 792737 7092 7880 8667 95 7243 8048 885338 6637 7375 8112 96 8895 9883 1087239 6940 7711 8482 97 5921 6579 723740 6789 7543 8297 98 6995 7772 855041 6789 7543 8297 99 6100 6778 745642 6940 7711 8482 100 6816 7574 833143 6637 7375 8112 101 6279 6977 767444 7092 7880 8667 102 6637 7375 811245 6486 7206 7927 103 6458 7176 789346 7243 8048 8853 104 6458 7176 789347 6334 7038 7742 105 6637 7375 811248 7394 8216 9038 106 6279 6977 767449 6183 6870 7557 107 6816 7574 833150 6128 6809 7489 108 6100 6778 745651 7119 7910 8701 109 6678 7421 816352 6293 6992 7691 110 7229 8033 8836


Table 3 Continued





1152640

3913 -165169601186561

2153217

3072 -128170241187266

3154940

2939 -017172156189371

4155573

6278 +024172859190145

5159649

4816 +286177388195127


(15-19](19-23](23-27](27-31](31-35]

(151-276](276-401](401-526](526-651](651-776]



17

3 16 22 49 65

2 7 15 21 27 35 48 64 81

1 6 10 14 20 26 34 47 63 80 TMC1

5 9 13 19 25 33 46 62 79 95 TMC2

4 8 12 18 24 32 45 61 78 94 108 TMC3

11 31 44 60 77 93 107 TMC4

23 30 43 59 76 92 106 115 TMC5

29 42 58 75 91 105 114

28 41 57 74 90 104 113

40 56 73 89 103 112

39 55 72 88 102

38 54 71 87 101 111

37 53 70 86 100 110

36 52 69 85 99 109

51 68 84 98

50 67 83 97

66 82 96




1 2263 3364 0325 1198 14392 356072 1801 3832 0292 0418 16222 109313 2170 3937 0381 0820 17565 208234 1950 2573 0169 0928 8710 360795 2136 4636 0267 1262 12531 27226





(165 + 128 + 017 + 024 + 286)5 = 124 (79)










Lead ($t) 1176 1689 1863 2064 2205 2381 2665Zinc ($t) 1236 1816 2014 2176 2344 2695 3244

173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80

5 9 13 19 25 33 46 62 79 954 8 12 18 24 32 45 61 78 94 108

11 31 44 60 77 93 10723 30 43 59 76 92 106 115

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96

(a)

173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80

5 9 13 19 25 33 46 62 79 954 8 12 18 24 32 45 61 78 94 108

11 31 44 60 77 93 10723 30 43 59 76 92 106 115

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96

(b)

173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80

5 9 13 19 25 33 46 62 79 954 8 12 18 24 32 45 61 78 94 108

11 31 44 60 77 93 10723 30 43 59 76 92 106 115

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96

(c)

173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80

5 9 13 19 25 33 46 62 79 954 8 12 18 24 32 45 61 78 94 108

11 31 44 60 77 93 10723 30 43 59 76 92 106 115

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96

(d)



173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80 TMC1

5 9 13 19 25 33 46 62 79 95 TMC2

4 8 12 18 24 32 45 61 78 94 108 TMC3

11 31 44 60 77 93 107 TMC4

23 30 43 59 76 92 106 115 TMC5

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96





Process Value




volatility 018491




volatility 016416





0

01

02

03

04

05

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC3 StdDev Pb

0

02

04

06

08

1

12

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC3 StdDev Zn

0

01

02

03

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC4 StdDev Pb

0

02

04

06

08

1

12

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC4 StedDev Zn

0

01

02

03

04

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

Number of blocks

TMC5 StdDev Pb

002040608

112141618

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

Number of blocks

TMC5 StdDev Zn

0

01

02

03

04

05

06

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC1 StdDev Pb

002040608

112141618

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC1 StdDev Zn

0

01

02

03

04

05

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Number of blocks

TMC2 StdDev Pb

0

02

04

06

08

1

12

14

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Number of blocks

TMC2 StdDev Zn










0

500

1000

1500

2000

2500

3000

3500

4000

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125

US$

t

Month

(a)

0

500

1000

1500

2000

2500

3000

3500

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125

US$

t

Month

(b)


0010203040506070809

1

500 1000 1500 2000 2500 3000 3500$t

VVLVLLM

HVHVVH

(a)

0010203040506070809

1

500 1000 1500 2000 2500 3000 3500 4000$t

VVLVLLM

HVHVVH

(b)




15104734 3944737 3044364 2443462 19244336591226 5356427 4305812 3576349 29263948228167 6904516 5704581 4871250 4096627

25583555 4336349 3366286 2691067 21178617190095 5863841 4730052 3916424 32015418951887 7536746 6246798 5320340 4472001

36213929 4891545 3857917 3139633 25243907923489 6519182 5317593 4456306 36948059796030 8298845 6935676 5958908 5056129

42976598 2108873 1450337 1038819 6860214063918 3133980 2351322 1834430 13747555275030 4267887 3358689 2754948 2200817

56198802 4899506 3882193 3166488 25554757882521 6503582 5321728 4466255 37120859723915 8256167 6917828 5949400 5056656



0

500

1000

1500

2000

2500

3000

3500

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Lead price


(a)

0

1

2

3

4

5

6

7

8

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Lead price

Range code

(b)

0

500

1000

1500

2000

2500

3000

3500

4000

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Zinc price


(c)

0

1

2

3

4

5

6

7

8

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Zinc price

Range code

(d)




119865 minus 119868 = (17561955 24060165 31322673)minus (12000000 13000000 15000000)

= (2561955 11060166 19322673) $(80)


4 Conclusion





LEAD ZINC

UNIT OPERATING COST


01234567

0 5 10 15 20 25 30 35 40 45 50 55 60

Rang

e cod

e val

ue

Month

01234567

0 5 10 15 20 25 30 35 40 45 50 55 60

Rang

e cod

e val

ue

Month

0500

100015002000250030003500

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

0500

1000150020002500300035004000

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

4648505254565860

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

0100000200000300000400000500000600000700000800000

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

ℎ

sim

Stimes４= aℎ

1

(９sum

=1

aℎ2 M

mcon

con

sim








0

100000

200000

300000

400000

500000

600000

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month


0

100000

200000

300000

400000

500000

600000

1 2 3 4 5

$t

Year




Data Availability




0

100000

200000

300000

400000

500000

600000

1 2 3 4 5

$t

Year


TMC 1year 4

TMC 4year 5

TMC 5year 3

TMC 3year 1

TMC 2year 2


References
























































Journal of












Journal of







Volume 2018






Volume 2018










119890119895 = minus 1ln (119873)

119873sum119894=1


(25)




119860+ = ( max⏟⏟⏟⏟⏟⏟⏟119894=12119873

119901119894119895 | 119895 isin 119869) ( min⏟⏟⏟⏟⏟⏟⏟119894=12119873

119901119894119895 | 119895 isin 1198691015840)= 119901+1 119901+2 119901+119870

(27)

119860minus = ( min⏟⏟⏟⏟⏟⏟⏟119894=12119873

119901119894119895 | 119895 isin 119869) ( max⏟⏟⏟⏟⏟⏟⏟119894=12119873


(28)

max⏟⏟⏟⏟⏟⏟⏟119894=12119873

119901119894119895 = max⏟⏟⏟⏟⏟⏟⏟119894=12119873

(119886119894119895 119887119894119895 119888119894119895) = max⏟⏟⏟⏟⏟⏟⏟119894=12119873

(119887119894119895) forall119895 isin [1 119870] (29)

min⏟⏟⏟⏟⏟⏟⏟119894=12119873

119901119894119895 = min⏟⏟⏟⏟⏟⏟⏟119894=12119873

(119886119894119895 119887119894119895 119888119894119895) = min⏟⏟⏟⏟⏟⏟⏟119894=12119873

(119887119894119895) forall119895 isin [1 119870] (30)

where



119889+119894 = 119870sum119895=1

119889 (119901119894119895 119901+119895 ) forall119894 isin [1119873] (31)

119889minus119894 = 119870sum119895=1




(33)




119879119872119862119892119903119900119908 = max (1198771198621198621 1198771198621198622 119877119862119862119873) (35)



where



119863119887119897119900119888119896 = [120593119898119895]119872times119870

=[[[[[[[[[[[[


]]]]]]]]]]]]

(37)

where








where










where


21 27 35 48

20 26 34 47

19 25 33 46

18 24 32 45 Waste

TMC1

TMC2

TMC3




where



1198861198941 = 119897119894sumℎ=1



1198861198942 = 1198752119894119860 119894 = 1198752119894119897119894 ∙ (1198872119890 ) forall119894 isin [1119873] (43)

where





1198861198943120574 = radic 1119897119894119897119894sumℎ=1


(44)



119862119905119890ℎ = 119888119905119890ℎ119895 119895=12119870

= 119888119905119890ℎ1 119888119905119890ℎ2 119888119905119890ℎ31 119888119905119890ℎ32 119888119905119890ℎ3120574 119895=2+120574 (45)

where



119888119905119890ℎ1 = 1119873119867sumℎ=1

119886ℎ1 (46)


119888119905119890ℎ2 = (4119890)21198902 = 16 (47)


119888119905119890ℎ3120574 = 1198861198943120574 (48)


119862 = 119862119895119895=12119870 = 1198621 1198622 11986231 11986232 1198623120574119895=2+120574 (49)

where










forall119894 isin [1119873](53)



119885 = max⏟⏟⏟⏟⏟⏟⏟forall119894isin[1119873]

119880119877119862119894= max⏟⏟⏟⏟⏟⏟⏟forall119894isin[1119873]


(54)





for 120585+1 119861119872119861 isin 119879119872119862119892119903119900119908120572

for 120585+2 119861119872119861 isin 119879119872119862119892119903119900119908120573

for 120585+3 119887ℎ isin 119879119872119862119892119903119900119908120572






subject to

119873sum119894=1

119887ℎ119894 = 1 forallℎ isin [1119867] (55)


(56)


where (⋃119897119894lt119867ℎ=1














119865 = (119880119883 119862) = Tsum120591=1

119872sum119898=1

119906120596120591119898 (119909120591 minus 119888119898)2 (58)

subject to

119872sum119898=1

119906120591119898 = 1 120591 = 1 2 T (59)

0 le 119906120591119898 le 1 119898 = 1 2 119872 120591 = 1 2 T (60)

0 lt Tsum120591=1

119906119894119898 lt 119898 119898 = 1 2 119872 (61)

where










119875 (119884)forall120591isin[1T]

=


(62)


119898 = 1 2 119872 (63)


119889119911119898 = 120572 (ln 119911119898 minus ln 119911119898) 119911119898119889120591 + 120590119911119898119889119882 (64)

where






(65)

where



119911119898(120591+1) = 1205730 + 1205731119911119898(120591) + 120576 (66)




3

44

555

4

5

67

6

5

444

3333333333

444444

33

4

3

22

333

4

33

4444

333333333

4

5

67

6

0

1

2

3

4

5

6

7

0 5 10 15 20 25 30 35 40 45 50 55 60

Rang

e co

de v

alue

Month



119911119898 (120591)forall120591isin[1T]

=


119898 = 2 3 119872 minus 1

(67)




(68)



18

95

137125

71

3321

020406080

100120140160





119889119888120591 = 120583119888120591119889120591 + 120590119889119882120591 (69)




|119862|119904119894119898119878timesT = 10038161003816100381610038161198881199041205911003816100381610038161003816119878timesT (71)






= 119898119888119900119899 ∙ 119898119898119903 ∙ 1003816100381610038161003816119901119904120591119898 1003816100381610038161003816119878timesT (72)



10038161003816100381610038161003816Vℎ10038161003816100381610038161003816119904119894119898119878timesT = 119886ℎ1 (Ysum120574=1


(73)



=1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(119886 119887 119888)ℎ11 (119886 119887 119888)ℎ12 (119886 119887 119888)ℎ1120591(119886 119887 119888)ℎ21 (119886 119887 119888)ℎ22 (119886 119887 119888)ℎ2120591 d

(119886 119887 119888)ℎ1199041 (119886 119887 119888)ℎ1199042 (119886 119887 119888)ℎ119878T

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816


(74)


forall120591isin[1T] =10038161003816100381610038161003816100381610038161003816100381610038161119878119878sum119904=1


(75)


119864 (119864 (Vℎ120591)119905)119905=[1119879] =100381610038161003816100381610038161003816100381610038161003816100381611212sum120591=1

119864 (Vℎ120591)1199051003816100381610038161003816100381610038161003816100381610038161003816119905=[1119879]



1003816100381610038161003816100381610038161003816(Vℎ119905 )1198891003816100381610038161003816100381610038161003816 =10038161003816100381610038161003816100381610038161003816100381610038161003816


10038161003816100381610038161003816100381610038161003816100381610038161003816 forallℎ isin [1119867] 119905 = [1 119879]

(77)


V119889119894119905 = 119897119894sumℎ=1


(78)


3 Numerical Example






Zinc















1 5301 5891 6480 59 6458 7176 78932 6844 7604 8365 60 6954 7727 84993 5081 5646 6210 61 6293 6992 76914 7064 7849 8634 62 7119 7910 87015 7615 8461 9307 63 6128 6809 74896 6954 7727 8499 64 7284 8094 89037 7725 8583 9442 65 5962 6625 72878 5715 6350 6984 66 8977 9974 109729 6747 7497 8247 67 6128 6809 748910 5508 6120 6732 68 7119 7910 870111 7367 8186 9004 69 6293 6992 769112 6871 7635 8398 70 6954 7727 849913 7491 8323 9156 71 6458 7176 789314 6747 7497 8247 72 6789 7543 829715 7615 8461 9307 73 6623 7359 809516 6623 7359 8095 74 6623 7359 809517 7615 8461 9307 75 6789 7543 829718 5907 6564 7220 76 6458 7176 789319 6871 7635 8398 77 6954 7727 849920 5715 6350 6984 78 6293 6992 769121 7064 7849 8634 79 7119 7910 870122 5522 6135 6749 80 6128 6809 748923 6816 7574 8331 81 7284 8094 890324 6678 7421 8163 82 9130 10144 1115825 7367 8186 9004 83 6334 7038 774226 6541 7268 7994 84 7243 8048 885327 7505 8339 9172 85 6486 7206 792728 6458 7176 7893 86 7092 7880 866729 6637 7375 8112 87 6637 7375 811230 6279 6977 7674 88 6940 7711 848231 6816 7574 8331 89 6789 7543 829732 6100 6778 7456 90 6789 7543 829733 6995 7772 8550 91 6940 7711 848234 5921 6579 7237 92 6637 7375 811235 7174 7971 8768 93 7092 7880 866736 6486 7206 7927 94 6486 7206 792737 7092 7880 8667 95 7243 8048 885338 6637 7375 8112 96 8895 9883 1087239 6940 7711 8482 97 5921 6579 723740 6789 7543 8297 98 6995 7772 855041 6789 7543 8297 99 6100 6778 745642 6940 7711 8482 100 6816 7574 833143 6637 7375 8112 101 6279 6977 767444 7092 7880 8667 102 6637 7375 811245 6486 7206 7927 103 6458 7176 789346 7243 8048 8853 104 6458 7176 789347 6334 7038 7742 105 6637 7375 811248 7394 8216 9038 106 6279 6977 767449 6183 6870 7557 107 6816 7574 833150 6128 6809 7489 108 6100 6778 745651 7119 7910 8701 109 6678 7421 816352 6293 6992 7691 110 7229 8033 8836


Table 3 Continued





1152640

3913 -165169601186561

2153217

3072 -128170241187266

3154940

2939 -017172156189371

4155573

6278 +024172859190145

5159649

4816 +286177388195127


(15-19](19-23](23-27](27-31](31-35]

(151-276](276-401](401-526](526-651](651-776]



17

3 16 22 49 65

2 7 15 21 27 35 48 64 81

1 6 10 14 20 26 34 47 63 80 TMC1

5 9 13 19 25 33 46 62 79 95 TMC2

4 8 12 18 24 32 45 61 78 94 108 TMC3

11 31 44 60 77 93 107 TMC4

23 30 43 59 76 92 106 115 TMC5

29 42 58 75 91 105 114

28 41 57 74 90 104 113

40 56 73 89 103 112

39 55 72 88 102

38 54 71 87 101 111

37 53 70 86 100 110

36 52 69 85 99 109

51 68 84 98

50 67 83 97

66 82 96




1 2263 3364 0325 1198 14392 356072 1801 3832 0292 0418 16222 109313 2170 3937 0381 0820 17565 208234 1950 2573 0169 0928 8710 360795 2136 4636 0267 1262 12531 27226





(165 + 128 + 017 + 024 + 286)5 = 124 (79)










Lead ($t) 1176 1689 1863 2064 2205 2381 2665Zinc ($t) 1236 1816 2014 2176 2344 2695 3244

173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80

5 9 13 19 25 33 46 62 79 954 8 12 18 24 32 45 61 78 94 108

11 31 44 60 77 93 10723 30 43 59 76 92 106 115

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96

(a)

173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80

5 9 13 19 25 33 46 62 79 954 8 12 18 24 32 45 61 78 94 108

11 31 44 60 77 93 10723 30 43 59 76 92 106 115

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96

(b)

173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80

5 9 13 19 25 33 46 62 79 954 8 12 18 24 32 45 61 78 94 108

11 31 44 60 77 93 10723 30 43 59 76 92 106 115

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96

(c)

173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80

5 9 13 19 25 33 46 62 79 954 8 12 18 24 32 45 61 78 94 108

11 31 44 60 77 93 10723 30 43 59 76 92 106 115

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96

(d)



