a belief-rule-based inference method for aggregate
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A belief-rule-based inference method for aggregateproduction planning under uncertaintyBin Li a , Hongwei Wang a , Jianbo Yang b , Min Guo a & Chao Qi aa Institute of Systems Engineering, Key Laboratory of Education Ministry for ImageProcessing and Intelligent Control, Huazhong University of Science and Technology, Wuhan,PR Chinab Manchester Business School, The University of Manchester, Manchester, UKVersion of record first published: 01 Feb 2012.
To cite this article: Bin Li , Hongwei Wang , Jianbo Yang , Min Guo & Chao Qi (2013): A belief-rule-based inference methodfor aggregate production planning under uncertainty, International Journal of Production Research, 51:1, 83-105
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International Journal of Production ResearchVol. 51, No. 1, 1 January 2013, 83–105
A belief-rule-based inference method for aggregate production planning under uncertainty
Bin Lia, Hongwei Wanga*, Jianbo Yangb, Min Guoa and Chao Qia
aInstitute of Systems Engineering, Key Laboratory of Education Ministry for Image Processing and Intelligent Control,Huazhong University of Science and Technology, Wuhan, PR China; bManchester Business School,
The University of Manchester, Manchester, UK
(Received 27 October 2010; final version received 7 December 2011)
Finding high-performance solutions for aggregate production planning (APP) poses a significant challenge forboth academics and practitioners alike. In real-world problems, severe demand fluctuations make forecastshardly reliable. Forecast errors can be biased and magnifying from immediate to distant periods, and unstabledemands are usually forecast in uncertain forms. For APP under uncertain demands, a new hierarchicalbelief-rule-based inference (BRBI) method is proposed. As an expert system with a belief-rule structure, BRBIcan assist decision-makers in planning production, workforce and inventory levels with correspondinginformation representation, causal inference and identification algorithms. Operational data and expertknowledge can be employed to construct, initialise, and adjust the belief-rule base (BRB). An inference enginealgorithm is developed to handle both deterministic and interval inputs. In order to make the methodapplicable to both continuous and discrete production settings, continuous mode and switching mode forBRBI are proposed using different transformation techniques. To approximate hidden patterns in APPsituations, simultaneous identification and two-step identification for structure and parameter of BRB aredeveloped. The two-step identification contains a belief k-means (BKM) clustering algorithm extended fromk-means and fuzzy c-means. BKM ensures that an optimal cluster can both facilitate human cognition andimprove accuracy of identification and inference. A paint-factory example is utilised to conduct comparativestudies and sensitivity analyses in deterministic forecast context, and an automotive production example isimplemented to illustrate BRBI’s advantage in interval forecast context and to contrast simultaneousidentification and two-step identification.
Keywords: aggregate production planning; uncertainty; evidential reasoning; belief-rule base;knowledge-based system
1. Introduction
Aggregate production planning (APP) is a medium-range planning for production rates, workforce sizes, inventorylevels, and, possibly, shipping rates in response to the fluctuating and uncertain demand in a production system.APP is for a definite planning horizon, usually 6–24 months, to determine the production of families of items on anaggregate basis with a time block of generally one month. It is conducted in a rolling horizon, by which forecasts aremoved forward and revised dynamically. As medium-range tactical planning in the hierarchical planning frameworkof an enterprise or a supply chain, it bridges the transition from the broad long-range strategic planning to thedetailed or specific short-range operational planning. It is the baseline for any further calculation of masterproduction scheduling, resources, capacity, and raw-material planning, and is essential for various kinds ofproduction processes, such as assembly line and continuous flow. In practice, there are many sources of uncertainty,e.g. environmental uncertainty and system uncertainty, which affect production processes (Ho 1989). Models forproduction planning that do not address uncertainty are believed to generate inferior or impractical planningdecisions as compared with models that explicitly account for uncertainty (Mula et al. 2006). Consequently, therehas been increasing interest in addressing uncertainty in production planning models over the past two decades(Graves 2011).
Derived from diverse sources, such as customer demand, production and supply capacity, lead times, price, andassociated cost, the inner and outer uncertainties in APP could appear or be evaluated in various forms, such asprobability and possibility. Stochastic programming is the classical approach to solve APP problems whereuncertainties are modelled by probabilistic distributions drawn from analysis and statistics of past sales data.
