hierarchical production planning and scheduling in a multi-product, multi-machine environment

This article was downloaded by: [Florida State University]On: 26 September 2014, At: 04:15Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number:1072954 Registered office: Mortimer House, 37-41 Mortimer Street,London W1T 3JH, UK

International Journal ofProduction ResearchPublication details, including instructions forauthors and subscription information:http://www.tandfonline.com/loi/tprs20

Hierarchical productionplanning and schedulingin a multi-product, multi-machine environmentM. M. Qiu & E.E. BurchPublished online: 15 Nov 2010.

To cite this article: M. M. Qiu & E.E. Burch (1997) Hierarchical productionplanning and scheduling in a multi-product, multi-machine environment,International Journal of Production Research, 35:11, 3023-3042, DOI:10.1080/002075497194273

To link to this article: http://dx.doi.org/10.1080/002075497194273

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy ofall the information (the “Content”) contained in the publicationson our platform. However, Taylor & Francis, our agents, and ourlicensors make no representations or warranties whatsoever as to theaccuracy, completeness, or suitability for any purpose of the Content.Any opinions and views expressed in this publication are the opinionsand views of the authors, and are not the views of or endorsed byTaylor & Francis. The accuracy of the Content should not be reliedupon and should be independently verified with primary sources ofinformation. Taylor and Francis shall not be liable for any losses,actions, claims, proceedings, demands, costs, expenses, damages,and other liabilities whatsoever or howsoever caused arising directlyor indirectly in connection with, in relation to or arising out of the useof the Content.

http://www.tandfonline.com/loi/tprs20

http://www.tandfonline.com/action/showCitFormats?doi=10.1080/002075497194273

http://dx.doi.org/10.1080/002075497194273

This article may be used for research, teaching, and private studypurposes. Any substantial or systematic reproduction, redistribution,reselling, loan, sub-licensing, systematic supply, or distribution in anyform to anyone is expressly forbidden. Terms & Conditions of accessand use can be found at http://www.tandfonline.com/page/terms-and-conditions

Dow

nloa

ded

by [

Flor

ida

Stat

e U

nive

rsity

] at

04:

15 2

6 Se

ptem

ber

2014

http://www.tandfonline.com/page/terms-and-conditions

http://www.tandfonline.com/page/terms-and-conditions

int. j. prod. res., 1997, vol. 35, no. 11, 3023± 3042

Hierarchical production planning and scheduling in a multi-product,multi-machine environment

M. M. QIU² * and E. E. BURCH³

A hierarchical production planning (HPP) and scheduling model is developed tosolve a real-world problem in ® bre manufacturing scheduling. The problem understudy requires determining production sequences in the presence of variable setupcosts in a multi-machine and multi-product environment. The concept of expectedsetup cost is developed to reduce the di� culty of addressing sequence-dependentsetup costs at the aggregate level. A mixed integer linear programming model(MILP) incorporates the logic of expert systems to minimize the number andcosts of machine setups and other associated costs. At the daily operationallevel, a set of heuristics is developed to manage contingencies, such as machinebreakdowns and changes in demand. The heuristics e� ectively utilize idle capacityand inventories to smooth production and avoid unnecessary extra setups,therefore maintaining system stability. Data provided by the ® bre plant is usedto validate the model. Compared to the actual schedule, results from the modelare more consistent with managerial priorities and substantial cost savings areobtained.

1. Introduction

In this article a very complex, real-world production planning and schedulingproblem is described. An hierarchical scheduling model is developed to solve theproblem. The model incorporates the use of mathematical programming, the logic ofexpert systems and the enormous computational power of the computer to solve theproblem.

The idea of hierarchical production planning (HPP) and scheduling was initiatedby Hax and Meal (1975). HPP has an intuitive appeal because management decisionmaking processes in the manufacturing environment have a natural hierarchicalstructure (Anthony 1965, Hax and Bitran 1979). the HPP approach recognizesand represents the planning process by a series of mathematical models or heuristics.The HPP model partitions the decision process into modules or subproblems withdi� erent planning time horizons. It also aggregates and disaggregates informationthrough the various hierarchical levels (Hax and Meal 1975). To ensure e� ectivedecision making, a strong linkage must exist between these models at each hier-archical level. As Bitran and Hax (1977) emphasize, the thrust of building an HPPmodel involves aggregating the product and investigating the interaction between thehierarchical stages. HPP can use either optimization models or heuristics at any levelin decision making. The classic models at the aggregate level are the l̀inear decisionrule’ (LDR) (Holt et al. 1955), the l̀inear programming model’ (LP) (Hansmann and

0020± 7543/97 $12. 00 Ñ 1997 Taylor & Francis Ltd.

Received November 1996.² School of Business and Economics, Longwood College, Farmville, VA 23909, USA.³ MBA Program, Clemson University, Clemson, SC 29634, USA.* To whom correspondence should addressed.

Dow

nloa

ded

by [

Flor

ida

Stat

e U

nive

rsity

] at

04:

15 2

6 Se

ptem

ber

2014

Hess 1960), the `management coe� cients model’ (MCM) (Bowman 1963),`parametric production planning’ (PPP) (Jones 1967), the `search decision rule(SDR) (Taubert 1968), and the `production switching heuristic’ (Mellichamp andLove 1978). There are also numerous extensions of these models presented in theliterature (Bergstrom and Smith 1970, Lasdon and Terjung 1979, Zipkin 1986).