173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80 TMC1

5 9 13 19 25 33 46 62 79 95 TMC2

4 8 12 18 24 32 45 61 78 94 108 TMC3

11 31 44 60 77 93 107 TMC4

23 30 43 59 76 92 106 115 TMC5

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96





Process Value




volatility 018491




volatility 016416





0

01

02

03

04

05

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC3 StdDev Pb

0

02

04

06

08

1

12

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC3 StdDev Zn

0

01

02

03

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC4 StdDev Pb

0

02

04

06

08

1

12

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC4 StedDev Zn

0

01

02

03

04

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

Number of blocks

TMC5 StdDev Pb

002040608

112141618

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

Number of blocks

TMC5 StdDev Zn

0

01

02

03

04

05

06

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC1 StdDev Pb

002040608

112141618

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC1 StdDev Zn

0

01

02

03

04

05

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Number of blocks

TMC2 StdDev Pb

0

02

04

06

08

1

12

14

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Number of blocks

TMC2 StdDev Zn










0

500

1000

1500

2000

2500

3000

3500

4000

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125

US$

t

Month

(a)

0

500

1000

1500

2000

2500

3000

3500

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125

US$

t

Month

(b)


0010203040506070809

1

500 1000 1500 2000 2500 3000 3500$t

VVLVLLM

HVHVVH

(a)

0010203040506070809

1

500 1000 1500 2000 2500 3000 3500 4000$t

VVLVLLM

HVHVVH

(b)




15104734 3944737 3044364 2443462 19244336591226 5356427 4305812 3576349 29263948228167 6904516 5704581 4871250 4096627

25583555 4336349 3366286 2691067 21178617190095 5863841 4730052 3916424 32015418951887 7536746 6246798 5320340 4472001

36213929 4891545 3857917 3139633 25243907923489 6519182 5317593 4456306 36948059796030 8298845 6935676 5958908 5056129

42976598 2108873 1450337 1038819 6860214063918 3133980 2351322 1834430 13747555275030 4267887 3358689 2754948 2200817

56198802 4899506 3882193 3166488 25554757882521 6503582 5321728 4466255 37120859723915 8256167 6917828 5949400 5056656



0

500

1000

1500

2000

2500

3000

3500

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Lead price


(a)

0

1

2

3

4

5

6

7

8

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Lead price

Range code

(b)

0

500

1000

1500

2000

2500

3000

3500

4000

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Zinc price


(c)

0

1

2

3

4

5

6

7

8

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Zinc price

Range code

(d)




119865 minus 119868 = (17561955 24060165 31322673)minus (12000000 13000000 15000000)

= (2561955 11060166 19322673) $(80)


4 Conclusion





LEAD ZINC

UNIT OPERATING COST


01234567

0 5 10 15 20 25 30 35 40 45 50 55 60

Rang

e cod

e val

ue

Month

01234567

0 5 10 15 20 25 30 35 40 45 50 55 60

Rang

e cod

e val

ue

Month

0500

100015002000250030003500

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

0500

1000150020002500300035004000

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

4648505254565860

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

0100000200000300000400000500000600000700000800000

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

ℎ

sim

Stimes４= aℎ

1

(９sum

=1

aℎ2 M

mcon

con

sim








0

100000

200000

300000

400000

500000

600000

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month


0

100000

200000

300000

400000

500000

600000

1 2 3 4 5

$t

Year




Data Availability




0

100000

200000

300000

400000

500000

600000

1 2 3 4 5

$t

Year


TMC 1year 4

TMC 4year 5

TMC 5year 3

TMC 3year 1

TMC 2year 2


References
























































Journal of












Journal of







Volume 2018






Volume 2018











where










where


21 27 35 48

20 26 34 47

19 25 33 46

18 24 32 45 Waste

TMC1

TMC2

TMC3




where



1198861198941 = 119897119894sumℎ=1



1198861198942 = 1198752119894119860 119894 = 1198752119894119897119894 ∙ (1198872119890 ) forall119894 isin [1119873] (43)

where





1198861198943120574 = radic 1119897119894119897119894sumℎ=1


(44)



119862119905119890ℎ = 119888119905119890ℎ119895 119895=12119870

= 119888119905119890ℎ1 119888119905119890ℎ2 119888119905119890ℎ31 119888119905119890ℎ32 119888119905119890ℎ3120574 119895=2+120574 (45)

where



119888119905119890ℎ1 = 1119873119867sumℎ=1

119886ℎ1 (46)


119888119905119890ℎ2 = (4119890)21198902 = 16 (47)


119888119905119890ℎ3120574 = 1198861198943120574 (48)


119862 = 119862119895119895=12119870 = 1198621 1198622 11986231 11986232 1198623120574119895=2+120574 (49)

where










forall119894 isin [1119873](53)



119885 = max⏟⏟⏟⏟⏟⏟⏟forall119894isin[1119873]

119880119877119862119894= max⏟⏟⏟⏟⏟⏟⏟forall119894isin[1119873]


(54)





for 120585+1 119861119872119861 isin 119879119872119862119892119903119900119908120572

for 120585+2 119861119872119861 isin 119879119872119862119892119903119900119908120573

for 120585+3 119887ℎ isin 119879119872119862119892119903119900119908120572






subject to

119873sum119894=1

119887ℎ119894 = 1 forallℎ isin [1119867] (55)


(56)


where (⋃119897119894lt119867ℎ=1














119865 = (119880119883 119862) = Tsum120591=1

119872sum119898=1

119906120596120591119898 (119909120591 minus 119888119898)2 (58)

subject to

119872sum119898=1

119906120591119898 = 1 120591 = 1 2 T (59)

0 le 119906120591119898 le 1 119898 = 1 2 119872 120591 = 1 2 T (60)

0 lt Tsum120591=1

119906119894119898 lt 119898 119898 = 1 2 119872 (61)

where










119875 (119884)forall120591isin[1T]

=


(62)


119898 = 1 2 119872 (63)


119889119911119898 = 120572 (ln 119911119898 minus ln 119911119898) 119911119898119889120591 + 120590119911119898119889119882 (64)

where






(65)

where



119911119898(120591+1) = 1205730 + 1205731119911119898(120591) + 120576 (66)




3

44

555

4

5

67

6

5

444

3333333333

444444

33

4

3

22

333

4

33

4444

333333333

4

5

67

6

0

1

2

3

4

5

6

7

0 5 10 15 20 25 30 35 40 45 50 55 60

Rang

e co

de v

alue

Month



119911119898 (120591)forall120591isin[1T]

=


119898 = 2 3 119872 minus 1

(67)




(68)



18

95

137125

71

3321

020406080

100120140160





119889119888120591 = 120583119888120591119889120591 + 120590119889119882120591 (69)




|119862|119904119894119898119878timesT = 10038161003816100381610038161198881199041205911003816100381610038161003816119878timesT (71)






= 119898119888119900119899 ∙ 119898119898119903 ∙ 1003816100381610038161003816119901119904120591119898 1003816100381610038161003816119878timesT (72)



10038161003816100381610038161003816Vℎ10038161003816100381610038161003816119904119894119898119878timesT = 119886ℎ1 (Ysum120574=1


(73)



=1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(119886 119887 119888)ℎ11 (119886 119887 119888)ℎ12 (119886 119887 119888)ℎ1120591(119886 119887 119888)ℎ21 (119886 119887 119888)ℎ22 (119886 119887 119888)ℎ2120591 d

(119886 119887 119888)ℎ1199041 (119886 119887 119888)ℎ1199042 (119886 119887 119888)ℎ119878T

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816


(74)


forall120591isin[1T] =10038161003816100381610038161003816100381610038161003816100381610038161119878119878sum119904=1


(75)


119864 (119864 (Vℎ120591)119905)119905=[1119879] =100381610038161003816100381610038161003816100381610038161003816100381611212sum120591=1

119864 (Vℎ120591)1199051003816100381610038161003816100381610038161003816100381610038161003816119905=[1119879]



1003816100381610038161003816100381610038161003816(Vℎ119905 )1198891003816100381610038161003816100381610038161003816 =10038161003816100381610038161003816100381610038161003816100381610038161003816


10038161003816100381610038161003816100381610038161003816100381610038161003816 forallℎ isin [1119867] 119905 = [1 119879]

(77)


V119889119894119905 = 119897119894sumℎ=1


(78)


3 Numerical Example






Zinc















1 5301 5891 6480 59 6458 7176 78932 6844 7604 8365 60 6954 7727 84993 5081 5646 6210 61 6293 6992 76914 7064 7849 8634 62 7119 7910 87015 7615 8461 9307 63 6128 6809 74896 6954 7727 8499 64 7284 8094 89037 7725 8583 9442 65 5962 6625 72878 5715 6350 6984 66 8977 9974 109729 6747 7497 8247 67 6128 6809 748910 5508 6120 6732 68 7119 7910 870111 7367 8186 9004 69 6293 6992 769112 6871 7635 8398 70 6954 7727 849913 7491 8323 9156 71 6458 7176 789314 6747 7497 8247 72 6789 7543 829715 7615 8461 9307 73 6623 7359 809516 6623 7359 8095 74 6623 7359 809517 7615 8461 9307 75 6789 7543 829718 5907 6564 7220 76 6458 7176 789319 6871 7635 8398 77 6954 7727 849920 5715 6350 6984 78 6293 6992 769121 7064 7849 8634 79 7119 7910 870122 5522 6135 6749 80 6128 6809 748923 6816 7574 8331 81 7284 8094 890324 6678 7421 8163 82 9130 10144 1115825 7367 8186 9004 83 6334 7038 774226 6541 7268 7994 84 7243 8048 885327 7505 8339 9172 85 6486 7206 792728 6458 7176 7893 86 7092 7880 866729 6637 7375 8112 87 6637 7375 811230 6279 6977 7674 88 6940 7711 848231 6816 7574 8331 89 6789 7543 829732 6100 6778 7456 90 6789 7543 829733 6995 7772 8550 91 6940 7711 848234 5921 6579 7237 92 6637 7375 811235 7174 7971 8768 93 7092 7880 866736 6486 7206 7927 94 6486 7206 792737 7092 7880 8667 95 7243 8048 885338 6637 7375 8112 96 8895 9883 1087239 6940 7711 8482 97 5921 6579 723740 6789 7543 8297 98 6995 7772 855041 6789 7543 8297 99 6100 6778 745642 6940 7711 8482 100 6816 7574 833143 6637 7375 8112 101 6279 6977 767444 7092 7880 8667 102 6637 7375 811245 6486 7206 7927 103 6458 7176 789346 7243 8048 8853 104 6458 7176 789347 6334 7038 7742 105 6637 7375 811248 7394 8216 9038 106 6279 6977 767449 6183 6870 7557 107 6816 7574 833150 6128 6809 7489 108 6100 6778 745651 7119 7910 8701 109 6678 7421 816352 6293 6992 7691 110 7229 8033 8836


Table 3 Continued





1152640

3913 -165169601186561

2153217

3072 -128170241187266

3154940

2939 -017172156189371

4155573

6278 +024172859190145

5159649

4816 +286177388195127


(15-19](19-23](23-27](27-31](31-35]

(151-276](276-401](401-526](526-651](651-776]



17

3 16 22 49 65

2 7 15 21 27 35 48 64 81

1 6 10 14 20 26 34 47 63 80 TMC1

5 9 13 19 25 33 46 62 79 95 TMC2

4 8 12 18 24 32 45 61 78 94 108 TMC3

11 31 44 60 77 93 107 TMC4

23 30 43 59 76 92 106 115 TMC5

29 42 58 75 91 105 114

28 41 57 74 90 104 113

40 56 73 89 103 112

39 55 72 88 102

38 54 71 87 101 111

37 53 70 86 100 110

36 52 69 85 99 109

51 68 84 98

50 67 83 97

66 82 96




1 2263 3364 0325 1198 14392 356072 1801 3832 0292 0418 16222 109313 2170 3937 0381 0820 17565 208234 1950 2573 0169 0928 8710 360795 2136 4636 0267 1262 12531 27226





(165 + 128 + 017 + 024 + 286)5 = 124 (79)










Lead ($t) 1176 1689 1863 2064 2205 2381 2665Zinc ($t) 1236 1816 2014 2176 2344 2695 3244

173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80

5 9 13 19 25 33 46 62 79 954 8 12 18 24 32 45 61 78 94 108

11 31 44 60 77 93 10723 30 43 59 76 92 106 115

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96

(a)

173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80

5 9 13 19 25 33 46 62 79 954 8 12 18 24 32 45 61 78 94 108

11 31 44 60 77 93 10723 30 43 59 76 92 106 115

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96

(b)

173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80

5 9 13 19 25 33 46 62 79 954 8 12 18 24 32 45 61 78 94 108

11 31 44 60 77 93 10723 30 43 59 76 92 106 115

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96

(c)

173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80

5 9 13 19 25 33 46 62 79 954 8 12 18 24 32 45 61 78 94 108

11 31 44 60 77 93 10723 30 43 59 76 92 106 115

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96

(d)



173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80 TMC1

5 9 13 19 25 33 46 62 79 95 TMC2

4 8 12 18 24 32 45 61 78 94 108 TMC3

11 31 44 60 77 93 107 TMC4

23 30 43 59 76 92 106 115 TMC5

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96





Process Value




volatility 018491




volatility 016416





0

01

02

03

04

05

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC3 StdDev Pb

0

02

04

06

08

1

12

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC3 StdDev Zn

0

01

02

03

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC4 StdDev Pb

0

02

04

06

08

1

12

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC4 StedDev Zn

0

01

02

03

04

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

Number of blocks

TMC5 StdDev Pb

002040608

112141618

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

Number of blocks

TMC5 StdDev Zn

0

01

02

03

04

05

06

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC1 StdDev Pb

002040608

112141618

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC1 StdDev Zn

0

01

02

03

04

05

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Number of blocks

TMC2 StdDev Pb

0

02

04

06

08

1

12

14

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Number of blocks

TMC2 StdDev Zn










0

500

1000

1500

2000

2500

3000

3500

4000

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125

US$

t

Month

(a)

0

500

1000

1500

2000

2500

3000

3500

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125

US$

t

Month

(b)


0010203040506070809

1

500 1000 1500 2000 2500 3000 3500$t

VVLVLLM

HVHVVH

(a)

0010203040506070809

1

500 1000 1500 2000 2500 3000 3500 4000$t

VVLVLLM

HVHVVH

(b)




15104734 3944737 3044364 2443462 19244336591226 5356427 4305812 3576349 29263948228167 6904516 5704581 4871250 4096627

25583555 4336349 3366286 2691067 21178617190095 5863841 4730052 3916424 32015418951887 7536746 6246798 5320340 4472001

36213929 4891545 3857917 3139633 25243907923489 6519182 5317593 4456306 36948059796030 8298845 6935676 5958908 5056129

42976598 2108873 1450337 1038819 6860214063918 3133980 2351322 1834430 13747555275030 4267887 3358689 2754948 2200817

56198802 4899506 3882193 3166488 25554757882521 6503582 5321728 4466255 37120859723915 8256167 6917828 5949400 5056656



0

500

1000

1500

2000

2500

3000

3500

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Lead price


(a)

0

1

2

3

4

5

6

7

8

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Lead price

Range code

(b)

0

500

1000

1500

2000

2500

3000

3500

4000

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Zinc price


(c)

0

1

2

3

4

5

6

7

8

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Zinc price

Range code

(d)




119865 minus 119868 = (17561955 24060165 31322673)minus (12000000 13000000 15000000)

= (2561955 11060166 19322673) $(80)


4 Conclusion





LEAD ZINC

UNIT OPERATING COST


01234567

0 5 10 15 20 25 30 35 40 45 50 55 60

Rang

e cod

e val

ue

Month

01234567

0 5 10 15 20 25 30 35 40 45 50 55 60

Rang

e cod

e val

ue

Month

0500

100015002000250030003500

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

0500

1000150020002500300035004000

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

4648505254565860

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

0100000200000300000400000500000600000700000800000

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

ℎ

sim

Stimes４= aℎ

1

(９sum

=1

aℎ2 M

mcon

con

sim








0

100000

200000

300000

400000

500000

600000

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month


0

100000

200000

300000

400000

500000

600000

1 2 3 4 5

$t

Year




Data Availability




0

100000

200000

300000

400000

500000

600000

1 2 3 4 5

$t

Year


TMC 1year 4

TMC 4year 5

TMC 5year 3

TMC 3year 1

TMC 2year 2


References
























































Journal of












Journal of







Volume 2018






Volume 2018










119862119905119890ℎ = 119888119905119890ℎ119895 119895=12119870

= 119888119905119890ℎ1 119888119905119890ℎ2 119888119905119890ℎ31 119888119905119890ℎ32 119888119905119890ℎ3120574 119895=2+120574 (45)

where



119888119905119890ℎ1 = 1119873119867sumℎ=1

119886ℎ1 (46)


119888119905119890ℎ2 = (4119890)21198902 = 16 (47)


119888119905119890ℎ3120574 = 1198861198943120574 (48)


119862 = 119862119895119895=12119870 = 1198621 1198622 11986231 11986232 1198623120574119895=2+120574 (49)

where










forall119894 isin [1119873](53)



119885 = max⏟⏟⏟⏟⏟⏟⏟forall119894isin[1119873]

119880119877119862119894= max⏟⏟⏟⏟⏟⏟⏟forall119894isin[1119873]


(54)





for 120585+1 119861119872119861 isin 119879119872119862119892119903119900119908120572

for 120585+2 119861119872119861 isin 119879119872119862119892119903119900119908120573

for 120585+3 119887ℎ isin 119879119872119862119892119903119900119908120572






subject to

119873sum119894=1

119887ℎ119894 = 1 forallℎ isin [1119867] (55)


(56)


where (⋃119897119894lt119867ℎ=1














119865 = (119880119883 119862) = Tsum120591=1

119872sum119898=1

119906120596120591119898 (119909120591 minus 119888119898)2 (58)

subject to

119872sum119898=1

119906120591119898 = 1 120591 = 1 2 T (59)

0 le 119906120591119898 le 1 119898 = 1 2 119872 120591 = 1 2 T (60)

0 lt Tsum120591=1

119906119894119898 lt 119898 119898 = 1 2 119872 (61)

where










119875 (119884)forall120591isin[1T]

=


(62)


119898 = 1 2 119872 (63)


119889119911119898 = 120572 (ln 119911119898 minus ln 119911119898) 119911119898119889120591 + 120590119911119898119889119882 (64)

where






(65)

where



119911119898(120591+1) = 1205730 + 1205731119911119898(120591) + 120576 (66)