*Corresponding author. Email: [email protected]
ISSN 0020–7543 print/ISSN 1366–588X online
� 2013 Taylor & Francis
http://dx.doi.org/10.1080/00207543.2011.652262
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Shorter and shorter product life cycles as well as growing innovation rates make demand extremely variable and thecollection of statistics required by stochastic models either unavailable or less and less reliable (Blackburn 1991,Giannoccaro et al. 2003, Sillekens et al. 2011). When uncertainty is due to the ambiguity of the availableinformation, the existence of conflicting evidence or measurement problems, possibility rather than probabilitytheory should be used to model uncertainty (Zimmermann 2000). Volatile customer tastes, technologicalinnovations, and reduced product life cycles, which reduce the amount of information available and make itobsolete more quickly, all necessitate alternative, non-probabilistic, theories of decision-making under uncertaintyin recent years (Bertsimas and Thiele 2006). Fuzzy programming, robust optimisation, rule-based models, etc. havebeen employed to cope with possibility expression of uncertainty, such as fuzzy, interval, ordinal, and belief values.However, most analytical models can only address one form of uncertainty, and presume a simple composition ofthe production process. It is unlikely that the production management will be satisfied with the results provided bythe analytical methods, because practical planning problems often involve various forms of uncertain or incompletedata (Zhang et al. 2011). For more complex processes with more than one type of uncertainty, the heuristic methodsbased on artificial intelligence are more appropriate and superior.
In practice, forecasts in the planning horizon are always wrong, and forecasts for distant periods are less reliablethan forecasts for immediate future (Kaminsky and Swaminathan 2001). The analytical programming approacheshave recognised inherent uncertainties, but they are all based on the assumption that demand forecasts are accurateand are equally weighted over the planning horizon. As a consequence, a decision for a forthcoming period can besignificantly affected by unreliable forecasts for distant periods. These methods incorporate various simplificationassumptions that limit their applicability, such as specialised cost functions. Managers ordinarily believe that thecost penalties associated with forcing the cost structure of their firms to fit the assumed particular cost structure aregreater than the cost penalties associated with using judgemental heuristic methods (Rinks 1982). The analyticalmathematical planning mechanisms also remain computationally prohibitive and insufficiently flexible in responseto changing production styles, highly dynamic variations, various resource constraints, and frequent re-planning(Meredith et al. 1994). The resulting production plans can be difficult to comprehend and hence difficult toimplement, which makes them unattractive even if efficiently computed (Graves 2011). Therefore, researchers andpractitioners, especially the latter, are tending to adopt heuristic and intelligent methods that are easier tounderstand and implemented, even though the results deduced are suboptimal.
Crucial information, necessary for building a realistic model, is usually hidden in historical data (Kilic et al.2007). There is also the empirical evidence that past management behaviours are always good and should be used todetermine coefficients for production and workforce rules (Rinks 1982). Motivated from the operational aspect ofan automotive production instance in which demand is forecast with uncertainty and distant forecasts are lessreliable than immediate forecasts, this paper proposes a hierarchical belief-rule-based inference (BRBI) method forAPP under uncertainty. The method is developed from the decision-support mechanism of BRBI methodologyusing the evidential reasoning (RIMER) (Yang et al. 2006a, 2007), which is derived on the basis of the evidentialreasoning (ER) approach (Yang and Sen 1994, Yang and Singh 1994, Yang 2001, Yang and Xu 2002a, 2002b) andrule-based expert system. RIMER provides a modelling and inference scheme with a belief-rule structure forproblems with various forms of uncertainty. It provides a transparent modelling and inference framework thatenables managers to intervene in the construction and updating of the belief-rule-base (BRB) using judgementalknowledge and operational data. The method is easy to understand and implement, and it requires littlecomputational effort. It has been used for graphite content detection (Yang et al. 2006a, 2007), pipeline-leakdetection (Xu et al. 2007, Zhou et al. 2009, 2011, Chen et al. 2011), inventory control (Li et al. 2011), clinicalguidelines (Kong et al. 2009), nuclear safeguards evaluation (Liu et al. 2009), consumer preference prediction (Wanget al. 2009), new product development (Tang et al. 2011), system reliability prediction (Hu et al. 2010), andgyroscopic drift prediction (Si et al. 2011).