Traditionally optimization models, especially linear programming (LP) and inte-ger programming models, predominate in the ® eld of operations research. In recentyears, expert systems have been integrated into mathematical models to handlejudgmental decisions. Expert systems apply human expertise to solve complex prob-lems. Incorporating qualitative components in HPP is crucial when human judgmentis required and objectives of decision-making are fuzzy. By comparing and contrast-ing the characteristics of optimization models and expert systems, Simon (1987)suggests the synthesis of these two. Mertens and Kanet (1986), Elam andKonsynski (1987), Baker (1990), and Reinschmidt et al. (1990) support this view.

2. Problem background

2.1. The hierarchical planning systemThe ® bre plant, whose production planning and scheduling system is studied, is a

manufacturing division of a large chemical company. The hierarchical ¯ ow of theproduction planning and control system is displayed in Fig. 1. The productionplanning and customer service (PPCS) department receives customers’ orders.Based on orders on hand, demand forecasts and historical data, each October theplanner prepares a one year budget which projects quarterly production, shipmentand inventory levels, and monthly production allocations by product. This yearlybudget plan is revised every quarter. The scheduler is responsible for assigningindividual products to machines for each month. Every Wednesday the manufactur-ing plant and the PPCS department agree on a feasible schedule for the upcomingweek. Often the weekly plan has to be modi® ed because of such contingencies asmachine breakdowns, changed order quantities, new orders, changes in due date,and raw materials shortage.

2.2. Manufacturing processesThe primary ® bre yarn manufacturing process which controls the entire produc-

tion process is spinning. Spinning machines are fed with polymer chips through ahopper (Fig. 2). The chips are melted in a spinning vessel by electrically heatedsteam. The molten polymer ¯ ows through a grid into a pool and, after passing aseries of meshes ® lled with pure sand, goes to the spinneret. The melt is pumpedthrough the system and on contact with air solidi® es immediately. A cross air blast isutilized to assist the solidi® cation of nylon. On exit from the spinneret the ® bre iscompletely free of water and is then treated with steam in a chamber. It is ® nallywound on bobbins. A spinneret has a speci® c number of holes, which equals thenumber of counts of the ® lament.

The chemical ® bre plant provides nylon ® bres for both industrial and domesticcarpets. The product line under study produces approximately thirty di� erent ® breyarns. They are made from three types of polymer chips: Types A, C and L, whichdi� er in degrees of transparency. Products can be grouped into families according tothe polymer chips they use. Changeovers to and from Polymer A products are mostcostly. Therefore, Polymer A products belong to one family and Polymer C and Lproducts belong to another family. Changeovers within a family (e.g. from Polymer

3024 M. M. Qiu and E. E. BurchD

ownl

oade

d by

[Fl

orid

a St

ate

Uni

vers

ity]

at 0

4:15

26

Sept

embe

r 20

14

C to Polymer L) are the next most costly. So, there are sub-families within the non-Polymer-A families.

From a manufacturing point of view, production is a batch process in a make-to-order environment. Production takes place on parallel, but non-identical, spinningmachines. Capacity is limited and in¯ exible. The total ® nished goods inventorycapacity (or target) equals approximately one and a ® fth months’ production.Inventory is reviewed on a continuous basis.

The ® bre manufacturing plant has four spinning machines: one Type-G and threeType-B. The Type-G machine is a newer model with twenty positions. A package of® bre is spun on each position (see Fig. 2). Each Type-B machine has twenty-eightpositions. Every spinning machine is divided into two splits, which are groups of

An HPP and scheduling model: a real world application 3025

Figure 1. Hierarchical planning ¯ ow.

Dow

nloa

ded

by [

Flor

ida

Stat

e U

nive

rsity

] at

04:

15 2

6 Se

ptem

ber

2014

positions. Each split can produce only one product at a time, but does not have torun at full capacity. Capacities of the spinning machines are as follows:

Type-G machine Type-B machinesSplit ID 1 2 3 4 5 6 7 8Number of positions 16 4 16 12 20 8 16 12

2.3. Machine setupMachine setup is an important issue in production control. Setup times and costs

are production sequence-dependent. Each setup a� ects temperature, which is acrucial environmental factor a� ecting ® bre quality; therefore, it is desirable toreduce the number of setups for both costs and quality reasons.

The di� erences in ® bre yarns require four signi® cantly di� erent levels of machinesetups (see Table 1):

(1) Changing positions. If the decision is made either to change polymer type toPolymer A or from Polymer A to another polymer type, the machine must bethoroughly cleaned. A supply hopper, a spinning vessel and a grid in each

3026 M. M. Qiu and E. E. Burch

Figure 2. Diagrammatic representation of a nylon spinning machine position (adapted fromMoncrie� 1975).

Dow

nloa

ded

by [

Flor

ida

Stat

e U

nive

rsity

] at

04:

15 2

6 Se

ptem

ber

2014

position are replaced with new ones. Making these positional changes takesfour hours and costs an average of $1868 per split.

(2) Changing polymer. When polymer ¯ ake is changed, the ¯ ake residue fromthe previous operation must be cleaned. Then the transfer hopper feeds thenew chips to the spinning machine. Changing polymer takes two hours andcosts an average of $700 per split. The spinneret packs must be changed when¯ ake is changed.

(3) Changing packs. A spinneret change is required when the production ischanged over to a product using the same polymer ¯ ake and ® nish as thepreviously scheduled product, but having a di� erent ® lament count.Changing packs takes about one and a half hours and costs $686 per spliton average.

(4) Changing ® nish. Finish change involves changing oil in the nylon polymerpool in the spinning vessel. It takes approximately thirty to forty-® ve min-utes. This changeover cost is ignored, since it is relatively small.