3

44

555

4

5

67

6

5

444

3333333333

444444

33

4

3

22

333

4

33

4444

333333333

4

5

67

6

0

1

2

3

4

5

6

7

0 5 10 15 20 25 30 35 40 45 50 55 60

Rang

e co

de v

alue

Month



119911119898 (120591)forall120591isin[1T]

=


119898 = 2 3 119872 minus 1

(67)




(68)



18

95

137125

71

3321

020406080

100120140160





119889119888120591 = 120583119888120591119889120591 + 120590119889119882120591 (69)




|119862|119904119894119898119878timesT = 10038161003816100381610038161198881199041205911003816100381610038161003816119878timesT (71)






= 119898119888119900119899 ∙ 119898119898119903 ∙ 1003816100381610038161003816119901119904120591119898 1003816100381610038161003816119878timesT (72)



10038161003816100381610038161003816Vℎ10038161003816100381610038161003816119904119894119898119878timesT = 119886ℎ1 (Ysum120574=1


(73)



=1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(119886 119887 119888)ℎ11 (119886 119887 119888)ℎ12 (119886 119887 119888)ℎ1120591(119886 119887 119888)ℎ21 (119886 119887 119888)ℎ22 (119886 119887 119888)ℎ2120591 d

(119886 119887 119888)ℎ1199041 (119886 119887 119888)ℎ1199042 (119886 119887 119888)ℎ119878T

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816


(74)


forall120591isin[1T] =10038161003816100381610038161003816100381610038161003816100381610038161119878119878sum119904=1


(75)


119864 (119864 (Vℎ120591)119905)119905=[1119879] =100381610038161003816100381610038161003816100381610038161003816100381611212sum120591=1

119864 (Vℎ120591)1199051003816100381610038161003816100381610038161003816100381610038161003816119905=[1119879]



1003816100381610038161003816100381610038161003816(Vℎ119905 )1198891003816100381610038161003816100381610038161003816 =10038161003816100381610038161003816100381610038161003816100381610038161003816


10038161003816100381610038161003816100381610038161003816100381610038161003816 forallℎ isin [1119867] 119905 = [1 119879]

(77)


V119889119894119905 = 119897119894sumℎ=1


(78)


3 Numerical Example






Zinc















1 5301 5891 6480 59 6458 7176 78932 6844 7604 8365 60 6954 7727 84993 5081 5646 6210 61 6293 6992 76914 7064 7849 8634 62 7119 7910 87015 7615 8461 9307 63 6128 6809 74896 6954 7727 8499 64 7284 8094 89037 7725 8583 9442 65 5962 6625 72878 5715 6350 6984 66 8977 9974 109729 6747 7497 8247 67 6128 6809 748910 5508 6120 6732 68 7119 7910 870111 7367 8186 9004 69 6293 6992 769112 6871 7635 8398 70 6954 7727 849913 7491 8323 9156 71 6458 7176 789314 6747 7497 8247 72 6789 7543 829715 7615 8461 9307 73 6623 7359 809516 6623 7359 8095 74 6623 7359 809517 7615 8461 9307 75 6789 7543 829718 5907 6564 7220 76 6458 7176 789319 6871 7635 8398 77 6954 7727 849920 5715 6350 6984 78 6293 6992 769121 7064 7849 8634 79 7119 7910 870122 5522 6135 6749 80 6128 6809 748923 6816 7574 8331 81 7284 8094 890324 6678 7421 8163 82 9130 10144 1115825 7367 8186 9004 83 6334 7038 774226 6541 7268 7994 84 7243 8048 885327 7505 8339 9172 85 6486 7206 792728 6458 7176 7893 86 7092 7880 866729 6637 7375 8112 87 6637 7375 811230 6279 6977 7674 88 6940 7711 848231 6816 7574 8331 89 6789 7543 829732 6100 6778 7456 90 6789 7543 829733 6995 7772 8550 91 6940 7711 848234 5921 6579 7237 92 6637 7375 811235 7174 7971 8768 93 7092 7880 866736 6486 7206 7927 94 6486 7206 792737 7092 7880 8667 95 7243 8048 885338 6637 7375 8112 96 8895 9883 1087239 6940 7711 8482 97 5921 6579 723740 6789 7543 8297 98 6995 7772 855041 6789 7543 8297 99 6100 6778 745642 6940 7711 8482 100 6816 7574 833143 6637 7375 8112 101 6279 6977 767444 7092 7880 8667 102 6637 7375 811245 6486 7206 7927 103 6458 7176 789346 7243 8048 8853 104 6458 7176 789347 6334 7038 7742 105 6637 7375 811248 7394 8216 9038 106 6279 6977 767449 6183 6870 7557 107 6816 7574 833150 6128 6809 7489 108 6100 6778 745651 7119 7910 8701 109 6678 7421 816352 6293 6992 7691 110 7229 8033 8836


Table 3 Continued





1152640

3913 -165169601186561

2153217

3072 -128170241187266

3154940

2939 -017172156189371

4155573

6278 +024172859190145

5159649

4816 +286177388195127


(15-19](19-23](23-27](27-31](31-35]

(151-276](276-401](401-526](526-651](651-776]



17

3 16 22 49 65

2 7 15 21 27 35 48 64 81

1 6 10 14 20 26 34 47 63 80 TMC1

5 9 13 19 25 33 46 62 79 95 TMC2

4 8 12 18 24 32 45 61 78 94 108 TMC3

11 31 44 60 77 93 107 TMC4

23 30 43 59 76 92 106 115 TMC5

29 42 58 75 91 105 114

28 41 57 74 90 104 113

40 56 73 89 103 112

39 55 72 88 102

38 54 71 87 101 111

37 53 70 86 100 110

36 52 69 85 99 109

51 68 84 98

50 67 83 97

66 82 96




1 2263 3364 0325 1198 14392 356072 1801 3832 0292 0418 16222 109313 2170 3937 0381 0820 17565 208234 1950 2573 0169 0928 8710 360795 2136 4636 0267 1262 12531 27226





(165 + 128 + 017 + 024 + 286)5 = 124 (79)










Lead ($t) 1176 1689 1863 2064 2205 2381 2665Zinc ($t) 1236 1816 2014 2176 2344 2695 3244

173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80

5 9 13 19 25 33 46 62 79 954 8 12 18 24 32 45 61 78 94 108

11 31 44 60 77 93 10723 30 43 59 76 92 106 115

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96

(a)

173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80

5 9 13 19 25 33 46 62 79 954 8 12 18 24 32 45 61 78 94 108

11 31 44 60 77 93 10723 30 43 59 76 92 106 115

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96

(b)

173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80

5 9 13 19 25 33 46 62 79 954 8 12 18 24 32 45 61 78 94 108

11 31 44 60 77 93 10723 30 43 59 76 92 106 115

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96

(c)

173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80

5 9 13 19 25 33 46 62 79 954 8 12 18 24 32 45 61 78 94 108

11 31 44 60 77 93 10723 30 43 59 76 92 106 115

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96

(d)



173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80 TMC1

5 9 13 19 25 33 46 62 79 95 TMC2

4 8 12 18 24 32 45 61 78 94 108 TMC3

11 31 44 60 77 93 107 TMC4

23 30 43 59 76 92 106 115 TMC5

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96





Process Value




volatility 018491




volatility 016416





0

01

02

03

04

05

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC3 StdDev Pb

0

02

04

06

08

1

12

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC3 StdDev Zn

0

01

02

03

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC4 StdDev Pb

0

02

04

06

08

1

12

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC4 StedDev Zn

0

01

02

03

04

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

Number of blocks

TMC5 StdDev Pb

002040608

112141618

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

Number of blocks

TMC5 StdDev Zn

0

01

02

03

04

05

06

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC1 StdDev Pb

002040608

112141618

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC1 StdDev Zn

0

01

02

03

04

05

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Number of blocks

TMC2 StdDev Pb

0

02

04

06

08

1

12

14

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Number of blocks

TMC2 StdDev Zn










0

500

1000

1500

2000

2500

3000

3500

4000

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125

US$

t

Month

(a)

0

500

1000

1500

2000

2500

3000

3500

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125

US$

t

Month

(b)


0010203040506070809

1

500 1000 1500 2000 2500 3000 3500$t

VVLVLLM

HVHVVH

(a)

0010203040506070809

1

500 1000 1500 2000 2500 3000 3500 4000$t

VVLVLLM

HVHVVH

(b)




15104734 3944737 3044364 2443462 19244336591226 5356427 4305812 3576349 29263948228167 6904516 5704581 4871250 4096627

25583555 4336349 3366286 2691067 21178617190095 5863841 4730052 3916424 32015418951887 7536746 6246798 5320340 4472001

36213929 4891545 3857917 3139633 25243907923489 6519182 5317593 4456306 36948059796030 8298845 6935676 5958908 5056129

42976598 2108873 1450337 1038819 6860214063918 3133980 2351322 1834430 13747555275030 4267887 3358689 2754948 2200817

56198802 4899506 3882193 3166488 25554757882521 6503582 5321728 4466255 37120859723915 8256167 6917828 5949400 5056656



0

500

1000

1500

2000

2500

3000

3500

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Lead price


(a)

0

1

2

3

4

5

6

7

8

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Lead price

Range code

(b)

0

500

1000

1500

2000

2500

3000

3500

4000

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Zinc price


(c)

0

1

2

3

4

5

6

7

8

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Zinc price

Range code

(d)




119865 minus 119868 = (17561955 24060165 31322673)minus (12000000 13000000 15000000)

= (2561955 11060166 19322673) $(80)


4 Conclusion





LEAD ZINC

UNIT OPERATING COST


01234567

0 5 10 15 20 25 30 35 40 45 50 55 60

Rang

e cod

e val

ue

Month

01234567

0 5 10 15 20 25 30 35 40 45 50 55 60

Rang

e cod

e val

ue

Month

0500

100015002000250030003500

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

0500

1000150020002500300035004000

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

4648505254565860

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

0100000200000300000400000500000600000700000800000

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

ℎ

sim

Stimes４= aℎ

1

(９sum

=1

aℎ2 M

mcon

con

sim








0

100000

200000

300000

400000

500000

600000

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month


0

100000

200000

300000

400000

500000

600000

1 2 3 4 5

$t

Year




Data Availability




0

100000

200000

300000

400000

500000

600000

1 2 3 4 5

$t

Year


TMC 1year 4

TMC 4year 5

TMC 5year 3

TMC 3year 1

TMC 2year 2


References
























































Journal of












Journal of







Volume 2018






Volume 2018












for 120585+1 119861119872119861 isin 119879119872119862119892119903119900119908120572

for 120585+2 119861119872119861 isin 119879119872119862119892119903119900119908120573

for 120585+3 119887ℎ isin 119879119872119862119892119903119900119908120572






subject to

119873sum119894=1

119887ℎ119894 = 1 forallℎ isin [1119867] (55)


(56)


where (⋃119897119894lt119867ℎ=1














119865 = (119880119883 119862) = Tsum120591=1

119872sum119898=1

119906120596120591119898 (119909120591 minus 119888119898)2 (58)

subject to

119872sum119898=1

119906120591119898 = 1 120591 = 1 2 T (59)

0 le 119906120591119898 le 1 119898 = 1 2 119872 120591 = 1 2 T (60)

0 lt Tsum120591=1

119906119894119898 lt 119898 119898 = 1 2 119872 (61)

where










119875 (119884)forall120591isin[1T]

=


(62)


119898 = 1 2 119872 (63)


119889119911119898 = 120572 (ln 119911119898 minus ln 119911119898) 119911119898119889120591 + 120590119911119898119889119882 (64)

where






(65)

where



119911119898(120591+1) = 1205730 + 1205731119911119898(120591) + 120576 (66)




3

44

555

4

5

67

6

5

444

3333333333

444444

33

4

3

22

333

4

33

4444

333333333

4

5

67

6

0

1

2

3

4

5

6

7

0 5 10 15 20 25 30 35 40 45 50 55 60

Rang

e co

de v

alue

Month



119911119898 (120591)forall120591isin[1T]

=


119898 = 2 3 119872 minus 1

(67)




(68)



18

95

137125

71

3321

020406080

100120140160





119889119888120591 = 120583119888120591119889120591 + 120590119889119882120591 (69)




|119862|119904119894119898119878timesT = 10038161003816100381610038161198881199041205911003816100381610038161003816119878timesT (71)






= 119898119888119900119899 ∙ 119898119898119903 ∙ 1003816100381610038161003816119901119904120591119898 1003816100381610038161003816119878timesT (72)



10038161003816100381610038161003816Vℎ10038161003816100381610038161003816119904119894119898119878timesT = 119886ℎ1 (Ysum120574=1


(73)



=1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(119886 119887 119888)ℎ11 (119886 119887 119888)ℎ12 (119886 119887 119888)ℎ1120591(119886 119887 119888)ℎ21 (119886 119887 119888)ℎ22 (119886 119887 119888)ℎ2120591 d

(119886 119887 119888)ℎ1199041 (119886 119887 119888)ℎ1199042 (119886 119887 119888)ℎ119878T

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816


(74)


forall120591isin[1T] =10038161003816100381610038161003816100381610038161003816100381610038161119878119878sum119904=1


(75)


119864 (119864 (Vℎ120591)119905)119905=[1119879] =100381610038161003816100381610038161003816100381610038161003816100381611212sum120591=1

119864 (Vℎ120591)1199051003816100381610038161003816100381610038161003816100381610038161003816119905=[1119879]



1003816100381610038161003816100381610038161003816(Vℎ119905 )1198891003816100381610038161003816100381610038161003816 =10038161003816100381610038161003816100381610038161003816100381610038161003816


10038161003816100381610038161003816100381610038161003816100381610038161003816 forallℎ isin [1119867] 119905 = [1 119879]

(77)


V119889119894119905 = 119897119894sumℎ=1


(78)


3 Numerical Example






Zinc















1 5301 5891 6480 59 6458 7176 78932 6844 7604 8365 60 6954 7727 84993 5081 5646 6210 61 6293 6992 76914 7064 7849 8634 62 7119 7910 87015 7615 8461 9307 63 6128 6809 74896 6954 7727 8499 64 7284 8094 89037 7725 8583 9442 65 5962 6625 72878 5715 6350 6984 66 8977 9974 109729 6747 7497 8247 67 6128 6809 748910 5508 6120 6732 68 7119 7910 870111 7367 8186 9004 69 6293 6992 769112 6871 7635 8398 70 6954 7727 849913 7491 8323 9156 71 6458 7176 789314 6747 7497 8247 72 6789 7543 829715 7615 8461 9307 73 6623 7359 809516 6623 7359 8095 74 6623 7359 809517 7615 8461 9307 75 6789 7543 829718 5907 6564 7220 76 6458 7176 789319 6871 7635 8398 77 6954 7727 849920 5715 6350 6984 78 6293 6992 769121 7064 7849 8634 79 7119 7910 870122 5522 6135 6749 80 6128 6809 748923 6816 7574 8331 81 7284 8094 890324 6678 7421 8163 82 9130 10144 1115825 7367 8186 9004 83 6334 7038 774226 6541 7268 7994 84 7243 8048 885327 7505 8339 9172 85 6486 7206 792728 6458 7176 7893 86 7092 7880 866729 6637 7375 8112 87 6637 7375 811230 6279 6977 7674 88 6940 7711 848231 6816 7574 8331 89 6789 7543 829732 6100 6778 7456 90 6789 7543 829733 6995 7772 8550 91 6940 7711 848234 5921 6579 7237 92 6637 7375 811235 7174 7971 8768 93 7092 7880 866736 6486 7206 7927 94 6486 7206 792737 7092 7880 8667 95 7243 8048 885338 6637 7375 8112 96 8895 9883 1087239 6940 7711 8482 97 5921 6579 723740 6789 7543 8297 98 6995 7772 855041 6789 7543 8297 99 6100 6778 745642 6940 7711 8482 100 6816 7574 833143 6637 7375 8112 101 6279 6977 767444 7092 7880 8667 102 6637 7375 811245 6486 7206 7927 103 6458 7176 789346 7243 8048 8853 104 6458 7176 789347 6334 7038 7742 105 6637 7375 811248 7394 8216 9038 106 6279 6977 767449 6183 6870 7557 107 6816 7574 833150 6128 6809 7489 108 6100 6778 745651 7119 7910 8701 109 6678 7421 816352 6293 6992 7691 110 7229 8033 8836


Table 3 Continued





1152640

3913 -165169601186561

2153217

3072 -128170241187266

3154940

2939 -017172156189371

4155573

6278 +024172859190145

5159649

4816 +286177388195127


(15-19](19-23](23-27](27-31](31-35]

(151-276](276-401](401-526](526-651](651-776]



17

3 16 22 49 65

2 7 15 21 27 35 48 64 81

1 6 10 14 20 26 34 47 63 80 TMC1

5 9 13 19 25 33 46 62 79 95 TMC2

4 8 12 18 24 32 45 61 78 94 108 TMC3

11 31 44 60 77 93 107 TMC4

23 30 43 59 76 92 106 115 TMC5

29 42 58 75 91 105 114

28 41 57 74 90 104 113

40 56 73 89 103 112

39 55 72 88 102

38 54 71 87 101 111

37 53 70 86 100 110

36 52 69 85 99 109

51 68 84 98

50 67 83 97

66 82 96




1 2263 3364 0325 1198 14392 356072 1801 3832 0292 0418 16222 109313 2170 3937 0381 0820 17565 208234 1950 2573 0169 0928 8710 360795 2136 4636 0267 1262 12531 27226





(165 + 128 + 017 + 024 + 286)5 = 124 (79)










Lead ($t) 1176 1689 1863 2064 2205 2381 2665Zinc ($t) 1236 1816 2014 2176 2344 2695 3244

173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80

5 9 13 19 25 33 46 62 79 954 8 12 18 24 32 45 61 78 94 108

11 31 44 60 77 93 10723 30 43 59 76 92 106 115

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96

(a)

173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80

5 9 13 19 25 33 46 62 79 954 8 12 18 24 32 45 61 78 94 108

11 31 44 60 77 93 10723 30 43 59 76 92 106 115

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96

(b)

173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80

5 9 13 19 25 33 46 62 79 954 8 12 18 24 32 45 61 78 94 108

11 31 44 60 77 93 10723 30 43 59 76 92 106 115

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96

(c)