Although various forms of forecast can be represented and transacted in the method, only deterministic andinterval styles are considered in this paper. The operational data in any style are transformed into belief distributionsin the ER framework as inputs for the hierarchical belief-rule bases. Production and workforce plans aresequentially inferred from the two sub-rule bases. An inference engine algorithm is developed to handle bothdeterministic and interval inputs. With a linguistic distribution structure and proper transformations, the BRBImethod could be constructed in either continuous mode in situations where any production/workforce level can beused or switching mode in situations where only a few production/workforce levels could be permitted. Structureidentification for BRB aims at determining the number and position of belief rules, i.e. the evaluation grades of theantecedents. Parameter identification for BRB involves approximating the BRB structures to the hidden patterns of
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a planning scheme. Simultaneous identification and two-step identification for structure and parameter of BRB areproposed, and the latter is analysed and verified to be more pertinent and efficient than the former. In the two-stepidentification, a BKM algorithm extended from k-means and fuzzy c-means (FCM) is developed to adjust the BRBstructure.
The remainder of the paper is organised as follows. Section 2 reviews the relevant literature about APP underuncertainty from multiple criteria and related options. Section 3 formulates the specific APP problem underuncertainty studied in this paper. Section 4 proposes the hierarchical BRBI method for APP under uncertainty.In Section 5, a paint factory example and an automotive production example are implemented to performcomparative studies and sensitivity analyses. The paper is concluded in Section 6.
2. Literature review
Since the well-known Holt, Modigliani, Muth, and Simon (HMMS) model with LDR (linear decision rule) for apaint factory was proposed (Holt et al. 1955), APP has attracted considerable attention from both practitioners andacademia, and researchers have developed many analytical and heuristic methods, with their own pros and cons, tosolve context-specific APP problems. In order to facilitate the understanding and use of various existing APPtechniques with constraints, Nam and Logendran (1992) compiled a bibliography of literature with an investigationof practical applications. Thompson et al. (1993) and Chen and Liao (2003) gave comparative studies ofmathematical and simplified APP strategies respectively for selection under conditions of uncertainty and cyclicproduct demands. Metaxiotis et al. (2002) surveyed the application of expert systems in production planning andscheduling. Mula et al. (2006) reviewed existing literatures on production planning under uncertainty. Graves (2011)reviewed and discussed how uncertainty should be handled in production planning based on personal practicalobservations in a variety of industrial contexts. We classify the APP methods under uncertainty in the followingcriteria as listed in Table 1, and then expatiate on extant options for those criteria with corresponding literature.
First, there is Uncertain Information Style. Plenty of models for production planning have been proposed to dealwith different forms of uncertainty such as probability and possibility. Bitran and Yanassee (1984) and Feiring(1991) considered production planning in stochastic environments where demand is stochastic with knowndistribution functions. Zanjani et al. (2010) proposed a multi-stage stochastic programming approach forproduction planning with uncertainty in quality of raw materials and consequently in process yields, as well asuncertainty in products demands. Zhang et al. (2011) developed a stochastic production planning model for aninternational enclosure manufacturing company with seasonal demand and market growth uncertainty. Leung et al.(2007) constructed a stochastic robust optimisation model to make a balance between solution robustness andmodel robustness under different economic growth scenarios. Aghezzaf et al. (2010) developed stochastic and robustoptimisation models to design robust tactical production plans in multi-stage production systems with stochasticfinished-product demands. Kanyalkar and Adil (2010) proposed a robust optimisation model for aggregate anddetailed planning of a multi-site procurement-production-distribution system.
Other forms of uncertainty, such as fuzziness, interval, and partial ignorance, are more common in APP thanprobability (Tang et al. 2000, Fung et al. 2003). Lee (1990) developed fuzzy programming models for APP withfuzzy workforce levels and fuzzy demands. Wang and Fang (1997) proposed a genetics-based fuzzy linearprogramming approach for APP in a fuzzy environment. Chen and Huang (2010) proposed a membership functionapproach to describe the fuzzy minimum total cost of APP problem with fuzzy input parameters. Fuzzy andpossibilistic programming methods were also extended to solve multi-product APP (Tang et al. 2000, Funget al. 2003, Wang and Liang 2005, Jamalnia and Soukhakian 2009), multi-site production planning
Table 1. Criteria and options for characterising APP methods under uncertainty.