Among the above four types of changes, changing positions is by far the mostexpensive and time consuming. Changing positions is required when productionchanges to or from Polymer A products. Therefore schedulers prefer to dedicatecertain splits to Polymer A products to reduce machine setup costs.

There are restrictions to product assignments and preferences for assigning spe-ci® c products to speci® c spinning machines. For these reasons any given product canonly be produced on two or three splits. Among the suitable splits for a givenproduct, one split may be preferred due to signi® cantly better quality.

2.4. Decision making processThe following decisions must be made by the schedulers for each ® bre yarn

product.

� Determination of a yearly production budget

� Determination of quarterly production, shipment, and inventory levels

� Determination of monthly production allocations

� Determination of spinning machine assignments for the upcoming month

� Determination of spinning machine schedules for the upcoming week, if con-tingency plans are necessary

Currently planning and scheduling at the plant are done subjectively, or by g̀utfeeling’. In 1993 this product line averaged approximately four setups every week,


Type ofchanges

Changingpositions

Changingpolymers

Changingpacks

Changing® nish

Other changesdone

simultaneously

Changingpolymers,packs, and

® nish

Changingpacksand

® nish

Changing® nish

None

Time required 4 hours 2 hours 1.5 hours 0.5 hours

Cost/split $1868 $700 $686 Ignored

Table 1. Four categories of spinning machine changeovers.

Dow

nloa

ded

by [

Flor

ida

Stat

e U

nive

rsity

] at

04:

15 2

6 Se

ptem

ber

2014

resulting in an annual setup cost of over $100 000. Direct costs include the cost oflabour for change-over and new parts of the equipment. Additionally intangiblecosts such as lost production and ® bre waste are incurred. The plant managementwishes to signi® cantly reduce the number of machine setups.

3. Methodology

Figure 1 illustrates the relationship between the organizational planning levelsand the proposed model’s hierarchical structure. Production quantities and inven-tory levels for each product are determined at the aggregate level of the model. Theyearly budget plan contains four quarterly production, shipment and inventorylevels and twelve monthly production allocations. In October, an aggregate MixedInteger Linear Programming (MILP) model can be run with a twelve month horizonto obtain a monthly production quantity for each product. Quarterly production,shipment and inventory levels for each product are obtained by summing the appro-priate monthly ® gures. The projected twelve months’ total production quantity foreach product is next year’s budget for that product. At the beginning of each quarterthe aggregate MILP model is run to obtain an updated quarterly plan.

A hierarchical planning system partitions overall planning and scheduling prob-lems into various levels of modules or subproblems (Hax and Meal 1975). Typicallyproduction quantities and inventory levels for product families are determined at theaggregate level (Leong et al. 1989). Family production schedules are determined atthe second level, i.e. the disaggregate level. Items within the families are sequenced atthe third level.

The conceptual modules of the proposed model, which is a deviation of thetypical HPP model are illustrated in Fig. 3. The aggregate level of the model deter-mines production quantities, inventory levels, and assigns production to machines(splits). Once production quantities have been determined and assigned to machines,monthly production sequences are developed at the dissaggregate level. Thesesequences actually determine the daily production sequence. Thus the daily opera-tional schedule is determined at the disaggregate level, but modi® ed, if necessary, atthe third level.

Due dates are not considered at the aggregate level, but at the disaggregate level.In general, if due dates are not considered at the higher level, they may con¯ ict withthe production quantities and the schedule for their families. This may cause unfea-sibility at the disaggreagate level. In this case, however, this is not a serious problem.In the chemical ® bre industry, demand is relatively stable and ® rms carry a consider-able amount of inventory. Most due dates at the ® bre plant are for the end of the


Figure 3. The conceptual modules of the proposed model.

Dow

nloa

ded

by [

Flor

ida

Stat

e U

nive

rsity

] at

04:

15 2

6 Se

ptem

ber

2014

month. Occasionally customers request early due dates. If any early due dates docause unfeasibility at the disaggregate level, the heuristics at the third level of themodel will resolve the problem.

At the third level of the model, heuristics are proposed to cope with unfeasibilityat the disaggregate level. The heuristics are also used to manage contingencies suchas new orders, increased order quantities, earlier due dates, machine breakdowns andraw materials shortage.

3.1. The aggregate modelAt the aggregate level the primary objective is to minimize the number and cost

of machine setups while satisfying demand. Very few applications in the literatureconsider setups at the aggregate level. Several researchers consider setups at thedisaggregate level (e.g. Oli� and Burch 1985, Toczylowski 1986, Mohanty andKulkarni 1987, Saad 1990). Hax and Golovin (1978) examine whether ignoringsetup costs at the aggregate planning level introduces suboptimization possibilities.They ® nd that high setup costs can a� ect the performance of the system.

Variable setup costs are basically sequence dependent and subject to constraintssuch as demand, capacity and due dates. At the aggregate level, when productionsequences are not determined, it is impossible to calculate the exact setup costs.

A new approach is proposed at this level: using a linear programming model tomimic human thinking. The aggregate model incorporates the logic of expert systemsto achieve managerial objectives. McBride and O’Leary (1993) de® ne four aspects ofthe interface between mathematical programming (MP) and expert systems (ES):(1) using ES to facilitate the use of MP, (2) using MP to ensure that ES generategood solutions, (3) formulating ES as MP, and (4) formulating parts of MP algo-rithms as ES problems.

Dhar and Ranganathan (1990) view rules (i.e. preferences and constraints) as aset of logical propositions. Dantzig (1963) in this classic book L inear Programmingand Extensions illustrates how logic propositions can be modelled as 0± 1 integerprograms: 1 denotes the condition is true, and 0 denotes the condition is false. Inthis way, the inference procedure becomes a symbolic calculation and manipulationof symbols (Hooker 1988).