173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80

5 9 13 19 25 33 46 62 79 954 8 12 18 24 32 45 61 78 94 108

11 31 44 60 77 93 10723 30 43 59 76 92 106 115

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96

(d)



173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80 TMC1

5 9 13 19 25 33 46 62 79 95 TMC2

4 8 12 18 24 32 45 61 78 94 108 TMC3

11 31 44 60 77 93 107 TMC4

23 30 43 59 76 92 106 115 TMC5

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96





Process Value




volatility 018491




volatility 016416





0

01

02

03

04

05

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC3 StdDev Pb

0

02

04

06

08

1

12

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC3 StdDev Zn

0

01

02

03

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC4 StdDev Pb

0

02

04

06

08

1

12

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC4 StedDev Zn

0

01

02

03

04

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

Number of blocks

TMC5 StdDev Pb

002040608

112141618

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

Number of blocks

TMC5 StdDev Zn

0

01

02

03

04

05

06

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC1 StdDev Pb

002040608

112141618

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC1 StdDev Zn

0

01

02

03

04

05

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Number of blocks

TMC2 StdDev Pb

0

02

04

06

08

1

12

14

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Number of blocks

TMC2 StdDev Zn










0

500

1000

1500

2000

2500

3000

3500

4000

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125

US$

t

Month

(a)

0

500

1000

1500

2000

2500

3000

3500

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125

US$

t

Month

(b)


0010203040506070809

1

500 1000 1500 2000 2500 3000 3500$t

VVLVLLM

HVHVVH

(a)

0010203040506070809

1

500 1000 1500 2000 2500 3000 3500 4000$t

VVLVLLM

HVHVVH

(b)




15104734 3944737 3044364 2443462 19244336591226 5356427 4305812 3576349 29263948228167 6904516 5704581 4871250 4096627

25583555 4336349 3366286 2691067 21178617190095 5863841 4730052 3916424 32015418951887 7536746 6246798 5320340 4472001

36213929 4891545 3857917 3139633 25243907923489 6519182 5317593 4456306 36948059796030 8298845 6935676 5958908 5056129

42976598 2108873 1450337 1038819 6860214063918 3133980 2351322 1834430 13747555275030 4267887 3358689 2754948 2200817

56198802 4899506 3882193 3166488 25554757882521 6503582 5321728 4466255 37120859723915 8256167 6917828 5949400 5056656



0

500

1000

1500

2000

2500

3000

3500

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Lead price


(a)

0

1

2

3

4

5

6

7

8

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Lead price

Range code

(b)

0

500

1000

1500

2000

2500

3000

3500

4000

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Zinc price


(c)

0

1

2

3

4

5

6

7

8

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Zinc price

Range code

(d)




119865 minus 119868 = (17561955 24060165 31322673)minus (12000000 13000000 15000000)

= (2561955 11060166 19322673) $(80)


4 Conclusion





LEAD ZINC

UNIT OPERATING COST


01234567

0 5 10 15 20 25 30 35 40 45 50 55 60

Rang

e cod

e val

ue

Month

01234567

0 5 10 15 20 25 30 35 40 45 50 55 60

Rang

e cod

e val

ue

Month

0500

100015002000250030003500

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

0500

1000150020002500300035004000

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

4648505254565860

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

0100000200000300000400000500000600000700000800000

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

ℎ

sim

Stimes４= aℎ

1

(９sum

=1

aℎ2 M

mcon

con

sim








0

100000

200000

300000

400000

500000

600000

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month


0

100000

200000

300000

400000

500000

600000

1 2 3 4 5

$t

Year




Data Availability




0

100000

200000

300000

400000

500000

600000

1 2 3 4 5

$t

Year


TMC 1year 4

TMC 4year 5

TMC 5year 3

TMC 3year 1

TMC 2year 2


References
























































Journal of












Journal of







Volume 2018






Volume 2018

















119865 = (119880119883 119862) = Tsum120591=1

119872sum119898=1

119906120596120591119898 (119909120591 minus 119888119898)2 (58)

subject to

119872sum119898=1

119906120591119898 = 1 120591 = 1 2 T (59)

0 le 119906120591119898 le 1 119898 = 1 2 119872 120591 = 1 2 T (60)

0 lt Tsum120591=1

119906119894119898 lt 119898 119898 = 1 2 119872 (61)

where










119875 (119884)forall120591isin[1T]

=


(62)


119898 = 1 2 119872 (63)


119889119911119898 = 120572 (ln 119911119898 minus ln 119911119898) 119911119898119889120591 + 120590119911119898119889119882 (64)

where






(65)

where



119911119898(120591+1) = 1205730 + 1205731119911119898(120591) + 120576 (66)




3

44

555

4

5

67

6

5

444

3333333333

444444

33

4

3

22

333

4

33

4444

333333333

4

5

67

6

0

1

2

3

4

5

6

7

0 5 10 15 20 25 30 35 40 45 50 55 60

Rang

e co

de v

alue

Month



119911119898 (120591)forall120591isin[1T]

=


119898 = 2 3 119872 minus 1

(67)




(68)



18

95

137125

71

3321

020406080

100120140160





119889119888120591 = 120583119888120591119889120591 + 120590119889119882120591 (69)




|119862|119904119894119898119878timesT = 10038161003816100381610038161198881199041205911003816100381610038161003816119878timesT (71)






= 119898119888119900119899 ∙ 119898119898119903 ∙ 1003816100381610038161003816119901119904120591119898 1003816100381610038161003816119878timesT (72)



10038161003816100381610038161003816Vℎ10038161003816100381610038161003816119904119894119898119878timesT = 119886ℎ1 (Ysum120574=1


(73)



=1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(119886 119887 119888)ℎ11 (119886 119887 119888)ℎ12 (119886 119887 119888)ℎ1120591(119886 119887 119888)ℎ21 (119886 119887 119888)ℎ22 (119886 119887 119888)ℎ2120591 d

(119886 119887 119888)ℎ1199041 (119886 119887 119888)ℎ1199042 (119886 119887 119888)ℎ119878T

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816


(74)


forall120591isin[1T] =10038161003816100381610038161003816100381610038161003816100381610038161119878119878sum119904=1


(75)


119864 (119864 (Vℎ120591)119905)119905=[1119879] =100381610038161003816100381610038161003816100381610038161003816100381611212sum120591=1

119864 (Vℎ120591)1199051003816100381610038161003816100381610038161003816100381610038161003816119905=[1119879]



1003816100381610038161003816100381610038161003816(Vℎ119905 )1198891003816100381610038161003816100381610038161003816 =10038161003816100381610038161003816100381610038161003816100381610038161003816


10038161003816100381610038161003816100381610038161003816100381610038161003816 forallℎ isin [1119867] 119905 = [1 119879]

(77)


V119889119894119905 = 119897119894sumℎ=1


(78)


3 Numerical Example






Zinc















1 5301 5891 6480 59 6458 7176 78932 6844 7604 8365 60 6954 7727 84993 5081 5646 6210 61 6293 6992 76914 7064 7849 8634 62 7119 7910 87015 7615 8461 9307 63 6128 6809 74896 6954 7727 8499 64 7284 8094 89037 7725 8583 9442 65 5962 6625 72878 5715 6350 6984 66 8977 9974 109729 6747 7497 8247 67 6128 6809 748910 5508 6120 6732 68 7119 7910 870111 7367 8186 9004 69 6293 6992 769112 6871 7635 8398 70 6954 7727 849913 7491 8323 9156 71 6458 7176 789314 6747 7497 8247 72 6789 7543 829715 7615 8461 9307 73 6623 7359 809516 6623 7359 8095 74 6623 7359 809517 7615 8461 9307 75 6789 7543 829718 5907 6564 7220 76 6458 7176 789319 6871 7635 8398 77 6954 7727 849920 5715 6350 6984 78 6293 6992 769121 7064 7849 8634 79 7119 7910 870122 5522 6135 6749 80 6128 6809 748923 6816 7574 8331 81 7284 8094 890324 6678 7421 8163 82 9130 10144 1115825 7367 8186 9004 83 6334 7038 774226 6541 7268 7994 84 7243 8048 885327 7505 8339 9172 85 6486 7206 792728 6458 7176 7893 86 7092 7880 866729 6637 7375 8112 87 6637 7375 811230 6279 6977 7674 88 6940 7711 848231 6816 7574 8331 89 6789 7543 829732 6100 6778 7456 90 6789 7543 829733 6995 7772 8550 91 6940 7711 848234 5921 6579 7237 92 6637 7375 811235 7174 7971 8768 93 7092 7880 866736 6486 7206 7927 94 6486 7206 792737 7092 7880 8667 95 7243 8048 885338 6637 7375 8112 96 8895 9883 1087239 6940 7711 8482 97 5921 6579 723740 6789 7543 8297 98 6995 7772 855041 6789 7543 8297 99 6100 6778 745642 6940 7711 8482 100 6816 7574 833143 6637 7375 8112 101 6279 6977 767444 7092 7880 8667 102 6637 7375 811245 6486 7206 7927 103 6458 7176 789346 7243 8048 8853 104 6458 7176 789347 6334 7038 7742 105 6637 7375 811248 7394 8216 9038 106 6279 6977 767449 6183 6870 7557 107 6816 7574 833150 6128 6809 7489 108 6100 6778 745651 7119 7910 8701 109 6678 7421 816352 6293 6992 7691 110 7229 8033 8836


Table 3 Continued





1152640

3913 -165169601186561

2153217

3072 -128170241187266

3154940

2939 -017172156189371

4155573

6278 +024172859190145

5159649

4816 +286177388195127


(15-19](19-23](23-27](27-31](31-35]

(151-276](276-401](401-526](526-651](651-776]



17

3 16 22 49 65

2 7 15 21 27 35 48 64 81

1 6 10 14 20 26 34 47 63 80 TMC1

5 9 13 19 25 33 46 62 79 95 TMC2

4 8 12 18 24 32 45 61 78 94 108 TMC3

11 31 44 60 77 93 107 TMC4

23 30 43 59 76 92 106 115 TMC5

29 42 58 75 91 105 114

28 41 57 74 90 104 113

40 56 73 89 103 112

39 55 72 88 102

38 54 71 87 101 111

37 53 70 86 100 110

36 52 69 85 99 109

51 68 84 98

50 67 83 97

66 82 96




1 2263 3364 0325 1198 14392 356072 1801 3832 0292 0418 16222 109313 2170 3937 0381 0820 17565 208234 1950 2573 0169 0928 8710 360795 2136 4636 0267 1262 12531 27226





(165 + 128 + 017 + 024 + 286)5 = 124 (79)










Lead ($t) 1176 1689 1863 2064 2205 2381 2665Zinc ($t) 1236 1816 2014 2176 2344 2695 3244

173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80

5 9 13 19 25 33 46 62 79 954 8 12 18 24 32 45 61 78 94 108

11 31 44 60 77 93 10723 30 43 59 76 92 106 115

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96

(a)

173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80

5 9 13 19 25 33 46 62 79 954 8 12 18 24 32 45 61 78 94 108

11 31 44 60 77 93 10723 30 43 59 76 92 106 115

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96

(b)

173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80

5 9 13 19 25 33 46 62 79 954 8 12 18 24 32 45 61 78 94 108

11 31 44 60 77 93 10723 30 43 59 76 92 106 115

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96

(c)

173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80

5 9 13 19 25 33 46 62 79 954 8 12 18 24 32 45 61 78 94 108

11 31 44 60 77 93 10723 30 43 59 76 92 106 115

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96

(d)



173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80 TMC1

5 9 13 19 25 33 46 62 79 95 TMC2

4 8 12 18 24 32 45 61 78 94 108 TMC3

11 31 44 60 77 93 107 TMC4

23 30 43 59 76 92 106 115 TMC5

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96





Process Value




volatility 018491




volatility 016416





0

01

02

03

04

05

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC3 StdDev Pb

0

02

04

06

08

1

12

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC3 StdDev Zn

0

01

02

03

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC4 StdDev Pb

0

02

04

06

08

1

12

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC4 StedDev Zn

0

01

02

03

04

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

Number of blocks

TMC5 StdDev Pb

002040608

112141618

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

Number of blocks

TMC5 StdDev Zn

0

01

02

03

04

05

06

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC1 StdDev Pb

002040608

112141618

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC1 StdDev Zn

0

01

02

03

04

05

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Number of blocks

TMC2 StdDev Pb

0

02

04

06

08

1

12

14

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Number of blocks

TMC2 StdDev Zn










0

500

1000

1500

2000

2500

3000

3500

4000

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125

US$

t

Month

(a)

0

500

1000

1500

2000

2500

3000

3500

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125

US$

t

Month

(b)


0010203040506070809

1

500 1000 1500 2000 2500 3000 3500$t

VVLVLLM

HVHVVH

(a)

0010203040506070809

1

500 1000 1500 2000 2500 3000 3500 4000$t

VVLVLLM

HVHVVH

(b)




15104734 3944737 3044364 2443462 19244336591226 5356427 4305812 3576349 29263948228167 6904516 5704581 4871250 4096627

25583555 4336349 3366286 2691067 21178617190095 5863841 4730052 3916424 32015418951887 7536746 6246798 5320340 4472001

36213929 4891545 3857917 3139633 25243907923489 6519182 5317593 4456306 36948059796030 8298845 6935676 5958908 5056129

42976598 2108873 1450337 1038819 6860214063918 3133980 2351322 1834430 13747555275030 4267887 3358689 2754948 2200817

56198802 4899506 3882193 3166488 25554757882521 6503582 5321728 4466255 37120859723915 8256167 6917828 5949400 5056656



0

500

1000

1500

2000

2500

3000

3500

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Lead price


(a)

0

1

2

3

4

5

6

7

8

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Lead price

Range code

(b)

0

500

1000

1500

2000

2500

3000

3500

4000

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Zinc price


(c)

0

1

2

3

4

5

6

7

8

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Zinc price

Range code

(d)




119865 minus 119868 = (17561955 24060165 31322673)minus (12000000 13000000 15000000)

= (2561955 11060166 19322673) $(80)


4 Conclusion





LEAD ZINC

UNIT OPERATING COST


01234567

0 5 10 15 20 25 30 35 40 45 50 55 60

Rang

e cod

e val

ue

Month

01234567

0 5 10 15 20 25 30 35 40 45 50 55 60

Rang

e cod

e val

ue

Month

0500

100015002000250030003500

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

0500

1000150020002500300035004000

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

4648505254565860

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

0100000200000300000400000500000600000700000800000

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

ℎ

sim

Stimes４= aℎ

1

(９sum

=1

aℎ2 M

mcon

con

sim








0

100000

200000

300000

400000

500000

600000

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month


0

100000

200000

300000

400000

500000

600000

1 2 3 4 5

$t

Year




Data Availability




0

100000

200000

300000

400000

500000

600000

1 2 3 4 5

$t

Year


TMC 1year 4

TMC 4year 5

TMC 5year 3

TMC 3year 1

TMC 2year 2


References
























































Journal of












Journal of







Volume 2018






Volume 2018













119875 (119884)forall120591isin[1T]

=


(62)


119898 = 1 2 119872 (63)


119889119911119898 = 120572 (ln 119911119898 minus ln 119911119898) 119911119898119889120591 + 120590119911119898119889119882 (64)

where






(65)

where



119911119898(120591+1) = 1205730 + 1205731119911119898(120591) + 120576 (66)




3

44

555

4

5

67

6

5

444

3333333333

444444

33

4

3

22

333

4

33

4444

333333333

4

5

67

6

0

1

2

3

4

5

6

7

0 5 10 15 20 25 30 35 40 45 50 55 60

Rang

e co

de v

alue

Month



119911119898 (120591)forall120591isin[1T]

=


119898 = 2 3 119872 minus 1

(67)




(68)



18

95

137125

71

3321

020406080

100120140160





119889119888120591 = 120583119888120591119889120591 + 120590119889119882120591 (69)




|119862|119904119894119898119878timesT = 10038161003816100381610038161198881199041205911003816100381610038161003816119878timesT (71)






= 119898119888119900119899 ∙ 119898119898119903 ∙ 1003816100381610038161003816119901119904120591119898 1003816100381610038161003816119878timesT (72)



10038161003816100381610038161003816Vℎ10038161003816100381610038161003816119904119894119898119878timesT = 119886ℎ1 (Ysum120574=1


(73)



=1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(119886 119887 119888)ℎ11 (119886 119887 119888)ℎ12 (119886 119887 119888)ℎ1120591(119886 119887 119888)ℎ21 (119886 119887 119888)ℎ22 (119886 119887 119888)ℎ2120591 d

(119886 119887 119888)ℎ1199041 (119886 119887 119888)ℎ1199042 (119886 119887 119888)ℎ119878T

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816


(74)


forall120591isin[1T] =10038161003816100381610038161003816100381610038161003816100381610038161119878119878sum119904=1


(75)


119864 (119864 (Vℎ120591)119905)119905=[1119879] =100381610038161003816100381610038161003816100381610038161003816100381611212sum120591=1

119864 (Vℎ120591)1199051003816100381610038161003816100381610038161003816100381610038161003816119905=[1119879]



1003816100381610038161003816100381610038161003816(Vℎ119905 )1198891003816100381610038161003816100381610038161003816 =10038161003816100381610038161003816100381610038161003816100381610038161003816


10038161003816100381610038161003816100381610038161003816100381610038161003816 forallℎ isin [1119867] 119905 = [1 119879]

(77)


V119889119894119905 = 119897119894sumℎ=1


(78)