Criteria Options
Uncertain information style Probability; possibilityModel style Analytical; simulation; heuristicForecast update style Forecast evolution; time-series update; Bayesian updateDependence with forecast model Dependent; independentPlanning style Adjustment; rolling planningTemporal demand information Future; future and historical
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(Torabi and Hassini 2009), multi-objective APP (Wang and Fang 2001), and hierarchical production planning(HPP) (Torabi et al. 2010) problems. Except randomness and fuzziness, demand in APP may be forecast in intervalstyle, i.e. only upper and lower bounds are given (Tang et al. 2000, Abass et al. 2010). Gfrerer and Zapfel (1995)discussed such an HPP problem, but only the detailed demand was predicted in interval form while aggregatedemand was supposed to be deterministic (single-point value). Abass et al. (2010) utilised a nonlinear intervalnumber programming (NINP) model (Jiang et al. 2008) for generalised production planning problem under intervaldemands and resources.
Second, there is Model Style. Analytical, simulating and heuristic model styles have been proposed for APPunder uncertainty. Stochastic programming (Bitran and Yanassee 1984, Feiring 1991, Zanjani et al. 2010, Zhanget al. 2011), robust optimisation (Leung et al. 2007, Aghezzaf et al. 2010, Kanyalkar and Adil 2010), fuzzyprogramming (Lee 1990, Wang and Fang 1997, Chen and Huang 2010), and nonlinear interval programming(Abass et al. 2010) are representative analytical methods for APP with stochastic, fuzzy, and interval styles ofuncertainty. To deal with complex and non-specific cost structures that usually preclude the analytical methods,researchers have proposed simulation methods in which parametric production planning (Jones 1967) andsimulation decision rule (Taubert 1968) are well known. Thompson and Davis (1990) developed a linearprogramming and Monte Carlo simulation integrated approach to model the uncertainties present in APP.Furthermore, to incorporate managers’ actual decision-making behaviour, heuristic methods have been advanced,such as management coefficients model (MCM) (Bowman 1963) and production switching heuristic (PSH)(Mellichamp and Love 1978, Hwang and Cha 1995). Rinks (1982) utilised a fuzzy logic system to model fuzzydemand and structure manager’s actual judgemental processes in APP. Turksen (1988) advocated usinginterval-valued membership functions to define fuzzy linguistic production rules. Ward et al. (1992) developed aC-language program based on Rinks’ fuzzy APP framework. Duchessi and O’Keefe (1990) introduced a prototypeof a knowledge-based approach for production planning.
Third, there is Forecast Update Style. In generating the aggregate and forecasting data to formulate theaggregate production plan, three forecast update styles have been proposed to exploit the demand structure (Sethiet al. 2005). The first style is forecast evolution. Graves et al. (1986, 1998) and Heath and Jackson (1994) studied theAPP problems under uncertainty and independently developed the martingale model of forecast evolution (MMFE)to characterise the forecast update process. Toktay and Wein (2001) utilised the MMFE to analyse aforecasting-production-inventory system with stationary demand. Zhou et al. (2007) proposed combining MMFEwith a forecast disaggregation model to accommodate production and inventory planning situations in whichaggregate and disaggregate forecasts are available at different time horizons. The second style is a Bayesian update.Kleindorfer and Kunreuther (1978) generated the demand in every period from a family of independentdistributions whose parameters are not specified with certainty. Filho (1999) modelled the cumulative demand by acompound Poisson process composed by independent random variables denoting customer orders. Kanyalkar andAdil (2010), Zanjani et al. (2010), and Zhang et al. (2011) proposed using stochastic scenarios to represent theuncertain demand process as new information is available to the decision-maker. The third style is a time-seriesupdate. Holt et al. (1955) proposed a moving average forecast method based on past orders to implement the LDR.Widiarta et al. (2009) studied the relative effectiveness of top-down and bottom-up strategies for forecasting theaggregate demand in a production-planning framework, in which the aggregate demand series is composed ofseveral correlated subaggregate items following stationary time-series processes.