The new approach uses optimization models to re¯ ect rules. Rules (or priorities)are applied in the objective function in terms of penalties, which implicitly minimizecosts. Instead of explicitly minimizing setup costs, the model implicitly minimizes thenumber and cost of machine setups. Instead of manipulating the productionsequences, the model manipulates the production system s̀tate’ . The LP approachassigns penalties to the undesired value of the system state variables and re¯ ects thepriorities of the objectives. According to management, machine preference has thehighest priority. Products are assigned to preferred splits for quality and productivityreasons. Also the scheduler prefers to dedicate certain splits to certain productfamilies in order to reduce setup costs. The ideal situation is a split dedicated to asingle product. The second priority is reducing setup costs, and the third priority is tomeet the total inventory target.

The notation used in the aggregate model follows.

Indices:i,L,p product, split, time period (or month) indices

An HPP and scheduling model: a real world application 3029D

ownl

oade

d by

[Fl

orid

a St

ate

Uni

vers

ity]

at 0

4:15

26

Sept

embe

r 20

14

Parameters:P1,P2 penaltiesP1 0 if product i is produced on preferred split L

9000 otherwiseP2 0 if inventory shortage Sp = 0

3000 otherwiseCLp total production capacity (in days) of split L in period pChi inventory holding cost in dollars per pound for product iRiL daily production rate in pounds of product i on split LNL number of positions on split LChÂiL Chi ´ RiL ´ NL= inventory holding cost in terms of product i produced on

split LCsiL setup cost of product i on split LDip demand (in pounds) for product i during period pIc total inventory target (in pounds) for period p

Decision variables:XiL p production quantity of product i (in days) on split L during period pY iL p 1 if XiL p > 0

0 otherwiseIiL p inventory level of product i (in days of production on split L) for period pSp inventory shortage in period p

3.1.1. Objective function

Min åN

i= 1 åM

L= 1 åT

p= 1

P1XiL p + åN

i= 1 åM

L= 1 åT

p= 1

CsiL Y iL p + åN

i= 1 åM

L= 1 åT

p= 1

ChÂiLIiL p + åT

p= 1

P2Sp

The objective function consists of production, setup, and inventory components.Production component: Actual production costs are omitted in the aggregate

model. Since the plant uses a standard production cost for each product, productioncosts are constants in the objective function. A decision variable XiL p represents thenumber of days that split L is devoted to producing product i. XiL p also serves as asystem state variable which identi® es if a product is assigned to its preferredmachine. A penalty is imposed on XiL p if a product is assigned to its non-preferredmachine. The continuous version of XiL p has considerable computational advantagesover the 0± 1 version.

Setup component: Kannan and Lyman (1992) show that family selection basedon individual job characteristics is as e� ective as selection based on family charac-teristics. Group technology heuristics are proposed to recognize similarities in setuprequirements in order to reduce setup frequency and costs (Vaithianathan andMcRoberis 1982, Mosier et al. 1984, Kannan and Lyman 1994).

Setup costs are sequence dependent. When production sequences are unknown,exact setup costs cannot be determined. In this study two classes, standard productsand other products, are used to de® ne expected setup costs. Standard products have apreferred machine. Other products are those that can be assigned to more than onemachine without preference. At the aggregate level although exactly what produc-tion will take place on each split is not known, the expected setup cost associated


ownl

oade

d by

[Fl

orid

a St

ate

Uni

vers

ity]

at 0

4:15

26

Sept

embe

r 20

14

with the splits is used to select a preferred split for an other product. The expectedsetup cost of the other product is calculated in proportion to the probability that thestandard products will be run on this particular split. For example, three standardproducts prefer split 1 and their annual forecast demands are 200, 300 and 500pounds respectively. The changeover costs from the standard products to theother product are $700, $686 and $686 respectively. Thus the expected setupcost of the other product on this particular split is $688.8 (or 700(0.2) +686(0.3) + 686(0.5)).

At ¯ at setup cost which is lower than all the other product setup costs is used forevery standard product to prevent extraneous setups over the planning horizon.Assuming the production of standard products will take place on the preferredsplits, expected setup costs are used to group the other products around the standardproducts such that the total setup cost can be reduced.

The second state variable, Y iL p, penalizes the state of one product being producedon more than one machine in the same time period. Actual setup costs of standardproducts are not included in the objective function because they are sequence-depen-dent and unknown at this stage.

Inventory component. A decision variable IiL p represents the amount of inven-tory of product i (in days of production on split L) at the end of period p. Theobjective at the aggregate level is to reduce the number and cost of setups.Minimizing setups con¯ icts with minimizing inventory. To ensure that reducingsetup costs has higher priority than reducing inventory costs and to take intangiblesetup costs into consideration, a weight (0.01) is imposed on inventory costs to makethem close to zero. Since inventory shortage will cause unfeasibility at the lowerlevels of the model and instability of the schedule, an inventory shortage variable Sp

(in units of pounds) is forced to be zero.

3.1.2. ConstraintThe constraints are of three functional types. Production constraints balance

demand and inventory requirements:

Ii0 + åM

L= 1RiLXiL - å

M

L= 1RiLIiL = Dil for all i

åM

L= 1RiLIi,L ,p- 1 + å

M

L= 1RiLXiL p - å

M

L= 1RiLIiL p = Dip for all i, p= 2,3, ...

The ® rst constraint is used for month 1 of the planning horizon because the begin-ning inventory is recorded in pounds. The second constraint is for the rest of theperiods of the planning horizon.