3 Numerical Example






Zinc















1 5301 5891 6480 59 6458 7176 78932 6844 7604 8365 60 6954 7727 84993 5081 5646 6210 61 6293 6992 76914 7064 7849 8634 62 7119 7910 87015 7615 8461 9307 63 6128 6809 74896 6954 7727 8499 64 7284 8094 89037 7725 8583 9442 65 5962 6625 72878 5715 6350 6984 66 8977 9974 109729 6747 7497 8247 67 6128 6809 748910 5508 6120 6732 68 7119 7910 870111 7367 8186 9004 69 6293 6992 769112 6871 7635 8398 70 6954 7727 849913 7491 8323 9156 71 6458 7176 789314 6747 7497 8247 72 6789 7543 829715 7615 8461 9307 73 6623 7359 809516 6623 7359 8095 74 6623 7359 809517 7615 8461 9307 75 6789 7543 829718 5907 6564 7220 76 6458 7176 789319 6871 7635 8398 77 6954 7727 849920 5715 6350 6984 78 6293 6992 769121 7064 7849 8634 79 7119 7910 870122 5522 6135 6749 80 6128 6809 748923 6816 7574 8331 81 7284 8094 890324 6678 7421 8163 82 9130 10144 1115825 7367 8186 9004 83 6334 7038 774226 6541 7268 7994 84 7243 8048 885327 7505 8339 9172 85 6486 7206 792728 6458 7176 7893 86 7092 7880 866729 6637 7375 8112 87 6637 7375 811230 6279 6977 7674 88 6940 7711 848231 6816 7574 8331 89 6789 7543 829732 6100 6778 7456 90 6789 7543 829733 6995 7772 8550 91 6940 7711 848234 5921 6579 7237 92 6637 7375 811235 7174 7971 8768 93 7092 7880 866736 6486 7206 7927 94 6486 7206 792737 7092 7880 8667 95 7243 8048 885338 6637 7375 8112 96 8895 9883 1087239 6940 7711 8482 97 5921 6579 723740 6789 7543 8297 98 6995 7772 855041 6789 7543 8297 99 6100 6778 745642 6940 7711 8482 100 6816 7574 833143 6637 7375 8112 101 6279 6977 767444 7092 7880 8667 102 6637 7375 811245 6486 7206 7927 103 6458 7176 789346 7243 8048 8853 104 6458 7176 789347 6334 7038 7742 105 6637 7375 811248 7394 8216 9038 106 6279 6977 767449 6183 6870 7557 107 6816 7574 833150 6128 6809 7489 108 6100 6778 745651 7119 7910 8701 109 6678 7421 816352 6293 6992 7691 110 7229 8033 8836


Table 3 Continued





1152640

3913 -165169601186561

2153217

3072 -128170241187266

3154940

2939 -017172156189371

4155573

6278 +024172859190145

5159649

4816 +286177388195127


(15-19](19-23](23-27](27-31](31-35]

(151-276](276-401](401-526](526-651](651-776]



17

3 16 22 49 65

2 7 15 21 27 35 48 64 81

1 6 10 14 20 26 34 47 63 80 TMC1

5 9 13 19 25 33 46 62 79 95 TMC2

4 8 12 18 24 32 45 61 78 94 108 TMC3

11 31 44 60 77 93 107 TMC4

23 30 43 59 76 92 106 115 TMC5

29 42 58 75 91 105 114

28 41 57 74 90 104 113

40 56 73 89 103 112

39 55 72 88 102

38 54 71 87 101 111

37 53 70 86 100 110

36 52 69 85 99 109

51 68 84 98

50 67 83 97

66 82 96




1 2263 3364 0325 1198 14392 356072 1801 3832 0292 0418 16222 109313 2170 3937 0381 0820 17565 208234 1950 2573 0169 0928 8710 360795 2136 4636 0267 1262 12531 27226





(165 + 128 + 017 + 024 + 286)5 = 124 (79)










Lead ($t) 1176 1689 1863 2064 2205 2381 2665Zinc ($t) 1236 1816 2014 2176 2344 2695 3244

173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80

5 9 13 19 25 33 46 62 79 954 8 12 18 24 32 45 61 78 94 108

11 31 44 60 77 93 10723 30 43 59 76 92 106 115

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96

(a)

173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80

5 9 13 19 25 33 46 62 79 954 8 12 18 24 32 45 61 78 94 108

11 31 44 60 77 93 10723 30 43 59 76 92 106 115

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96

(b)

173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80

5 9 13 19 25 33 46 62 79 954 8 12 18 24 32 45 61 78 94 108

11 31 44 60 77 93 10723 30 43 59 76 92 106 115

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96

(c)

173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80

5 9 13 19 25 33 46 62 79 954 8 12 18 24 32 45 61 78 94 108

11 31 44 60 77 93 10723 30 43 59 76 92 106 115

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96

(d)



173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80 TMC1

5 9 13 19 25 33 46 62 79 95 TMC2

4 8 12 18 24 32 45 61 78 94 108 TMC3

11 31 44 60 77 93 107 TMC4

23 30 43 59 76 92 106 115 TMC5

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96





Process Value




volatility 018491




volatility 016416





0

01

02

03

04

05

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC3 StdDev Pb

0

02

04

06

08

1

12

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC3 StdDev Zn

0

01

02

03

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC4 StdDev Pb

0

02

04

06

08

1

12

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC4 StedDev Zn

0

01

02

03

04

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

Number of blocks

TMC5 StdDev Pb

002040608

112141618

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

Number of blocks

TMC5 StdDev Zn

0

01

02

03

04

05

06

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC1 StdDev Pb

002040608

112141618

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC1 StdDev Zn

0

01

02

03

04

05

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Number of blocks

TMC2 StdDev Pb

0

02

04

06

08

1

12

14

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Number of blocks

TMC2 StdDev Zn










0

500

1000

1500

2000

2500

3000

3500

4000

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125

US$

t

Month

(a)

0

500

1000

1500

2000

2500

3000

3500

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125

US$

t

Month

(b)


0010203040506070809

1

500 1000 1500 2000 2500 3000 3500$t

VVLVLLM

HVHVVH

(a)

0010203040506070809

1

500 1000 1500 2000 2500 3000 3500 4000$t

VVLVLLM

HVHVVH

(b)




15104734 3944737 3044364 2443462 19244336591226 5356427 4305812 3576349 29263948228167 6904516 5704581 4871250 4096627

25583555 4336349 3366286 2691067 21178617190095 5863841 4730052 3916424 32015418951887 7536746 6246798 5320340 4472001

36213929 4891545 3857917 3139633 25243907923489 6519182 5317593 4456306 36948059796030 8298845 6935676 5958908 5056129

42976598 2108873 1450337 1038819 6860214063918 3133980 2351322 1834430 13747555275030 4267887 3358689 2754948 2200817

56198802 4899506 3882193 3166488 25554757882521 6503582 5321728 4466255 37120859723915 8256167 6917828 5949400 5056656



0

500

1000

1500

2000

2500

3000

3500

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Lead price


(a)

0

1

2

3

4

5

6

7

8

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Lead price

Range code

(b)

0

500

1000

1500

2000

2500

3000

3500

4000

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Zinc price


(c)

0

1

2

3

4

5

6

7

8

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Zinc price

Range code

(d)




119865 minus 119868 = (17561955 24060165 31322673)minus (12000000 13000000 15000000)

= (2561955 11060166 19322673) $(80)


4 Conclusion





LEAD ZINC

UNIT OPERATING COST


01234567

0 5 10 15 20 25 30 35 40 45 50 55 60

Rang

e cod

e val

ue

Month

01234567

0 5 10 15 20 25 30 35 40 45 50 55 60

Rang

e cod

e val

ue

Month

0500

100015002000250030003500

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

0500

1000150020002500300035004000

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

4648505254565860

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

0100000200000300000400000500000600000700000800000

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

ℎ

sim

Stimes４= aℎ

1

(９sum

=1

aℎ2 M

mcon

con

sim








0

100000

200000

300000

400000

500000

600000

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month


0

100000

200000

300000

400000

500000

600000

1 2 3 4 5

$t

Year




Data Availability




0

100000

200000

300000

400000

500000

600000

1 2 3 4 5

$t

Year


TMC 1year 4

TMC 4year 5

TMC 5year 3

TMC 3year 1

TMC 2year 2


References
























































Journal of












Journal of







Volume 2018






Volume 2018









3

44

555

4

5

67

6

5

444

3333333333

444444

33

4

3

22

333

4

33

4444

333333333

4

5

67

6

0

1

2

3

4

5

6

7

0 5 10 15 20 25 30 35 40 45 50 55 60

Rang

e co

de v

alue

Month



119911119898 (120591)forall120591isin[1T]

=


119898 = 2 3 119872 minus 1

(67)




(68)



18

95

137125

71

3321

020406080

100120140160





119889119888120591 = 120583119888120591119889120591 + 120590119889119882120591 (69)




|119862|119904119894119898119878timesT = 10038161003816100381610038161198881199041205911003816100381610038161003816119878timesT (71)






= 119898119888119900119899 ∙ 119898119898119903 ∙ 1003816100381610038161003816119901119904120591119898 1003816100381610038161003816119878timesT (72)



10038161003816100381610038161003816Vℎ10038161003816100381610038161003816119904119894119898119878timesT = 119886ℎ1 (Ysum120574=1


(73)



=1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(119886 119887 119888)ℎ11 (119886 119887 119888)ℎ12 (119886 119887 119888)ℎ1120591(119886 119887 119888)ℎ21 (119886 119887 119888)ℎ22 (119886 119887 119888)ℎ2120591 d

(119886 119887 119888)ℎ1199041 (119886 119887 119888)ℎ1199042 (119886 119887 119888)ℎ119878T

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816


(74)


forall120591isin[1T] =10038161003816100381610038161003816100381610038161003816100381610038161119878119878sum119904=1


(75)


119864 (119864 (Vℎ120591)119905)119905=[1119879] =100381610038161003816100381610038161003816100381610038161003816100381611212sum120591=1

119864 (Vℎ120591)1199051003816100381610038161003816100381610038161003816100381610038161003816119905=[1119879]



1003816100381610038161003816100381610038161003816(Vℎ119905 )1198891003816100381610038161003816100381610038161003816 =10038161003816100381610038161003816100381610038161003816100381610038161003816


10038161003816100381610038161003816100381610038161003816100381610038161003816 forallℎ isin [1119867] 119905 = [1 119879]

(77)


V119889119894119905 = 119897119894sumℎ=1


(78)


3 Numerical Example






Zinc















1 5301 5891 6480 59 6458 7176 78932 6844 7604 8365 60 6954 7727 84993 5081 5646 6210 61 6293 6992 76914 7064 7849 8634 62 7119 7910 87015 7615 8461 9307 63 6128 6809 74896 6954 7727 8499 64 7284 8094 89037 7725 8583 9442 65 5962 6625 72878 5715 6350 6984 66 8977 9974 109729 6747 7497 8247 67 6128 6809 748910 5508 6120 6732 68 7119 7910 870111 7367 8186 9004 69 6293 6992 769112 6871 7635 8398 70 6954 7727 849913 7491 8323 9156 71 6458 7176 789314 6747 7497 8247 72 6789 7543 829715 7615 8461 9307 73 6623 7359 809516 6623 7359 8095 74 6623 7359 809517 7615 8461 9307 75 6789 7543 829718 5907 6564 7220 76 6458 7176 789319 6871 7635 8398 77 6954 7727 849920 5715 6350 6984 78 6293 6992 769121 7064 7849 8634 79 7119 7910 870122 5522 6135 6749 80 6128 6809 748923 6816 7574 8331 81 7284 8094 890324 6678 7421 8163 82 9130 10144 1115825 7367 8186 9004 83 6334 7038 774226 6541 7268 7994 84 7243 8048 885327 7505 8339 9172 85 6486 7206 792728 6458 7176 7893 86 7092 7880 866729 6637 7375 8112 87 6637 7375 811230 6279 6977 7674 88 6940 7711 848231 6816 7574 8331 89 6789 7543 829732 6100 6778 7456 90 6789 7543 829733 6995 7772 8550 91 6940 7711 848234 5921 6579 7237 92 6637 7375 811235 7174 7971 8768 93 7092 7880 866736 6486 7206 7927 94 6486 7206 792737 7092 7880 8667 95 7243 8048 885338 6637 7375 8112 96 8895 9883 1087239 6940 7711 8482 97 5921 6579 723740 6789 7543 8297 98 6995 7772 855041 6789 7543 8297 99 6100 6778 745642 6940 7711 8482 100 6816 7574 833143 6637 7375 8112 101 6279 6977 767444 7092 7880 8667 102 6637 7375 811245 6486 7206 7927 103 6458 7176 789346 7243 8048 8853 104 6458 7176 789347 6334 7038 7742 105 6637 7375 811248 7394 8216 9038 106 6279 6977 767449 6183 6870 7557 107 6816 7574 833150 6128 6809 7489 108 6100 6778 745651 7119 7910 8701 109 6678 7421 816352 6293 6992 7691 110 7229 8033 8836


Table 3 Continued





1152640

3913 -165169601186561

2153217

3072 -128170241187266

3154940

2939 -017172156189371

4155573

6278 +024172859190145

5159649

4816 +286177388195127


(15-19](19-23](23-27](27-31](31-35]

(151-276](276-401](401-526](526-651](651-776]



17

3 16 22 49 65

2 7 15 21 27 35 48 64 81

1 6 10 14 20 26 34 47 63 80 TMC1

5 9 13 19 25 33 46 62 79 95 TMC2

4 8 12 18 24 32 45 61 78 94 108 TMC3

11 31 44 60 77 93 107 TMC4

23 30 43 59 76 92 106 115 TMC5

29 42 58 75 91 105 114

28 41 57 74 90 104 113

40 56 73 89 103 112

39 55 72 88 102

38 54 71 87 101 111

37 53 70 86 100 110

36 52 69 85 99 109

51 68 84 98

50 67 83 97

66 82 96




1 2263 3364 0325 1198 14392 356072 1801 3832 0292 0418 16222 109313 2170 3937 0381 0820 17565 208234 1950 2573 0169 0928 8710 360795 2136 4636 0267 1262 12531 27226





(165 + 128 + 017 + 024 + 286)5 = 124 (79)










Lead ($t) 1176 1689 1863 2064 2205 2381 2665Zinc ($t) 1236 1816 2014 2176 2344 2695 3244

173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80

5 9 13 19 25 33 46 62 79 954 8 12 18 24 32 45 61 78 94 108

11 31 44 60 77 93 10723 30 43 59 76 92 106 115

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96

(a)

173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80

5 9 13 19 25 33 46 62 79 954 8 12 18 24 32 45 61 78 94 108

11 31 44 60 77 93 10723 30 43 59 76 92 106 115

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96

(b)

173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80

5 9 13 19 25 33 46 62 79 954 8 12 18 24 32 45 61 78 94 108

11 31 44 60 77 93 10723 30 43 59 76 92 106 115

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96

(c)

173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80

5 9 13 19 25 33 46 62 79 954 8 12 18 24 32 45 61 78 94 108

11 31 44 60 77 93 10723 30 43 59 76 92 106 115

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96

(d)



173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80 TMC1

5 9 13 19 25 33 46 62 79 95 TMC2

4 8 12 18 24 32 45 61 78 94 108 TMC3

11 31 44 60 77 93 107 TMC4

23 30 43 59 76 92 106 115 TMC5

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96





Process Value




volatility 018491




volatility 016416





0

01

02

03

04

05

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC3 StdDev Pb

0

02

04

06

08

1

12

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC3 StdDev Zn

0

01

02

03

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC4 StdDev Pb

0

02

04

06

08

1

12

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC4 StedDev Zn

0

01

02

03

04

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

Number of blocks

TMC5 StdDev Pb

002040608

112141618

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

Number of blocks

TMC5 StdDev Zn

0

01

02

03

04

05

06

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC1 StdDev Pb

002040608

112141618

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC1 StdDev Zn

0

01

02

03

04

05

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Number of blocks

TMC2 StdDev Pb

0

02

04

06

08

1

12

14

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Number of blocks

TMC2 StdDev Zn










0

500

1000

1500

2000

2500

3000

3500

4000

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125

US$

t

Month

(a)

0

500

1000

1500

2000

2500

3000

3500

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125

US$

t

Month

(b)


0010203040506070809

1

500 1000 1500 2000 2500 3000 3500$t

VVLVLLM

HVHVVH

(a)

0010203040506070809

1

500 1000 1500 2000 2500 3000 3500 4000$t

VVLVLLM

HVHVVH

(b)




15104734 3944737 3044364 2443462 19244336591226 5356427 4305812 3576349 29263948228167 6904516 5704581 4871250 4096627

25583555 4336349 3366286 2691067 21178617190095 5863841 4730052 3916424 32015418951887 7536746 6246798 5320340 4472001

36213929 4891545 3857917 3139633 25243907923489 6519182 5317593 4456306 36948059796030 8298845 6935676 5958908 5056129

42976598 2108873 1450337 1038819 6860214063918 3133980 2351322 1834430 13747555275030 4267887 3358689 2754948 2200817

56198802 4899506 3882193 3166488 25554757882521 6503582 5321728 4466255 37120859723915 8256167 6917828 5949400 5056656



0

500

1000

1500

2000

2500

3000

3500

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Lead price


(a)

0

1

2

3

4

5

6

7

8

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Lead price

Range code

(b)

0

500

1000

1500

2000

2500

3000

3500

4000

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Zinc price


(c)

0

1

2

3

4

5

6

7

8

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Zinc price

Range code

(d)




119865 minus 119868 = (17561955 24060165 31322673)minus (12000000 13000000 15000000)

= (2561955 11060166 19322673) $(80)


4 Conclusion





LEAD ZINC

UNIT OPERATING COST


01234567

0 5 10 15 20 25 30 35 40 45 50 55 60

Rang

e cod

e val

ue

Month

01234567

0 5 10 15 20 25 30 35 40 45 50 55 60

Rang

e cod

e val

ue

Month

0500

100015002000250030003500

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

0500

1000150020002500300035004000

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

4648505254565860

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

0100000200000300000400000500000600000700000800000

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

ℎ

sim

Stimes４= aℎ

1

(９sum

=1

aℎ2 M

mcon

con

sim








0

100000

200000

300000

400000

500000

600000

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month


0

100000

200000

300000

400000

500000

600000

1 2 3 4 5

$t

Year




Data Availability




0

100000

200000

300000

400000

500000

600000

1 2 3 4 5

$t

Year


TMC 1year 4

TMC 4year 5

TMC 5year 3

TMC 3year 1

TMC 2year 2


References
























































Journal of












Journal of







Volume 2018






Volume 2018












= 119898119888119900119899 ∙ 119898119898119903 ∙ 1003816100381610038161003816119901119904120591119898 1003816100381610038161003816119878timesT (72)