In addition to the three criteria listed above, there are three other criteria that are not definitely declared in theliterature but fundamental to solving APP problems: Dependence with Forecast Model, Planning Style, and TemporalDemand Information. Almost all the planning models constructed based on stochastic demand are dependent uponthe specific demand forecast models employed, and the planning models constructed on possibilistic demand are notdependent upon the specific demand forecast models employed. Although the forecast methods are massive andcustomised for various circumstances, the forecast error will always exist owing to the non-stationary fluctuatingdemand, so production planning methods connected with specific forecast models do not have essential and generalmeanings. There are two planning styles to confront uncertainty in APP: adjustment based on original planned dataand rolling horizon implementation of a planning model (Magee and Boodman 1967, Silver et al. 1998). The formeris seldom used because the adjustment factor is hard to determine in the balance between adjustment magnitude andbuffer inventory, and the analytical adjustment rule could not deal with various forms of uncertainty. In contrast,the latter is commonly utilised for its flexibility and comprehensive reaction. The analytical, simulating, andheuristic methods reviewed in this paper are all implemented in the latter planning style. The temporal demandinformation required to establish the production planning models includes historical information, namely historical
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sales and historical forecast demand, and future information, namely future forecast demand. The majority of theAPP techniques are confined to future information without consideration of historical information. However, toperfect production planning decisions, both historical and future information should be taken into account toperform their respective functions, which is advocated by PSH and knowledge-based methods.
In this paper, the specialised options concerning these criteria are as follows. Unstable demand is forecast innon-probabilistic or possibilistic style, and particularly deterministic and interval styles are considered. Both of thesekinds of information are transformed into unified belief distributions in the ER framework. The BRBI method is anartificial-intelligence-based heuristic technique, and the BRB can be constructed, initialised, and adjusted by anexpert’s knowledge and operational data. This paper is not dedicated to study the demand forecast updateproblems, but proposes APP strategies with general and unspecific demand forecast and updating models byincorporating the critical regulation that forecast error is magnifying and biased from immediate to distant periods.The proposed method is also independent of the underlying demand forecast model. The planning is conducted in arolling style. Both historical sales and forecast information are utilised in the BRBI method to give comprehensiveand reliable production and workforce decisions.
3. Problem formulations
The main task of APP is to determine the optimum production, workforce, and inventory levels in response todemand fluctuation over a finite time horizon. Inventory level is generally treated as a derived result afterdetermining the two previous variables. The APP problem under uncertainty considered in this paper has twospecific characteristics. First, the demand forecast error is magnifying from immediate to distant periods, and thisregulation could not be formalised analytically. Second, the fluctuating demand will be forecast with uncertainforms, such as interval, partial ignorance, and fuzziness. Various forms of information including precise data,random numbers and subjective judgements can be consistently modelled in the ER framework (Yang 2001, Wanget al. 2006, Yang et al. 2006a). However, deterministic and interval types are only considered in this paper, and theconcrete transformation techniques are provided in Appendix A. A value v can be consistently represented in thefollowing distributed belief structure within the ER framework:
SðvÞ ¼ fðHn, �nÞ, n ¼ 1, 2, . . . ,Ng, ð1Þ
where Hn is the evaluation grade, �n is the belief degree to the corresponding evaluation grade Hn, and N is thenumber is the evaluation grades. For instance, an assessment for the forecast demand of the coming period may befðlarge, 0:7Þ, ðmedium, 0:3Þ, ðsmall, 0Þg, that is the demand is forecast to be a large volume with 70% belief degree anda medium volume with 30% belief degree.
Two perspectives with different ways of utilising demand information are expressed in Equations (2) and (3)respectively. Methods in Perspective 1 are based on the tacit assumption that demand forecasts are accurate andequally weighted, and can be used to derive a series of decisions for a definite planning horizon. This perspectiveresults means that decision-making for the coming months can be significantly affected by long-term forecasts, butforecasts for distant months are less reliable than forecasts for the immediate months. To overcome thisshortcoming, the other perspective is provided, such as PSH and knowledge-based methods. In Perspective 2,decision rules for production rate and workforce level are initialised by judgemental knowledge and can be refinedby historical sales over certain periods. Only the forecast demand of the forthcoming period is used to activatedecision rules to infer the decision variables for the forthcoming period. The BRBI method is constructed inPerspective 2, and hence fit for situations with increasing forecast errors. Through the minimisation of total relatedcosts, we can obtain the optimal production and workforce levels with application of the BRBI method.