Capacity constraints limit production quantities and inventory levels during eachperiod:

åN

i= 1

XiL p £ CLp for all L, p.

The aggregate inventory target constrains the use of inventories to smooth pro-duction. The inventory target Ic is set based on the physical warehouse capacity.


ownl

oade

d by

[Fl

orid

a St

ate

Uni

vers

ity]

at 0

4:15

26

Sept

embe

r 20

14

åN

i= 1 åM

L= 1RiLIiL p + Sp = Ic for all p.

The inventory shortage variable Sp is utilized to avoid unfeasibility at the lower levelof the model and maintain stability of the schedule. If we use

åN

i= 1 åM

L= 1RiLIiL p £ Ic

as an inventory constraint, the model will ® rst use up inventories to minimize thetotal cost.

Finally a set of constraints converts all XiL p into 0± 1 integer variables so that inthe objective function setup costs can be imposed on machine setups:

M Y iL p ³ XiL p for all i, L, p

M is a number big enough to force Y iL p = 1 when XiL p /= 0.

3.1.3. Priorities and weighting scaleHeavy penalties are imposed on products assigned to non-preferred splits and to

inventory shortages. Values of P1 and P2 are chosen under two considerations. First,they re¯ ect the intangible costs that are signi® cantly higher than the other costs.Secondly, between machine preference and schedule stability, the former has a rela-tively higher priority. In this application the weighting was ® ne-tuned (Franz 1989)with historical data to achieve satisfactory solutions.

The priorities are not rigid; they can be overridden by the corresponding con-straints to avoid unfeasibilities. For example, when restricted by capacity, the modelmay either split a production batch into two or more lots to be produced on di� erentsplits at the same time, or assign production to a non-preferred split.

3.2. The disaggregate modelGiven forecast demand, machine capacities, and the total inventory target, the

aggregate model determines the monthly lot size, machine (or split) assignment andthe ending inventory level for each individual product. Once the products have beenassigned to the splits, it is possible to run a set of network models to obtain the dailyschedule of the splits separately. In other words, at the aggregate level the modelgroups products onto the splits in the light of reducing setups. At the disaggregatelevel the model determines the sequence in which products are produced on theirassigned splits. Thus the daily production schedule for the month is determined atthis level.

The notation used in the disaggregate model follows.

Indices:i, j product indexL split indexp time period (or day) index

Parameters:Di demand for product i (in days)CL total production capacity (in days) of split LCij changeover cost (in dollars) from product i to product j


ownl

oade

d by

[Fl

orid

a St

ate

Uni

vers

ity]

at 0

4:15

26

Sept

embe

r 20

14

Decision variables:

Y iL p 1 if production of product i takes place on split L on day p0 otherwise

ZijL p 1 if a changeover from product i to product j occurs on split L on day p0 otherwise

Min åN

i= 1 åN

j= 1 åM

L= 1 åT

p= 1

CijZijL p

s.t.

åM

L= 1 åT

p= 1

YiL p = Di for all i (1)

åN

i= 1 åT

p= 1

Y iL p = CL for all L (2)

åN

i= 1

YiL p = 1 for all L, p (3)

Y iL p = åN

j= 1

ZijL p for all i,L, p (4)

Y j,L p- 1 = åN

j= 1

ZjiL p for all i, L, p (5)

åN

i= 1 åN

j= 1

ZijL p = 1 for all L, p (6)

YiL p Î (0,1) for all i, L, pZijL p Î (0,1) for all i, j, L, p

The objective function minimizes total setup cost. The model uses the output ofthe aggregate model, (i.e. desired production quantities Di and machine assignments)and the initial machine status (i.e. which products were last run in the previousperiod) as input.

All decision variables are 0± 1 integer variables. The variable ZijL p is de® ned as achangeover from product j to product i on split L at the beginning of day p.ZijL p = 1, if the changeover occurs; ZijL p = 0, otherwise. Since a production change-over can occur any weekday, a 0± 1 variable is de® ned for the possible changeover ofeach day.

Constraint (1) balances the production. Demand in constraint (1) is the produc-tion rate determined at the aggregate level. Constraint (2) is the capacity limitation.Constraint (3) restricts the splits to producing only one product at a given point intime. To change from product j to product i at the beginning of day p requires twoconditions: (a) production of product i takes place on split L during day p (constraint4); (b) Product j must be the product run on split L during the previous period


ownl

oade

d by

[Fl

orid

a St

ate

Uni

vers

ity]

at 0

4:15

26

Sept

embe

r 20

14

(constraint 5). Note that ZiiL p = ZijL p, when i = j. Constraint (6) restricts each splitto one change-over per day.

3.3. The daily operation heuristicsDue to unpredictable daily contingencies, however, sometimes the daily schedule

developed at level two must be modi® ed to deal with these contingencies. Suchcontingencies include machine breakdowns, changes in orders and raw materialsshortage. The heuristics developed in this section address these contingencies bythe use of idle capacity and inventory to smooth production and avoid unnecessarysetups.

A set of heuristics is developed to facilitate planning for these contingencies. Asneeded, the scheduler can use any of the heuristics developed for a machine break-down, an increased order quantity, an earlier due date, raw material shortage and anew order. Each of the heuristics employs the same logic. They are very similar withonly minor modi® cations. The heuristics for machine breakdown is used as anexample.

When a machine breaks down, capacity is reduced. The scheduler follows severalguidelines to adjust the day-to-day operational schedule:

(1) Use machine idle and operation slack times. If there is enough machine idletime and all the due dates can be met, the scheduler simply postpones theoriginal plan.