10038161003816100381610038161003816Vℎ10038161003816100381610038161003816119904119894119898119878timesT = 119886ℎ1 (Ysum120574=1


(73)



=1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(119886 119887 119888)ℎ11 (119886 119887 119888)ℎ12 (119886 119887 119888)ℎ1120591(119886 119887 119888)ℎ21 (119886 119887 119888)ℎ22 (119886 119887 119888)ℎ2120591 d

(119886 119887 119888)ℎ1199041 (119886 119887 119888)ℎ1199042 (119886 119887 119888)ℎ119878T

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816


(74)


forall120591isin[1T] =10038161003816100381610038161003816100381610038161003816100381610038161119878119878sum119904=1


(75)


119864 (119864 (Vℎ120591)119905)119905=[1119879] =100381610038161003816100381610038161003816100381610038161003816100381611212sum120591=1

119864 (Vℎ120591)1199051003816100381610038161003816100381610038161003816100381610038161003816119905=[1119879]



1003816100381610038161003816100381610038161003816(Vℎ119905 )1198891003816100381610038161003816100381610038161003816 =10038161003816100381610038161003816100381610038161003816100381610038161003816


10038161003816100381610038161003816100381610038161003816100381610038161003816 forallℎ isin [1119867] 119905 = [1 119879]

(77)


V119889119894119905 = 119897119894sumℎ=1


(78)


3 Numerical Example






Zinc















1 5301 5891 6480 59 6458 7176 78932 6844 7604 8365 60 6954 7727 84993 5081 5646 6210 61 6293 6992 76914 7064 7849 8634 62 7119 7910 87015 7615 8461 9307 63 6128 6809 74896 6954 7727 8499 64 7284 8094 89037 7725 8583 9442 65 5962 6625 72878 5715 6350 6984 66 8977 9974 109729 6747 7497 8247 67 6128 6809 748910 5508 6120 6732 68 7119 7910 870111 7367 8186 9004 69 6293 6992 769112 6871 7635 8398 70 6954 7727 849913 7491 8323 9156 71 6458 7176 789314 6747 7497 8247 72 6789 7543 829715 7615 8461 9307 73 6623 7359 809516 6623 7359 8095 74 6623 7359 809517 7615 8461 9307 75 6789 7543 829718 5907 6564 7220 76 6458 7176 789319 6871 7635 8398 77 6954 7727 849920 5715 6350 6984 78 6293 6992 769121 7064 7849 8634 79 7119 7910 870122 5522 6135 6749 80 6128 6809 748923 6816 7574 8331 81 7284 8094 890324 6678 7421 8163 82 9130 10144 1115825 7367 8186 9004 83 6334 7038 774226 6541 7268 7994 84 7243 8048 885327 7505 8339 9172 85 6486 7206 792728 6458 7176 7893 86 7092 7880 866729 6637 7375 8112 87 6637 7375 811230 6279 6977 7674 88 6940 7711 848231 6816 7574 8331 89 6789 7543 829732 6100 6778 7456 90 6789 7543 829733 6995 7772 8550 91 6940 7711 848234 5921 6579 7237 92 6637 7375 811235 7174 7971 8768 93 7092 7880 866736 6486 7206 7927 94 6486 7206 792737 7092 7880 8667 95 7243 8048 885338 6637 7375 8112 96 8895 9883 1087239 6940 7711 8482 97 5921 6579 723740 6789 7543 8297 98 6995 7772 855041 6789 7543 8297 99 6100 6778 745642 6940 7711 8482 100 6816 7574 833143 6637 7375 8112 101 6279 6977 767444 7092 7880 8667 102 6637 7375 811245 6486 7206 7927 103 6458 7176 789346 7243 8048 8853 104 6458 7176 789347 6334 7038 7742 105 6637 7375 811248 7394 8216 9038 106 6279 6977 767449 6183 6870 7557 107 6816 7574 833150 6128 6809 7489 108 6100 6778 745651 7119 7910 8701 109 6678 7421 816352 6293 6992 7691 110 7229 8033 8836


Table 3 Continued





1152640

3913 -165169601186561

2153217

3072 -128170241187266

3154940

2939 -017172156189371

4155573

6278 +024172859190145

5159649

4816 +286177388195127


(15-19](19-23](23-27](27-31](31-35]

(151-276](276-401](401-526](526-651](651-776]



17

3 16 22 49 65

2 7 15 21 27 35 48 64 81

1 6 10 14 20 26 34 47 63 80 TMC1

5 9 13 19 25 33 46 62 79 95 TMC2

4 8 12 18 24 32 45 61 78 94 108 TMC3

11 31 44 60 77 93 107 TMC4

23 30 43 59 76 92 106 115 TMC5

29 42 58 75 91 105 114

28 41 57 74 90 104 113

40 56 73 89 103 112

39 55 72 88 102

38 54 71 87 101 111

37 53 70 86 100 110

36 52 69 85 99 109

51 68 84 98

50 67 83 97

66 82 96




1 2263 3364 0325 1198 14392 356072 1801 3832 0292 0418 16222 109313 2170 3937 0381 0820 17565 208234 1950 2573 0169 0928 8710 360795 2136 4636 0267 1262 12531 27226





(165 + 128 + 017 + 024 + 286)5 = 124 (79)










Lead ($t) 1176 1689 1863 2064 2205 2381 2665Zinc ($t) 1236 1816 2014 2176 2344 2695 3244

173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80

5 9 13 19 25 33 46 62 79 954 8 12 18 24 32 45 61 78 94 108

11 31 44 60 77 93 10723 30 43 59 76 92 106 115

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96

(a)

173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80

5 9 13 19 25 33 46 62 79 954 8 12 18 24 32 45 61 78 94 108

11 31 44 60 77 93 10723 30 43 59 76 92 106 115

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96

(b)

173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80

5 9 13 19 25 33 46 62 79 954 8 12 18 24 32 45 61 78 94 108

11 31 44 60 77 93 10723 30 43 59 76 92 106 115

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96

(c)

173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80

5 9 13 19 25 33 46 62 79 954 8 12 18 24 32 45 61 78 94 108

11 31 44 60 77 93 10723 30 43 59 76 92 106 115

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96

(d)



173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80 TMC1

5 9 13 19 25 33 46 62 79 95 TMC2

4 8 12 18 24 32 45 61 78 94 108 TMC3

11 31 44 60 77 93 107 TMC4

23 30 43 59 76 92 106 115 TMC5

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96





Process Value




volatility 018491




volatility 016416





0

01

02

03

04

05

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC3 StdDev Pb

0

02

04

06

08

1

12

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC3 StdDev Zn

0

01

02

03

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC4 StdDev Pb

0

02

04

06

08

1

12

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC4 StedDev Zn

0

01

02

03

04

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

Number of blocks

TMC5 StdDev Pb

002040608

112141618

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

Number of blocks

TMC5 StdDev Zn

0

01

02

03

04

05

06

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC1 StdDev Pb

002040608

112141618

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC1 StdDev Zn

0

01

02

03

04

05

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Number of blocks

TMC2 StdDev Pb

0

02

04

06

08

1

12

14

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Number of blocks

TMC2 StdDev Zn










0

500

1000

1500

2000

2500

3000

3500

4000

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125

US$

t

Month

(a)

0

500

1000

1500

2000

2500

3000

3500

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125

US$

t

Month

(b)


0010203040506070809

1

500 1000 1500 2000 2500 3000 3500$t

VVLVLLM

HVHVVH

(a)

0010203040506070809

1

500 1000 1500 2000 2500 3000 3500 4000$t

VVLVLLM

HVHVVH

(b)




15104734 3944737 3044364 2443462 19244336591226 5356427 4305812 3576349 29263948228167 6904516 5704581 4871250 4096627

25583555 4336349 3366286 2691067 21178617190095 5863841 4730052 3916424 32015418951887 7536746 6246798 5320340 4472001

36213929 4891545 3857917 3139633 25243907923489 6519182 5317593 4456306 36948059796030 8298845 6935676 5958908 5056129

42976598 2108873 1450337 1038819 6860214063918 3133980 2351322 1834430 13747555275030 4267887 3358689 2754948 2200817

56198802 4899506 3882193 3166488 25554757882521 6503582 5321728 4466255 37120859723915 8256167 6917828 5949400 5056656



0

500

1000

1500

2000

2500

3000

3500

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Lead price


(a)

0

1

2

3

4

5

6

7

8

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Lead price

Range code

(b)

0

500

1000

1500

2000

2500

3000

3500

4000

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Zinc price


(c)

0

1

2

3

4

5

6

7

8

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Zinc price

Range code

(d)




119865 minus 119868 = (17561955 24060165 31322673)minus (12000000 13000000 15000000)

= (2561955 11060166 19322673) $(80)


4 Conclusion





LEAD ZINC

UNIT OPERATING COST


01234567

0 5 10 15 20 25 30 35 40 45 50 55 60

Rang

e cod

e val

ue

Month

01234567

0 5 10 15 20 25 30 35 40 45 50 55 60

Rang

e cod

e val

ue

Month

0500

100015002000250030003500

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

0500

1000150020002500300035004000

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

4648505254565860

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

0100000200000300000400000500000600000700000800000

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

ℎ

sim

Stimes４= aℎ

1

(９sum

=1

aℎ2 M

mcon

con

sim








0

100000

200000

300000

400000

500000

600000

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month


0

100000

200000

300000

400000

500000

600000

1 2 3 4 5

$t

Year




Data Availability




0

100000

200000

300000

400000

500000

600000

1 2 3 4 5

$t

Year


TMC 1year 4

TMC 4year 5

TMC 5year 3

TMC 3year 1

TMC 2year 2


References
























































Journal of












Journal of







Volume 2018






Volume 2018











Zinc















1 5301 5891 6480 59 6458 7176 78932 6844 7604 8365 60 6954 7727 84993 5081 5646 6210 61 6293 6992 76914 7064 7849 8634 62 7119 7910 87015 7615 8461 9307 63 6128 6809 74896 6954 7727 8499 64 7284 8094 89037 7725 8583 9442 65 5962 6625 72878 5715 6350 6984 66 8977 9974 109729 6747 7497 8247 67 6128 6809 748910 5508 6120 6732 68 7119 7910 870111 7367 8186 9004 69 6293 6992 769112 6871 7635 8398 70 6954 7727 849913 7491 8323 9156 71 6458 7176 789314 6747 7497 8247 72 6789 7543 829715 7615 8461 9307 73 6623 7359 809516 6623 7359 8095 74 6623 7359 809517 7615 8461 9307 75 6789 7543 829718 5907 6564 7220 76 6458 7176 789319 6871 7635 8398 77 6954 7727 849920 5715 6350 6984 78 6293 6992 769121 7064 7849 8634 79 7119 7910 870122 5522 6135 6749 80 6128 6809 748923 6816 7574 8331 81 7284 8094 890324 6678 7421 8163 82 9130 10144 1115825 7367 8186 9004 83 6334 7038 774226 6541 7268 7994 84 7243 8048 885327 7505 8339 9172 85 6486 7206 792728 6458 7176 7893 86 7092 7880 866729 6637 7375 8112 87 6637 7375 811230 6279 6977 7674 88 6940 7711 848231 6816 7574 8331 89 6789 7543 829732 6100 6778 7456 90 6789 7543 829733 6995 7772 8550 91 6940 7711 848234 5921 6579 7237 92 6637 7375 811235 7174 7971 8768 93 7092 7880 866736 6486 7206 7927 94 6486 7206 792737 7092 7880 8667 95 7243 8048 885338 6637 7375 8112 96 8895 9883 1087239 6940 7711 8482 97 5921 6579 723740 6789 7543 8297 98 6995 7772 855041 6789 7543 8297 99 6100 6778 745642 6940 7711 8482 100 6816 7574 833143 6637 7375 8112 101 6279 6977 767444 7092 7880 8667 102 6637 7375 811245 6486 7206 7927 103 6458 7176 789346 7243 8048 8853 104 6458 7176 789347 6334 7038 7742 105 6637 7375 811248 7394 8216 9038 106 6279 6977 767449 6183 6870 7557 107 6816 7574 833150 6128 6809 7489 108 6100 6778 745651 7119 7910 8701 109 6678 7421 816352 6293 6992 7691 110 7229 8033 8836


Table 3 Continued





1152640

3913 -165169601186561

2153217

3072 -128170241187266

3154940

2939 -017172156189371

4155573

6278 +024172859190145

5159649

4816 +286177388195127


(15-19](19-23](23-27](27-31](31-35]

(151-276](276-401](401-526](526-651](651-776]



17

3 16 22 49 65

2 7 15 21 27 35 48 64 81

1 6 10 14 20 26 34 47 63 80 TMC1

5 9 13 19 25 33 46 62 79 95 TMC2

4 8 12 18 24 32 45 61 78 94 108 TMC3

11 31 44 60 77 93 107 TMC4

23 30 43 59 76 92 106 115 TMC5

29 42 58 75 91 105 114

28 41 57 74 90 104 113

40 56 73 89 103 112

39 55 72 88 102

38 54 71 87 101 111

37 53 70 86 100 110

36 52 69 85 99 109

51 68 84 98

50 67 83 97

66 82 96




1 2263 3364 0325 1198 14392 356072 1801 3832 0292 0418 16222 109313 2170 3937 0381 0820 17565 208234 1950 2573 0169 0928 8710 360795 2136 4636 0267 1262 12531 27226





(165 + 128 + 017 + 024 + 286)5 = 124 (79)










Lead ($t) 1176 1689 1863 2064 2205 2381 2665Zinc ($t) 1236 1816 2014 2176 2344 2695 3244

173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80

5 9 13 19 25 33 46 62 79 954 8 12 18 24 32 45 61 78 94 108

11 31 44 60 77 93 10723 30 43 59 76 92 106 115

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96

(a)

173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80

5 9 13 19 25 33 46 62 79 954 8 12 18 24 32 45 61 78 94 108

11 31 44 60 77 93 10723 30 43 59 76 92 106 115

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96

(b)

173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80

5 9 13 19 25 33 46 62 79 954 8 12 18 24 32 45 61 78 94 108

11 31 44 60 77 93 10723 30 43 59 76 92 106 115

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96

(c)

173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80

5 9 13 19 25 33 46 62 79 954 8 12 18 24 32 45 61 78 94 108

11 31 44 60 77 93 10723 30 43 59 76 92 106 115

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96

(d)



173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80 TMC1

5 9 13 19 25 33 46 62 79 95 TMC2

4 8 12 18 24 32 45 61 78 94 108 TMC3

11 31 44 60 77 93 107 TMC4

23 30 43 59 76 92 106 115 TMC5

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96





Process Value




volatility 018491




volatility 016416





0

01

02

03

04

05

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC3 StdDev Pb

0

02

04

06

08

1

12

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC3 StdDev Zn

0

01

02

03

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC4 StdDev Pb

0

02

04

06

08

1

12

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC4 StedDev Zn

0

01

02

03

04

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

Number of blocks

TMC5 StdDev Pb

002040608

112141618

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

Number of blocks

TMC5 StdDev Zn

0

01

02

03

04

05

06

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC1 StdDev Pb

002040608

112141618

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC1 StdDev Zn

0

01

02

03

04

05

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Number of blocks

TMC2 StdDev Pb

0

02

04

06

08

1

12

14

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Number of blocks

TMC2 StdDev Zn










0

500

1000

1500

2000

2500

3000

3500

4000

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125

US$

t

Month

(a)

0

500

1000

1500

2000

2500

3000

3500

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125

US$

t

Month

(b)


0010203040506070809

1

500 1000 1500 2000 2500 3000 3500$t

VVLVLLM

HVHVVH

(a)

0010203040506070809

1

500 1000 1500 2000 2500 3000 3500 4000$t

VVLVLLM

HVHVVH

(b)




15104734 3944737 3044364 2443462 19244336591226 5356427 4305812 3576349 29263948228167 6904516 5704581 4871250 4096627

25583555 4336349 3366286 2691067 21178617190095 5863841 4730052 3916424 32015418951887 7536746 6246798 5320340 4472001

36213929 4891545 3857917 3139633 25243907923489 6519182 5317593 4456306 36948059796030 8298845 6935676 5958908 5056129

42976598 2108873 1450337 1038819 6860214063918 3133980 2351322 1834430 13747555275030 4267887 3358689 2754948 2200817

56198802 4899506 3882193 3166488 25554757882521 6503582 5321728 4466255 37120859723915 8256167 6917828 5949400 5056656



0

500

1000

1500

2000

2500

3000

3500

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Lead price


(a)

0

1

2

3

4

5

6

7

8

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Lead price

Range code

(b)

0

500

1000

1500

2000

2500

3000

3500

4000

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Zinc price


(c)

0

1

2

3

4

5

6

7

8

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Zinc price

Range code

(d)




119865 minus 119868 = (17561955 24060165 31322673)minus (12000000 13000000 15000000)

= (2561955 11060166 19322673) $(80)