Perspective 1 : TC1t ¼ f1 Pn,Wnðn ¼ t, . . . , tþ T� 1ÞjDf
n ðn ¼ t, . . . , tþ T� 1Þ� �
ð2Þ
Perspective 2 : TC2t ¼ f2 Pn,Wnðn ¼ t, . . . , tþ T� 1ÞjDnðn ¼ t� T, . . . , t� 1Þ,Df
t
� �, ð3Þ
where TC1t and TC2
t are total costs, Pt and Wt are the production rate and workforce level at month t, Dt and Dft are
the real demand and forecast demand at month t, and T is the planning and simulation horizon. The concrete costcomponents constituting the total costs must be established under the designated problems considered. Note thatthe BRBI method is not restricted to any specific cost structures.
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The inventory equation affected by production and demand is:
It ¼ It�1 þ Pt �Qt ð4Þ
where It is the inventory level at the end of month t, and Qt is the demand quantity, explicitly Qt ¼ Dft in Perspective
1 and Qt¼Dt in Perspective 2.The boundary or capacity constraints for the decision variables are given as follows:
Pt � Pt � Pt ð5Þ
Wt �Wt �Wt ð6Þ
�VPt � Pt � Pt�1 � VPt ð7Þ
�VWt �Wt �Wt�1 � VWt ð8Þ
Pt,Wt,Pt,Wt,VPt,VPt,VWt,VWt � 0, ð9Þ
where Pt and Wt are the permissible minimum production rate and workforce level at month t, Pt and Wt areavailable the production rate and workforce level capacities at month t, VPt and VPt are the permissible decrementand increment of production rate from month t� 1 to t, VWt and VWt are the permissible decrement and incrementof workforce level from month t� 1 to t, and all variables are non-negative.
4. Belief-rule based inference method for APP
With the two specific characteristics as stated in the above section, the APP problem under uncertainty considered inthis paper invalidates the traditional analytical and simulating approaches. In a decision-making process, one mayneed to introduce intelligent computing techniques in order to capture the structure and behaviour of a system thatis highly nonlinear and uncertain (Kilic et al. 2007). From our point of view, the following principles should beabided by when selecting decision models for APP: (a) the sophistication of models should match the complexity ofa specific decision situation; (b) the techniques should emphasise practical importance rather than just theoreticalmerit; (c) the models should be adaptable to different planning strategies; and (d) the models should be able to dealwith different forms of uncertainties. Based on these considerations, we propose the BRBI method for APP underuncertainty. BRBI is a hybrid modelling and inference scheme in which subjective knowledge or system behaviourcan be described using belief-rule-based natural language. Belief rule has been proposed recently as an efficient toolfor uncertain and nonlinear modelling with reasonable precision while allowing linguistic interpretability (Yanget al. 2007), and has been successfully utilised in many complex decision-making domains where traditionalanalytical methods do not work well.
4.1 Hierarchical BRBI framework for APP
In LDR and MCM, production rate Pt and workforce levelWt for month t are computed from the future Tmonths’forecast demands (fDf
i ji ¼ t, . . . , tþ T� 1g), workforce level Wt�1 at month t� 1, and inventory level It�1 at montht� 1. In PSH, production rate Pt and workforce level Wt are determined by forecast demand Df
t and inventory levelIt�1 in a switching heuristic model. In Rinks’ fuzzy logic models for APP, production rate Pt and change inworkforce level DWt are derived from forecast demand Df
t, workforce level Wt�1, and inventory level It�1. LDR andMCM are typical cases of Perspective 1, and PSH and fuzzy logic model are representatives of Perspective 2. Thereare two insufficient considerations in all these methods. First, production rates Pt and workforce level Wt arededuced from the same input variables with the same pattern, but in practice, Pt is determined beforehand, andWt isderived based on Pt (Silver et al. 1998, Sillekens et al. 2011). That is, the two decision variables should be determinedsequentially rather than in parallel. Second, Bowman (1963) argues that the determination of Pt should take theprevious month’s production rate Pt�1 into consideration. The reference of past decision can ensure that the newdecision does not change independently of the past decision. To address these concerns, we propose the followinghierarchical BRBI framework for APP.