(2) Use inventory. If there is not enough idle capacity or any due date cannot bemet, the scheduler will use inventory to satisfy demand. Production to replacethe consumed inventories will be scheduled at the nearest feasible future time.This production will be treated as a new order and assigned a due date at theend of the future period.

In essence this is a combinatorial optimization problem. A set of multiple objec-tive models is developed to solve the problem. The primary objective is to maximizethe number of products whose demand can be satis® ed either by producing in thescheduled sequence or by utilizing their inventories. The secondary objective is tominimize the number of products whose inventory is used. These two subproblemshave common restrictions such as meeting due dates with the reduced capacity. Ifthey are solved individually, the solutions may con¯ ict with each other and are thenmeaningless.

The multi-objective models combine these subproblems so that a set of consistentsolutions can be achieved. Once an MILP model is solved at the higher level, thesolution becomes one of the constraints of the lower level models. All the otherconstraints at the higher level are passed on to the lower level models. There canbe as many subproblems as managerial objectives.

A set of 0± 1 integer linear programming models is used, and an alternative logicbased on the experienced scheulers is adopted. The 0± 1 LP model ® rst checks if allthe due dates can be met by exhausting inventories available to reduce the products’processing times. Then it checks how much inventory has to be used. The LP modeltries to focus on using fewer products’ inventory. The model yields a solution con-sistent with the schedulers’ objectives.

The heuristic to reschedule for machine breakdowns is presented below:


ownl

oade

d by

[Fl

orid

a St

ate

Uni

vers

ity]

at 0

4:15

26

Sept

embe

r 20

14

Step 1. Calculate the remaining processing time for the product running at the timeof machine break-down. Remaining processing time equals the project com-pletion time minus the time at which the machine breaks down.

Step 2. Check how much inventory is available for every product in the sequence.Determine the amount of inventory that is available to substitute for theproduct’s processing time. Since there may be a need to make up for theinventory, the 0± 1 LP model will determine whether the inventory has to beused at this point.

Step 3. Using multiple objective 0± 1 integer programming models, determine whichproducts can remain in the scheduled sequence, which products’ inventoryneeds to be used and which products have to be taken out of the scheduleand to be reassigned. Capacity is expressed in terms of machine time (indays) available. The results from the higher level LP models are carried overto the lower level models as part of their constraints.

The primary objective. The primary objective is to maximize the number of productswhose demand can be satis® ed either by production on this scheduled split or byutilizing their inventory.

The following notations are used in the model:

Parameters:Pi Processing time required or remaining processing time of product i (in

days).Ai Inventory available to reduce processing time for product i (in days)Ai £ Pi The inventory used to reduce the product’s processing time should not

exceed the product’s required processing timeN Number of products a� ected by the break-downDDi Due date of product iDdn The day at which a machine breaks downTdn The estimated machine down time (in days).

Decision variables:Xi 1 Demand of product i can be satis® ed either on split L or by inventory

0 otherwiseIi 1 Inventory of product i is used to reduce its processing time

0 otherwise

Max åN

i= 1

Xi (1)s.t.

(P1XI - A1II) £ DD1 - Ddn - Tdn

å2

i= 1

(PiXi - AiIi) £ DD2 - Ddn - Tdn

..

.

åN

i= 1

(PiXi - AiIi) £ DDN - Ddn - Tdn

Xi ³ Ii for all the products affected by the breakdown on this split.


ownl

oade

d by

[Fl

orid

a St

ate

Uni

vers

ity]

at 0

4:15

26

Sept

embe

r 20

14

(PiXi - AiIi) is the required processing time for product i. If Xi = 1 and Ii = 0,(PiXi - AiIi) = Pi; if Xi = 1 and Ii = 1, (PiXi - AiIi) = Pi - Ai. Constraining Xi ³ Ii

assures that the inventory is used to substitute for the corresponding product’sprocessing time. The production sequence is ® xed. Starting from the second productin the sequence, the total processing time required (i.e. including all predecessors’processing time but excluding the time satis® ed by their inventory) must meet the duedate requirement.

The secondary objective. The objective function of the second model is to minimizethe number of products whose inventories are being used..

Once the maximum number of products whose demand can be satis® ed in thisscheduled sequence has been obtained, the next consideration is to utilize whichproduct’s inventory. The chance to improve the primary objective exists becauseof possible alternative optimum solutions, which means, physically, di� erent prod-ucts’ inventory can be used while maintaining the primary optimal solution. Ourobjective is to use those products’ inventory which will result in the minimumnumber of products whose inventories are being used.

Max åN

i= 1

Ii (2)

s.t.

åN

i= 1

Xi = Mmax

å1

i= 1

(P1Xi - A1Ii) £ DD1 - Ddn - Tdn

å2

i= 1

(PiXi - AiIi) £ DD2 - Ddn - Tdn

..

.

åN

i= 1

(PiXi - AiIi) £ DDN - Ddn - Tdn

Xi ³ Ii for all i

Xi Î (0,1),Ii Î (0,1).

The number of products whose demand can be satisi® ed either by producing inthis sequence or by utilizing their inventory, Mmax , is the solution from model (1).

3.4. Further discussion on multiple period schedulingThe above heuristic for machine breakdown only considers a single period. When

implementing a multiple period production plan, the scheduler may have more roomto use idle capacity and available inventories over the longer time frame to maintainthe system stability. For a multiple planning period scheduling, the following rulesare necessary supplements for applying the heuristic:


ownl

oade

d by

[Fl

orid

a St

ate

Uni

vers

ity]

at 0

4:15

26

Sept

embe

r 20

14

(Note: The scheduler begins to examine his schedule for next month in the middle ofthe month. So in the supplementary rules, the next month is referred to as the currentmonth.)