4 Conclusion





LEAD ZINC

UNIT OPERATING COST


01234567

0 5 10 15 20 25 30 35 40 45 50 55 60

Rang

e cod

e val

ue

Month

01234567

0 5 10 15 20 25 30 35 40 45 50 55 60

Rang

e cod

e val

ue

Month

0500

100015002000250030003500

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

0500

1000150020002500300035004000

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

4648505254565860

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

0100000200000300000400000500000600000700000800000

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

ℎ

sim

Stimes４= aℎ

1

(９sum

=1

aℎ2 M

mcon

con

sim








0

100000

200000

300000

400000

500000

600000

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month


0

100000

200000

300000

400000

500000

600000

1 2 3 4 5

$t

Year




Data Availability




0

100000

200000

300000

400000

500000

600000

1 2 3 4 5

$t

Year


TMC 1year 4

TMC 4year 5

TMC 5year 3

TMC 3year 1

TMC 2year 2


References
























































Journal of












Journal of







Volume 2018






Volume 2018











1 5301 5891 6480 59 6458 7176 78932 6844 7604 8365 60 6954 7727 84993 5081 5646 6210 61 6293 6992 76914 7064 7849 8634 62 7119 7910 87015 7615 8461 9307 63 6128 6809 74896 6954 7727 8499 64 7284 8094 89037 7725 8583 9442 65 5962 6625 72878 5715 6350 6984 66 8977 9974 109729 6747 7497 8247 67 6128 6809 748910 5508 6120 6732 68 7119 7910 870111 7367 8186 9004 69 6293 6992 769112 6871 7635 8398 70 6954 7727 849913 7491 8323 9156 71 6458 7176 789314 6747 7497 8247 72 6789 7543 829715 7615 8461 9307 73 6623 7359 809516 6623 7359 8095 74 6623 7359 809517 7615 8461 9307 75 6789 7543 829718 5907 6564 7220 76 6458 7176 789319 6871 7635 8398 77 6954 7727 849920 5715 6350 6984 78 6293 6992 769121 7064 7849 8634 79 7119 7910 870122 5522 6135 6749 80 6128 6809 748923 6816 7574 8331 81 7284 8094 890324 6678 7421 8163 82 9130 10144 1115825 7367 8186 9004 83 6334 7038 774226 6541 7268 7994 84 7243 8048 885327 7505 8339 9172 85 6486 7206 792728 6458 7176 7893 86 7092 7880 866729 6637 7375 8112 87 6637 7375 811230 6279 6977 7674 88 6940 7711 848231 6816 7574 8331 89 6789 7543 829732 6100 6778 7456 90 6789 7543 829733 6995 7772 8550 91 6940 7711 848234 5921 6579 7237 92 6637 7375 811235 7174 7971 8768 93 7092 7880 866736 6486 7206 7927 94 6486 7206 792737 7092 7880 8667 95 7243 8048 885338 6637 7375 8112 96 8895 9883 1087239 6940 7711 8482 97 5921 6579 723740 6789 7543 8297 98 6995 7772 855041 6789 7543 8297 99 6100 6778 745642 6940 7711 8482 100 6816 7574 833143 6637 7375 8112 101 6279 6977 767444 7092 7880 8667 102 6637 7375 811245 6486 7206 7927 103 6458 7176 789346 7243 8048 8853 104 6458 7176 789347 6334 7038 7742 105 6637 7375 811248 7394 8216 9038 106 6279 6977 767449 6183 6870 7557 107 6816 7574 833150 6128 6809 7489 108 6100 6778 745651 7119 7910 8701 109 6678 7421 816352 6293 6992 7691 110 7229 8033 8836


Table 3 Continued





1152640

3913 -165169601186561

2153217

3072 -128170241187266

3154940

2939 -017172156189371

4155573

6278 +024172859190145

5159649

4816 +286177388195127


(15-19](19-23](23-27](27-31](31-35]

(151-276](276-401](401-526](526-651](651-776]



17

3 16 22 49 65

2 7 15 21 27 35 48 64 81

1 6 10 14 20 26 34 47 63 80 TMC1

5 9 13 19 25 33 46 62 79 95 TMC2

4 8 12 18 24 32 45 61 78 94 108 TMC3

11 31 44 60 77 93 107 TMC4

23 30 43 59 76 92 106 115 TMC5

29 42 58 75 91 105 114

28 41 57 74 90 104 113

40 56 73 89 103 112

39 55 72 88 102

38 54 71 87 101 111

37 53 70 86 100 110

36 52 69 85 99 109

51 68 84 98

50 67 83 97

66 82 96




1 2263 3364 0325 1198 14392 356072 1801 3832 0292 0418 16222 109313 2170 3937 0381 0820 17565 208234 1950 2573 0169 0928 8710 360795 2136 4636 0267 1262 12531 27226





(165 + 128 + 017 + 024 + 286)5 = 124 (79)










Lead ($t) 1176 1689 1863 2064 2205 2381 2665Zinc ($t) 1236 1816 2014 2176 2344 2695 3244

173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80

5 9 13 19 25 33 46 62 79 954 8 12 18 24 32 45 61 78 94 108

11 31 44 60 77 93 10723 30 43 59 76 92 106 115

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96

(a)

173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80

5 9 13 19 25 33 46 62 79 954 8 12 18 24 32 45 61 78 94 108

11 31 44 60 77 93 10723 30 43 59 76 92 106 115

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96

(b)

173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80

5 9 13 19 25 33 46 62 79 954 8 12 18 24 32 45 61 78 94 108

11 31 44 60 77 93 10723 30 43 59 76 92 106 115

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96

(c)

173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80

5 9 13 19 25 33 46 62 79 954 8 12 18 24 32 45 61 78 94 108

11 31 44 60 77 93 10723 30 43 59 76 92 106 115

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96

(d)



173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80 TMC1

5 9 13 19 25 33 46 62 79 95 TMC2

4 8 12 18 24 32 45 61 78 94 108 TMC3

11 31 44 60 77 93 107 TMC4

23 30 43 59 76 92 106 115 TMC5

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96





Process Value




volatility 018491




volatility 016416





0

01

02

03

04

05

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC3 StdDev Pb

0

02

04

06

08

1

12

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC3 StdDev Zn

0

01

02

03

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC4 StdDev Pb

0

02

04

06

08

1

12

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC4 StedDev Zn

0

01

02

03

04

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

Number of blocks

TMC5 StdDev Pb

002040608

112141618

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

Number of blocks

TMC5 StdDev Zn

0

01

02

03

04

05

06

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC1 StdDev Pb

002040608

112141618

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC1 StdDev Zn

0

01

02

03

04

05

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Number of blocks

TMC2 StdDev Pb

0

02

04

06

08

1

12

14

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Number of blocks

TMC2 StdDev Zn










0

500

1000

1500

2000

2500

3000

3500

4000

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125

US$

t

Month

(a)

0

500

1000

1500

2000

2500

3000

3500

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125

US$

t

Month

(b)


0010203040506070809

1

500 1000 1500 2000 2500 3000 3500$t

VVLVLLM

HVHVVH

(a)

0010203040506070809

1

500 1000 1500 2000 2500 3000 3500 4000$t

VVLVLLM

HVHVVH

(b)




15104734 3944737 3044364 2443462 19244336591226 5356427 4305812 3576349 29263948228167 6904516 5704581 4871250 4096627

25583555 4336349 3366286 2691067 21178617190095 5863841 4730052 3916424 32015418951887 7536746 6246798 5320340 4472001

36213929 4891545 3857917 3139633 25243907923489 6519182 5317593 4456306 36948059796030 8298845 6935676 5958908 5056129

42976598 2108873 1450337 1038819 6860214063918 3133980 2351322 1834430 13747555275030 4267887 3358689 2754948 2200817

56198802 4899506 3882193 3166488 25554757882521 6503582 5321728 4466255 37120859723915 8256167 6917828 5949400 5056656



0

500

1000

1500

2000

2500

3000

3500

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Lead price


(a)

0

1

2

3

4

5

6

7

8

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Lead price

Range code

(b)

0

500

1000

1500

2000

2500

3000

3500

4000

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Zinc price


(c)

0

1

2

3

4

5

6

7

8

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Zinc price

Range code

(d)




119865 minus 119868 = (17561955 24060165 31322673)minus (12000000 13000000 15000000)

= (2561955 11060166 19322673) $(80)


4 Conclusion





LEAD ZINC

UNIT OPERATING COST


01234567

0 5 10 15 20 25 30 35 40 45 50 55 60

Rang

e cod

e val

ue

Month

01234567

0 5 10 15 20 25 30 35 40 45 50 55 60

Rang

e cod

e val

ue

Month

0500

100015002000250030003500

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

0500

1000150020002500300035004000

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

4648505254565860

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

0100000200000300000400000500000600000700000800000

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

ℎ

sim

Stimes４= aℎ

1

(９sum

=1

aℎ2 M

mcon

con

sim








0

100000

200000

300000

400000

500000

600000

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month


0

100000

200000

300000

400000

500000

600000

1 2 3 4 5

$t

Year




Data Availability




0

100000

200000

300000

400000

500000

600000

1 2 3 4 5

$t

Year


TMC 1year 4

TMC 4year 5

TMC 5year 3

TMC 3year 1

TMC 2year 2


References
























































Journal of












Journal of







Volume 2018






Volume 2018









Table 3 Continued





1152640

3913 -165169601186561

2153217

3072 -128170241187266

3154940

2939 -017172156189371

4155573

6278 +024172859190145

5159649

4816 +286177388195127


(15-19](19-23](23-27](27-31](31-35]

(151-276](276-401](401-526](526-651](651-776]



17

3 16 22 49 65

2 7 15 21 27 35 48 64 81

1 6 10 14 20 26 34 47 63 80 TMC1

5 9 13 19 25 33 46 62 79 95 TMC2

4 8 12 18 24 32 45 61 78 94 108 TMC3

11 31 44 60 77 93 107 TMC4

23 30 43 59 76 92 106 115 TMC5

29 42 58 75 91 105 114

28 41 57 74 90 104 113

40 56 73 89 103 112

39 55 72 88 102

38 54 71 87 101 111

37 53 70 86 100 110

36 52 69 85 99 109

51 68 84 98

50 67 83 97

66 82 96




1 2263 3364 0325 1198 14392 356072 1801 3832 0292 0418 16222 109313 2170 3937 0381 0820 17565 208234 1950 2573 0169 0928 8710 360795 2136 4636 0267 1262 12531 27226





(165 + 128 + 017 + 024 + 286)5 = 124 (79)










Lead ($t) 1176 1689 1863 2064 2205 2381 2665Zinc ($t) 1236 1816 2014 2176 2344 2695 3244

173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80

5 9 13 19 25 33 46 62 79 954 8 12 18 24 32 45 61 78 94 108

11 31 44 60 77 93 10723 30 43 59 76 92 106 115

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96

(a)

173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80

5 9 13 19 25 33 46 62 79 954 8 12 18 24 32 45 61 78 94 108

11 31 44 60 77 93 10723 30 43 59 76 92 106 115

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96

(b)

173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80

5 9 13 19 25 33 46 62 79 954 8 12 18 24 32 45 61 78 94 108

11 31 44 60 77 93 10723 30 43 59 76 92 106 115

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96

(c)

173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80

5 9 13 19 25 33 46 62 79 954 8 12 18 24 32 45 61 78 94 108

11 31 44 60 77 93 10723 30 43 59 76 92 106 115

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96

(d)



173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80 TMC1

5 9 13 19 25 33 46 62 79 95 TMC2

4 8 12 18 24 32 45 61 78 94 108 TMC3

11 31 44 60 77 93 107 TMC4

23 30 43 59 76 92 106 115 TMC5

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96





Process Value




volatility 018491




volatility 016416





0

01

02

03

04

05

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC3 StdDev Pb

0

02

04

06

08

1

12

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC3 StdDev Zn

0

01

02

03

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC4 StdDev Pb

0

02

04

06

08

1

12

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC4 StedDev Zn

0

01

02

03

04

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

Number of blocks

TMC5 StdDev Pb

002040608

112141618

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

Number of blocks

TMC5 StdDev Zn

0

01

02

03

04

05

06

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC1 StdDev Pb

002040608

112141618

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC1 StdDev Zn

0

01

02

03

04

05

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Number of blocks

TMC2 StdDev Pb

0

02

04

06

08

1

12

14

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Number of blocks

TMC2 StdDev Zn










0

500

1000

1500

2000

2500

3000

3500

4000

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125

US$

t

Month

(a)

0

500

1000

1500

2000

2500

3000

3500

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125

US$

t

Month

(b)


0010203040506070809

1

500 1000 1500 2000 2500 3000 3500$t

VVLVLLM

HVHVVH

(a)

0010203040506070809

1

500 1000 1500 2000 2500 3000 3500 4000$t

VVLVLLM

HVHVVH

(b)




15104734 3944737 3044364 2443462 19244336591226 5356427 4305812 3576349 29263948228167 6904516 5704581 4871250 4096627

25583555 4336349 3366286 2691067 21178617190095 5863841 4730052 3916424 32015418951887 7536746 6246798 5320340 4472001

36213929 4891545 3857917 3139633 25243907923489 6519182 5317593 4456306 36948059796030 8298845 6935676 5958908 5056129

42976598 2108873 1450337 1038819 6860214063918 3133980 2351322 1834430 13747555275030 4267887 3358689 2754948 2200817

56198802 4899506 3882193 3166488 25554757882521 6503582 5321728 4466255 37120859723915 8256167 6917828 5949400 5056656



0

500

1000

1500

2000

2500

3000

3500

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Lead price


(a)

0

1

2

3

4

5

6

7

8

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Lead price

Range code

(b)

0

500

1000

1500

2000

2500

3000

3500

4000

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Zinc price


(c)

0

1

2

3

4

5

6

7

8

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Zinc price

Range code

(d)




119865 minus 119868 = (17561955 24060165 31322673)minus (12000000 13000000 15000000)

= (2561955 11060166 19322673) $(80)


4 Conclusion





LEAD ZINC

UNIT OPERATING COST


01234567

0 5 10 15 20 25 30 35 40 45 50 55 60

Rang

e cod

e val

ue

Month

01234567

0 5 10 15 20 25 30 35 40 45 50 55 60

Rang

e cod

e val

ue

Month

0500

100015002000250030003500

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

0500

1000150020002500300035004000

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

4648505254565860

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

0100000200000300000400000500000600000700000800000

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

ℎ

sim

Stimes４= aℎ

1

(９sum

=1

aℎ2 M

mcon

con

sim








0

100000

200000

300000

400000

500000

600000

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month


0

100000

200000

300000

400000

500000

600000

1 2 3 4 5

$t

Year




Data Availability




0

100000

200000

300000

400000

500000

600000

1 2 3 4 5

$t

Year


TMC 1year 4

TMC 4year 5

TMC 5year 3

TMC 3year 1

TMC 2year 2


References
























































Journal of












Journal of







Volume 2018






Volume 2018









17

3 16 22 49 65

2 7 15 21 27 35 48 64 81

1 6 10 14 20 26 34 47 63 80 TMC1

5 9 13 19 25 33 46 62 79 95 TMC2

4 8 12 18 24 32 45 61 78 94 108 TMC3

11 31 44 60 77 93 107 TMC4

23 30 43 59 76 92 106 115 TMC5

29 42 58 75 91 105 114

28 41 57 74 90 104 113

40 56 73 89 103 112

39 55 72 88 102

38 54 71 87 101 111

37 53 70 86 100 110

36 52 69 85 99 109

51 68 84 98

50 67 83 97

66 82 96




1 2263 3364 0325 1198 14392 356072 1801 3832 0292 0418 16222 109313 2170 3937 0381 0820 17565 208234 1950 2573 0169 0928 8710 360795 2136 4636 0267 1262 12531 27226





(165 + 128 + 017 + 024 + 286)5 = 124 (79)










Lead ($t) 1176 1689 1863 2064 2205 2381 2665Zinc ($t) 1236 1816 2014 2176 2344 2695 3244

173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80

5 9 13 19 25 33 46 62 79 954 8 12 18 24 32 45 61 78 94 108

11 31 44 60 77 93 10723 30 43 59 76 92 106 115

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96

(a)

173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80

5 9 13 19 25 33 46 62 79 954 8 12 18 24 32 45 61 78 94 108

11 31 44 60 77 93 10723 30 43 59 76 92 106 115

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96

(b)

173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80

5 9 13 19 25 33 46 62 79 954 8 12 18 24 32 45 61 78 94 108

11 31 44 60 77 93 10723 30 43 59 76 92 106 115

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96

(c)

173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80

5 9 13 19 25 33 46 62 79 954 8 12 18 24 32 45 61 78 94 108

11 31 44 60 77 93 10723 30 43 59 76 92 106 115

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96

(d)



173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80 TMC1

5 9 13 19 25 33 46 62 79 95 TMC2

4 8 12 18 24 32 45 61 78 94 108 TMC3

11 31 44 60 77 93 107 TMC4

23 30 43 59 76 92 106 115 TMC5

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96





Process Value




volatility 018491




volatility 016416





0

01

02

03

04

05

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC3 StdDev Pb

0

02

04

06

08

1

12

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC3 StdDev Zn

0

01

02

03

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC4 StdDev Pb

0

02

04

06

08

1

12

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC4 StedDev Zn

0

01

02

03

04

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

Number of blocks

TMC5 StdDev Pb

002040608

112141618

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

Number of blocks

TMC5 StdDev Zn

0

01

02

03

04

05

06

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC1 StdDev Pb

002040608

112141618

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC1 StdDev Zn

0

01

02

03

04

05

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Number of blocks

TMC2 StdDev Pb

0

02

04

06

08

1

12

14

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Number of blocks

TMC2 StdDev Zn










0

500

1000

1500

2000

2500

3000

3500

4000

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125

US$

t

Month

(a)

0

500

1000

1500

2000

2500

3000

3500

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125

US$

t

Month

(b)


0010203040506070809

1

500 1000 1500 2000 2500 3000 3500$t

VVLVLLM

HVHVVH

(a)

0010203040506070809

1

500 1000 1500 2000 2500 3000 3500 4000$t

VVLVLLM

HVHVVH

(b)




15104734 3944737 3044364 2443462 19244336591226 5356427 4305812 3576349 29263948228167 6904516 5704581 4871250 4096627

25583555 4336349 3366286 2691067 21178617190095 5863841 4730052 3916424 32015418951887 7536746 6246798 5320340 4472001

36213929 4891545 3857917 3139633 25243907923489 6519182 5317593 4456306 36948059796030 8298845 6935676 5958908 5056129

42976598 2108873 1450337 1038819 6860214063918 3133980 2351322 1834430 13747555275030 4267887 3358689 2754948 2200817

56198802 4899506 3882193 3166488 25554757882521 6503582 5321728 4466255 37120859723915 8256167 6917828 5949400 5056656



0

500

1000

1500

2000

2500

3000

3500

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Lead price


(a)

0

1

2

3

4

5

6

7

8

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Lead price

Range code

(b)

0

500

1000

1500

2000

2500

3000

3500

4000

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Zinc price


(c)

0

1

2

3

4

5

6

7

8

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Zinc price

Range code

(d)




119865 minus 119868 = (17561955 24060165 31322673)minus (12000000 13000000 15000000)

= (2561955 11060166 19322673) $(80)