(1) Before Step 1 of the heuristic, create a capacity cushion. Use a portion of theidle capacity (10 days at most, according to experience from machine down-time) in the last period to start producing this period’s demand once theassignment for last period is completed. From experience, the schedulerwill not advance the production if there is 15 days or more idle capacity inthis period.

(2) Inventory policy. In addition to Step 2 of the heuristic, according to theproduction balance equation, the production quantity can be partitionedinto two parts ± to satisfy demand and to change inventory level. That is:

P = D + D Iwhere P = Production quantity

D = DemandD I = Ending inventory - Beginning inventory

As long as the reduced ending inventory is greater than or equal to zero forthis period and each of the following periods, demand can be satis® ed.Therefore, the amount of inventory available to substitute for the corre-sponding product’s processing time should be either less than or equal tothe processing time or an amount that will not bring the ending inventorylevels in this and following periods below zero, whichever is smaller.

(3) If ending inventory is greater than or equal to zero in this period but will beshort in any following period, use idle capacity available to make up theconsumed inventory.

The above set of rules is a supplement to all the heuristics dealing with contin-gencies when they are used for multiple period scheduling. Unless necessary, thescheduler will not run the aggregate and disaggregate model more than once everythree months.

4. Model validation

The proposed hierarchical model was developed with historical data provided bythe ® bre plant. Another set of data was obtained from the plant to validate theproposed model. The model’s ability was tested using the plant’s actual daily opera-tional schedule during January± April, 1994.

Results show that the aggregate level of the model assigns products to splits atleast as well as the experienced scheduler. The comparison of machine assignments isdisplayed in Table 2. Eleven out of seventeen standard products (i.e. products withmachine preference) were assigned to their preferred split by the model, while onlynine were assigned to their preferred splits by the experienced scheduler. The pro-duction lot of Product 18 was split onto Split 3 and Split 8 in Month 2 in the actualschedule, but it did not happen when the model was used.

Results show that the model is e� ective in utilizing idle capacity and inventory inaddition to its ability for machine assignment. This resulted in a signi® cant reductionin the number of setups. The entire model developed a schedule (see Fig. 4) that


ownl

oade

d by

[Fl

orid

a St

ate

Uni

vers

ity]

at 0

4:15

26

Sept

embe

r 20

14

required 20 setups at a cost of $13 824 during a four month period. The schedulers’actual schedule (see Fig. 5) used 39 setups at a cost of $22 448. This amounts to anexpected annual savings of $25872 (savings in intangible costs were not included).

Although reducing inventory carrying costs was not our primary goal, we exam-ined the impact of the proposed model on total inventory carrying cost. The endinginventory levels given by the model were more consistent with the inventory target.Since the beginning and ending inventory levels were set to the actual inventorylevels for testing, only the ® rst three months’ resulting ending inventory levelswere compared with actual inventory levels. The average monthly inventory givenby the model was 2400 167 pounds versus 2 612 867 pounds actual monthly averageinventory level. Consequently the model’ s total inventory carrying cost for the ® rstthree months period was $209 400 compared to $229 600 actual total inventorycarrying cost. This amounts to an expected annual savings of $80 800.

5. Conclusions

This paper demonstrates the combination of a hierarchical production planningand scheduling framework, optimization models and the logic of expert systems tosolve a complex real world scheduling problem. A unique contribution of thisresearch is the use of the concept of expected setup costs to consider sequencedependent setup costs at the aggregate level of the model. An optimization modelincorporates the logic of expert systems to assign products to machines at the aggre-gate level. Then at the disaggregate level, production is sequenced on each splitseparately to achieve e� ciency. Heuristics are developed at the third level of the


Product ID Preferred splitSplit assigned

by modelSplit assignedby schedulers

1 No preference 5 52 5 5 54 5 5, 6 55 1 1 16 5 5 57 5 5 48 1 1 19 5 6 5

10 6 5, 6 5, 611 1 1 112 6 5 514 No preference 4 4, 615 2 2 216 8 3, 8 7, 818 3 3 3, 819 4 4, 8 4, 820 No preference 4 4, 621 No preference 6 4, 622 No preference 4 323 2 2 224 8 8 7, 825 7 7 7, 8

Table 2. Comparison of results between model’s and actual daily operational schedules.

Dow

nloa

ded

by [

Flor

ida

Stat

e U

nive

rsity

] at

04:

15 2

6 Se

ptem

ber

2014


Figure 4. Model’s daily operational schedule (Jan± April, 1994).

Dow

nloa

ded

by [

Flor

ida

Stat

e U

nive

rsity

] at

04:

15 2

6 Se

ptem

ber

2014


Figure 5. Actual daily operational schedule (Jan± April, 1994).

Dow

nloa

ded

by [

Flor

ida

Stat

e U

nive

rsity

] at

04:

15 2

6 Se

ptem

ber

2014

model to meet the ¯ exibility requirement due to contingencies in the manufacturingenvironment. Results indicate signi® cant savings can be expected.

References

Anthony, R. N., 1965, Planning and Control Systems: A Framework for Analysis (Boston,MA: Graduate School of Business Administration, Harvard University).

Baker, T. E., 1990, Integrating AI/OR/Database technologies for production planning andscheduling. Proceedings of the Fourth International Conference on Expert Systems inProduction and Operations Management, pp. 101± 110.