4 Conclusion





LEAD ZINC

UNIT OPERATING COST


01234567

0 5 10 15 20 25 30 35 40 45 50 55 60

Rang

e cod

e val

ue

Month

01234567

0 5 10 15 20 25 30 35 40 45 50 55 60

Rang

e cod

e val

ue

Month

0500

100015002000250030003500

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

0500

1000150020002500300035004000

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

4648505254565860

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

0100000200000300000400000500000600000700000800000

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

ℎ

sim

Stimes４= aℎ

1

(９sum

=1

aℎ2 M

mcon

con

sim








0

100000

200000

300000

400000

500000

600000

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month


0

100000

200000

300000

400000

500000

600000

1 2 3 4 5

$t

Year




Data Availability




0

100000

200000

300000

400000

500000

600000

1 2 3 4 5

$t

Year


TMC 1year 4

TMC 4year 5

TMC 5year 3

TMC 3year 1

TMC 2year 2


References
























































Journal of












Journal of







Volume 2018






Volume 2018











Lead ($t) 1176 1689 1863 2064 2205 2381 2665Zinc ($t) 1236 1816 2014 2176 2344 2695 3244

173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80

5 9 13 19 25 33 46 62 79 954 8 12 18 24 32 45 61 78 94 108

11 31 44 60 77 93 10723 30 43 59 76 92 106 115

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96

(a)

173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80

5 9 13 19 25 33 46 62 79 954 8 12 18 24 32 45 61 78 94 108

11 31 44 60 77 93 10723 30 43 59 76 92 106 115

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96

(b)

173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80

5 9 13 19 25 33 46 62 79 954 8 12 18 24 32 45 61 78 94 108

11 31 44 60 77 93 10723 30 43 59 76 92 106 115

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96

(c)

173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80

5 9 13 19 25 33 46 62 79 954 8 12 18 24 32 45 61 78 94 108

11 31 44 60 77 93 10723 30 43 59 76 92 106 115

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96

(d)



173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80 TMC1

5 9 13 19 25 33 46 62 79 95 TMC2

4 8 12 18 24 32 45 61 78 94 108 TMC3

11 31 44 60 77 93 107 TMC4

23 30 43 59 76 92 106 115 TMC5

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96





Process Value




volatility 018491




volatility 016416





0

01

02

03

04

05

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC3 StdDev Pb

0

02

04

06

08

1

12

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC3 StdDev Zn

0

01

02

03

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC4 StdDev Pb

0

02

04

06

08

1

12

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC4 StedDev Zn

0

01

02

03

04

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

Number of blocks

TMC5 StdDev Pb

002040608

112141618

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

Number of blocks

TMC5 StdDev Zn

0

01

02

03

04

05

06

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC1 StdDev Pb

002040608

112141618

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC1 StdDev Zn

0

01

02

03

04

05

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Number of blocks

TMC2 StdDev Pb

0

02

04

06

08

1

12

14

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Number of blocks

TMC2 StdDev Zn










0

500

1000

1500

2000

2500

3000

3500

4000

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125

US$

t

Month

(a)

0

500

1000

1500

2000

2500

3000

3500

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125

US$

t

Month

(b)


0010203040506070809

1

500 1000 1500 2000 2500 3000 3500$t

VVLVLLM

HVHVVH

(a)

0010203040506070809

1

500 1000 1500 2000 2500 3000 3500 4000$t

VVLVLLM

HVHVVH

(b)




15104734 3944737 3044364 2443462 19244336591226 5356427 4305812 3576349 29263948228167 6904516 5704581 4871250 4096627

25583555 4336349 3366286 2691067 21178617190095 5863841 4730052 3916424 32015418951887 7536746 6246798 5320340 4472001

36213929 4891545 3857917 3139633 25243907923489 6519182 5317593 4456306 36948059796030 8298845 6935676 5958908 5056129

42976598 2108873 1450337 1038819 6860214063918 3133980 2351322 1834430 13747555275030 4267887 3358689 2754948 2200817

56198802 4899506 3882193 3166488 25554757882521 6503582 5321728 4466255 37120859723915 8256167 6917828 5949400 5056656



0

500

1000

1500

2000

2500

3000

3500

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Lead price


(a)

0

1

2

3

4

5

6

7

8

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Lead price

Range code

(b)

0

500

1000

1500

2000

2500

3000

3500

4000

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Zinc price


(c)

0

1

2

3

4

5

6

7

8

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Zinc price

Range code

(d)




119865 minus 119868 = (17561955 24060165 31322673)minus (12000000 13000000 15000000)

= (2561955 11060166 19322673) $(80)


4 Conclusion





LEAD ZINC

UNIT OPERATING COST


01234567

0 5 10 15 20 25 30 35 40 45 50 55 60

Rang

e cod

e val

ue

Month

01234567

0 5 10 15 20 25 30 35 40 45 50 55 60

Rang

e cod

e val

ue

Month

0500

100015002000250030003500

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

0500

1000150020002500300035004000

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

4648505254565860

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

0100000200000300000400000500000600000700000800000

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

ℎ

sim

Stimes４= aℎ

1

(９sum

=1

aℎ2 M

mcon

con

sim








0

100000

200000

300000

400000

500000

600000

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month


0

100000

200000

300000

400000

500000

600000

1 2 3 4 5

$t

Year




Data Availability




0

100000

200000

300000

400000

500000

600000

1 2 3 4 5

$t

Year


TMC 1year 4

TMC 4year 5

TMC 5year 3

TMC 3year 1

TMC 2year 2


References
























































Journal of












Journal of







Volume 2018






Volume 2018









173 16 22 49 652 7 15 21 27 35 48 64 811 6 10 14 20 26 34 47 63 80 TMC1

5 9 13 19 25 33 46 62 79 95 TMC2

4 8 12 18 24 32 45 61 78 94 108 TMC3

11 31 44 60 77 93 107 TMC4

23 30 43 59 76 92 106 115 TMC5

29 42 58 75 91 105 11428 41 57 74 90 104 113

40 56 73 89 103 11239 55 72 88 10238 54 71 87 101 11137 53 70 86 100 11036 52 69 85 99 109

51 68 84 9850 67 83 97

66 82 96





Process Value




volatility 018491




volatility 016416





0

01

02

03

04

05

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC3 StdDev Pb

0

02

04

06

08

1

12

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC3 StdDev Zn

0

01

02

03

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC4 StdDev Pb

0

02

04

06

08

1

12

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC4 StedDev Zn

0

01

02

03

04

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

Number of blocks

TMC5 StdDev Pb

002040608

112141618

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

Number of blocks

TMC5 StdDev Zn

0

01

02

03

04

05

06

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC1 StdDev Pb

002040608

112141618

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC1 StdDev Zn

0

01

02

03

04

05

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Number of blocks

TMC2 StdDev Pb

0

02

04

06

08

1

12

14

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Number of blocks

TMC2 StdDev Zn










0

500

1000

1500

2000

2500

3000

3500

4000

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125

US$

t

Month

(a)

0

500

1000

1500

2000

2500

3000

3500

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125

US$

t

Month

(b)


0010203040506070809

1

500 1000 1500 2000 2500 3000 3500$t

VVLVLLM

HVHVVH

(a)

0010203040506070809

1

500 1000 1500 2000 2500 3000 3500 4000$t

VVLVLLM

HVHVVH

(b)




15104734 3944737 3044364 2443462 19244336591226 5356427 4305812 3576349 29263948228167 6904516 5704581 4871250 4096627

25583555 4336349 3366286 2691067 21178617190095 5863841 4730052 3916424 32015418951887 7536746 6246798 5320340 4472001

36213929 4891545 3857917 3139633 25243907923489 6519182 5317593 4456306 36948059796030 8298845 6935676 5958908 5056129

42976598 2108873 1450337 1038819 6860214063918 3133980 2351322 1834430 13747555275030 4267887 3358689 2754948 2200817

56198802 4899506 3882193 3166488 25554757882521 6503582 5321728 4466255 37120859723915 8256167 6917828 5949400 5056656



0

500

1000

1500

2000

2500

3000

3500

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Lead price


(a)

0

1

2

3

4

5

6

7

8

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Lead price

Range code

(b)

0

500

1000

1500

2000

2500

3000

3500

4000

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Zinc price


(c)

0

1

2

3

4

5

6

7

8

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Zinc price

Range code

(d)




119865 minus 119868 = (17561955 24060165 31322673)minus (12000000 13000000 15000000)

= (2561955 11060166 19322673) $(80)


4 Conclusion





LEAD ZINC

UNIT OPERATING COST


01234567

0 5 10 15 20 25 30 35 40 45 50 55 60

Rang

e cod

e val

ue

Month

01234567

0 5 10 15 20 25 30 35 40 45 50 55 60

Rang

e cod

e val

ue

Month

0500

100015002000250030003500

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

0500

1000150020002500300035004000

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

4648505254565860

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

0100000200000300000400000500000600000700000800000

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

ℎ

sim

Stimes４= aℎ

1

(９sum

=1

aℎ2 M

mcon

con

sim








0

100000

200000

300000

400000

500000

600000

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month


0

100000

200000

300000

400000

500000

600000

1 2 3 4 5

$t

Year




Data Availability




0

100000

200000

300000

400000

500000

600000

1 2 3 4 5

$t

Year


TMC 1year 4

TMC 4year 5

TMC 5year 3

TMC 3year 1

TMC 2year 2


References
























































Journal of












Journal of







Volume 2018






Volume 2018









0

01

02

03

04

05

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC3 StdDev Pb

0

02

04

06

08

1

12

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC3 StdDev Zn

0

01

02

03

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC4 StdDev Pb

0

02

04

06

08

1

12

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC4 StedDev Zn

0

01

02

03

04

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

Number of blocks

TMC5 StdDev Pb

002040608

112141618

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

Number of blocks

TMC5 StdDev Zn

0

01

02

03

04

05

06

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC1 StdDev Pb

002040608

112141618

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of blocks

TMC1 StdDev Zn

0

01

02

03

04

05

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Number of blocks

TMC2 StdDev Pb

0

02

04

06

08

1

12

14

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Number of blocks

TMC2 StdDev Zn










0

500

1000

1500

2000

2500

3000

3500

4000

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125

US$

t

Month

(a)

0

500

1000

1500

2000

2500

3000

3500

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125

US$

t

Month

(b)


0010203040506070809

1

500 1000 1500 2000 2500 3000 3500$t

VVLVLLM

HVHVVH

(a)

0010203040506070809

1

500 1000 1500 2000 2500 3000 3500 4000$t

VVLVLLM

HVHVVH

(b)




15104734 3944737 3044364 2443462 19244336591226 5356427 4305812 3576349 29263948228167 6904516 5704581 4871250 4096627

25583555 4336349 3366286 2691067 21178617190095 5863841 4730052 3916424 32015418951887 7536746 6246798 5320340 4472001

36213929 4891545 3857917 3139633 25243907923489 6519182 5317593 4456306 36948059796030 8298845 6935676 5958908 5056129

42976598 2108873 1450337 1038819 6860214063918 3133980 2351322 1834430 13747555275030 4267887 3358689 2754948 2200817

56198802 4899506 3882193 3166488 25554757882521 6503582 5321728 4466255 37120859723915 8256167 6917828 5949400 5056656



0

500

1000

1500

2000

2500

3000

3500

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Lead price


(a)

0

1

2

3

4

5

6

7

8

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Lead price

Range code

(b)

0

500

1000

1500

2000

2500

3000

3500

4000

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Zinc price


(c)

0

1

2

3

4

5

6

7

8

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Zinc price

Range code

(d)




119865 minus 119868 = (17561955 24060165 31322673)minus (12000000 13000000 15000000)

= (2561955 11060166 19322673) $(80)


4 Conclusion





LEAD ZINC

UNIT OPERATING COST


01234567

0 5 10 15 20 25 30 35 40 45 50 55 60

Rang

e cod

e val

ue

Month

01234567

0 5 10 15 20 25 30 35 40 45 50 55 60

Rang

e cod

e val

ue

Month

0500

100015002000250030003500

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

0500

1000150020002500300035004000

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

4648505254565860

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

0100000200000300000400000500000600000700000800000

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

ℎ

sim

Stimes４= aℎ

1

(９sum

=1

aℎ2 M

mcon

con

sim








0

100000

200000

300000

400000

500000

600000

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month


0

100000

200000

300000

400000

500000

600000

1 2 3 4 5

$t

Year




Data Availability




0

100000

200000

300000

400000

500000

600000

1 2 3 4 5

$t

Year


TMC 1year 4

TMC 4year 5

TMC 5year 3

TMC 3year 1

TMC 2year 2


References
























































Journal of












Journal of







Volume 2018






Volume 2018









0

500

1000

1500

2000

2500

3000

3500

4000

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125

US$

t

Month

(a)

0

500

1000

1500

2000

2500

3000

3500

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125

US$

t

Month

(b)


0010203040506070809

1

500 1000 1500 2000 2500 3000 3500$t

VVLVLLM

HVHVVH

(a)

0010203040506070809

1

500 1000 1500 2000 2500 3000 3500 4000$t

VVLVLLM

HVHVVH

(b)




15104734 3944737 3044364 2443462 19244336591226 5356427 4305812 3576349 29263948228167 6904516 5704581 4871250 4096627

25583555 4336349 3366286 2691067 21178617190095 5863841 4730052 3916424 32015418951887 7536746 6246798 5320340 4472001

36213929 4891545 3857917 3139633 25243907923489 6519182 5317593 4456306 36948059796030 8298845 6935676 5958908 5056129

42976598 2108873 1450337 1038819 6860214063918 3133980 2351322 1834430 13747555275030 4267887 3358689 2754948 2200817

56198802 4899506 3882193 3166488 25554757882521 6503582 5321728 4466255 37120859723915 8256167 6917828 5949400 5056656



0

500

1000

1500

2000

2500

3000

3500

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Lead price


(a)

0

1

2

3

4

5

6

7

8

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Lead price

Range code

(b)

0

500

1000

1500

2000

2500

3000

3500

4000

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Zinc price


(c)

0

1

2

3

4

5

6

7

8

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Zinc price

Range code

(d)




119865 minus 119868 = (17561955 24060165 31322673)minus (12000000 13000000 15000000)

= (2561955 11060166 19322673) $(80)


4 Conclusion





LEAD ZINC

UNIT OPERATING COST


01234567

0 5 10 15 20 25 30 35 40 45 50 55 60

Rang

e cod

e val

ue

Month

01234567

0 5 10 15 20 25 30 35 40 45 50 55 60

Rang

e cod

e val

ue

Month

0500

100015002000250030003500

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

0500

1000150020002500300035004000

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

4648505254565860

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

0100000200000300000400000500000600000700000800000

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

ℎ

sim

Stimes４= aℎ

1

(９sum

=1

aℎ2 M

mcon

con

sim








0

100000

200000

300000

400000

500000

600000

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month


0

100000

200000

300000

400000

500000

600000

1 2 3 4 5

$t

Year




Data Availability




0

100000

200000

300000

400000

500000

600000

1 2 3 4 5

$t

Year


TMC 1year 4

TMC 4year 5

TMC 5year 3

TMC 3year 1

TMC 2year 2


References
























































Journal of












Journal of







Volume 2018






Volume 2018









0

500

1000

1500

2000

2500

3000

3500

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Lead price


(a)

0

1

2

3

4

5

6

7

8

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Lead price

Range code

(b)

0

500

1000

1500

2000

2500

3000

3500

4000

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Zinc price


(c)

0

1

2

3

4

5

6

7

8

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

$t

Week

Zinc price

Range code

(d)




119865 minus 119868 = (17561955 24060165 31322673)minus (12000000 13000000 15000000)

= (2561955 11060166 19322673) $(80)


4 Conclusion





LEAD ZINC

UNIT OPERATING COST


01234567

0 5 10 15 20 25 30 35 40 45 50 55 60

Rang

e cod

e val

ue

Month

01234567

0 5 10 15 20 25 30 35 40 45 50 55 60

Rang

e cod

e val

ue

Month

0500

100015002000250030003500

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

0500

1000150020002500300035004000

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

4648505254565860

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

0100000200000300000400000500000600000700000800000

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

ℎ

sim

Stimes４= aℎ

1

(９sum

=1

aℎ2 M

mcon

con

sim








0

100000

200000

300000

400000

500000

600000

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month


0

100000

200000

300000

400000

500000

600000

1 2 3 4 5

$t

Year




Data Availability




0

100000

200000

300000

400000

500000

600000

1 2 3 4 5

$t

Year


TMC 1year 4

TMC 4year 5

TMC 5year 3

TMC 3year 1

TMC 2year 2


References
























































Journal of












Journal of







Volume 2018






Volume 2018









LEAD ZINC

UNIT OPERATING COST


01234567

0 5 10 15 20 25 30 35 40 45 50 55 60

Rang

e cod

e val

ue

Month

01234567

0 5 10 15 20 25 30 35 40 45 50 55 60

Rang

e cod

e val

ue

Month

0500

100015002000250030003500

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

0500

1000150020002500300035004000

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

4648505254565860

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

0100000200000300000400000500000600000700000800000

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month

ℎ

sim

Stimes４= aℎ

1

(９sum

=1

aℎ2 M

mcon

con

sim








0

100000

200000

300000

400000

500000

600000

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month


0

100000

200000

300000

400000

500000

600000

1 2 3 4 5

$t

Year




Data Availability




0

100000

200000

300000

400000

500000

600000

1 2 3 4 5

$t

Year


TMC 1year 4

TMC 4year 5

TMC 5year 3

TMC 3year 1

TMC 2year 2


References
























































Journal of












Journal of







Volume 2018






Volume 2018









0

100000

200000

300000

400000

500000

600000

0 5 10 15 20 25 30 35 40 45 50 55 60

$t

Month


0

100000

200000

300000

400000

500000

600000

1 2 3 4 5

$t

Year




Data Availability




0

100000

200000

300000

400000

500000

600000

1 2 3 4 5

$t

Year


TMC 1year 4

TMC 4year 5

TMC 5year 3

TMC 3year 1

TMC 2year 2


References
























































Journal of












Journal of







Volume 2018






Volume 2018


























































Journal of












Journal of







Volume 2018






Volume 2018








long-term room and pillar mine production planning based...

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