Bergstrom, G. L., and Smith, B. E., 1970, Multi-item production planning ± an extension ofthe HMMS rules. Management Science, 16 (1), 614± 629.

Bitran, G. R., and Hax, A. C., 1977, On the design of hierarchical production planningsystems. Decision Sciences, 8 (1), 28± 55.

Bowman, E. H., 1963, Consistency and optimality in managerial decision making.Management Science, 9 (2), 310± 321.

Dantz ig , G., 1963, L inear Programming and Extensions, (Princeton, NJ: PrincetonUniversity Press).

Dhar, V., and Ranganathan, N., 1990, Integer programming vs expert systems: an experi-mental comparison. Communications of the ACM, 33 (3), 323± 336.

Elam, J. C., and Konsynski, B., 1987, Using arti® cial intelligence techniques to enhance thecapabilities of model management systems. Decision Sciences, 18, 487± 502.

Franz , L. S., 1989, Data driven modeling: an application in scheduling. Decision Sciences, 20(2), 359± 377.

Hansmann, F., and Hess, S. W., 1960, A linear programming approach to production andemployment scheduling. Management Science, 1 (1), 46± 51.

Hax , A. C., and Bitran, G. R., 1979, Hierarchical planning systems ± a production applica-tion. In Disaggregation: Problems in Manufacturing and Service Organizations, L. R.Rizman, L. J. Krajewski, W. L. Berry, S. H. Goodman, S. T. Hardy and L. D. Vitt (eds)(Boston, MA: Martinus Nijho� ), pp. 69± 93.

Hax , A. C., and Golovin, J. J., 1978, Hierarchical production planning systems. In Studies inOperations Management, A. C. Hax (ed.) (Amsterdam: Elsevier), pp. 400± 428.

Hax , A. C., and Meal, H. C., 1975, Hierarchical integration of production planning andscheduling. In Studies in Management Sciences: vol. I, Logistics M. A. Geisler (ed.) (NewYork: Elsevier), pp. 53± 69.

Holt, C. C., Modigliani, F., and Simon, H. A., 1955, A linear decision rule for productionand employment scheduling. Management Science, 2 (1), 1± 29.

Hooker, J. N., 1988, A quantitative approach to logical inference. Decision Support Systems,4, 45± 69.

Jones, C. H., 1967, Parametric production planning. Management Science, 13 (11), 843± 866.Kannan, V. R., and Lyman, S. B., 1992, A comparison of family and job based priority

schemes in group scheduling. Proceedings of the National Decision Sciences Institute,San Francisco, California, pp. 1454± 1456.

Kannan, V. R., and Lyman, S. B., 1994, Impact of family-based scheduling on transferbatches in a job shop manufacturing cell. International Journal of ProductionResearch, 32 (12), 2777± 2794.

Lasdon, L. S., and Terjung, R. C., 1979, An e� cient algorithm for multi-item scheduling.Operations Research, 19 (4), 946± 969.

Leong, G. K., Oliff, M. D., and Markland, R. E., 1989, Improved hierarchical productionplanning. Journal of Operations Management, 8 (3), 90± 114.

McBride, R. D., and O’Leary, D. E., 1993, The use of mathematical programming witharti® cial intelligence and expert systems. European Journal of Operational Research, 70,1± 15.

Mellichamp, J. M., and Love, R. M., 1978, Production switching heuristics for the aggregateplanning problem. Management Science, 24 (12), 1242± 1251.

Mertens, P., and Kanet, J. J., 1986, Expert systems in production management: an assess-ment. Journal of Operations Management, 6 (4), 393± 404.


ownl

oade

d by

[Fl

orid

a St

ate

Uni

vers

ity]

at 0

4:15

26

Sept

embe

r 20

14

Mohanty, R. P., and Kulkarni, R. V, 1987, Hierarchical production planning: comparisonof some heuristics. Engineering Costs and Production Economics, 11, 203± 214.

Moncrieff, R. W., 1975, Man-made Fibers (New York: Wiley).Mosier, C. T., Elvers, D. A., and Kelly, D., 1984, Analysis of group technology scheduling

heuristics. International Journal of Production Research, 22 (5), 857± 875.Oliff, M. D., and Burch, E. E., 1985, Multiproduct production scheduling at Owens-

Corning Fiberglas. Interfaces, 15 (5), 25± 34.Reinschmidt, K. F., Slater, J. H., and Finn, G. A., 1990, Expert systems for plant sched-

uling using linear programming. Proceedings of the Fourth International Conference onExpert Systems in Production and Operations Management, pp. 198± 210.

Saad , G. H., 1990, Hierarchical production-planning systems: extensions and modi® cations.Journal of the Operational Research Society, 41 (7), 609± 624.

Simon, H. A., 1987, Two heads are better than one: the collaboration between AI and OR.Interfaces, 17 (4), 8± 15.

Taubert, W. H., 1968, A search decision rule for the aggregate scheduling problem.Management Science, 14 (6), 343± 359.

Tocz ylowski, E., 1986, On aggregation of items in the single-stage lot size schedulingproblem. Large Scale Systems, 10, 157± 164.

Vaithianathan, R., and McRoberis, K. L., 1982, On scheduling in a GT environment.Journal of Manufacturing Systems, 1, 149± 155.

Zipkin, D. H., 1986, Models for design and control of stochastic, multi-item batch productionsystems. Operations Research, 91± 104.

3042 An HPP and scheduling model: a real world applicationD

ownl

oade

d by

[Fl

orid

a St

ate

Uni

vers

ity]

at 0

4:15

26

Sept

embe

r 20

14

hierarchical production planning and scheduling in a multi-product, multi-machine environment

Documents