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Figure 1 implies that, in the first stage, Pt is inferred from Dft, It�1, Wt�1, and Pt�1, and in the second stage, Wt is
inferred from Pt and Wt�1. The causal inference functions of Pt and Wt can be represented as follows:
Pt ¼ �ðDft, It�1,Wt�1,Pt�1Þ ð10Þ
Wt ¼ �ðPt,Wt�1Þ: ð11Þ
In the belief-rule scheme, these functions can be seen as two sub-BRBs. For Sub-rule-base 1, the antecedent
attributes are Dft, It�1, Wt�1, and Pt�1, and the consequent attribute is Pt. For Sub-rule-base 2, the antecedent
attributes are Pt and Wt�1, and the consequent attribute is Wt. Suppose the numbers of evaluation grades for Dft,
It�1, Wt, and Pt are J1, J2, J3, and J4 respectively. The evaluation grades and corresponding belief degrees of
belief-rules’ antecedent and consequent attributes in the ER framework can be represented as follows:
G1,at ðD
ft, It�1,Wt�1,Pt�1Þ ¼ fðHi, ji ,�i, ji Þj ji ¼ 1, 2, . . . , Ji; i ¼ 1, 2, . . . , 4g ð12Þ
G1,ct ðPtÞ ¼ fðH4, j4 ,�
1r1, j4Þj j4 ¼ 1, 2, . . . , J4; r1 ¼ 1, 2, . . . ,R1g ð13Þ
G2,at ðPt,Wt�1Þ ¼ fðH4, j4 ,�4, j4 Þ; ðH3, j3 ,�3, j3Þj ji ¼ 1, 2, . . . , Ji; i ¼ 3, 4g ð14Þ
G2,ct ðWtÞ ¼ fðH3, j3 ,�
2r2, j3Þj j3 ¼ 1, 2, . . . , J3; r2 ¼ 1, 2, . . . ,R2g, ð15Þ
where Hi, ji (ji ¼ 1, 2, . . . , Ji; i ¼ 1, 2, . . . , 4) represent the evaluation grades for Dft, It�1, Wt, and Pt respectively. �i, ji
( ji ¼ 1, 2, . . . , Ji; i ¼ 1, 2, . . . , 4) are belief degrees assessed toHi, ji of the four kinds of antecedent variables. �1r1, j4
and
�2r2, j3 are belief degrees assessed to H4, j4 andH3, j3 of the two consequent variables. R1 and R2 are the total number of
belief rules in the two sub-belief rule bases and satisfy R1 ¼ J1 � J2 � J3 � J4 and R2 ¼ J4 � J3. The utility function for
each evaluation grade is defined by �ðHi, jiÞ ¼ Ui, ji (ji ¼ 1, 2, . . . , Ji; i ¼ 1, 2, . . . , 4).The formulations of belief rules for Sub-rule-base 1 and Sub-rule-base 2 are as follows:
IF Dft is H1, j1 , It�1 is H2, j2 ,Wt�1 is H3, j3 , and Pt�1 is H4, j4 ,
THEN Pt is fðH4,1,�1r1,1Þ, ðH4,2,�
1r1,2Þ, . . . , ðH4,J4 ,�
1r1,J4Þg, WITH �1r1 ,
ð16Þ
where jk ¼ 1, 2, . . . , Jk for ðk ¼ 1, . . . , 4Þ, r1 ¼ 1, 2, . . . ,R1, and �1r1is the weight of r1th belief-rule.
IF Pt is H4, j4 and Wt�1 is H3, j3 , THEN Wt is fðH3,1,�2r2,1Þ, ðH3,2,�
2r2,2Þ, . . . , ðH3,J3 ,�
2r2,J3Þg, WITH �2r2 , ð17Þ
where jk ¼ 1, 2, . . . , Jk for ðk ¼ 4, 3Þ, r2 ¼ 1, 2, . . . ,R2, �2r2is the weight of r2th belief-rule.
The two sub-rule bases constitute a hierarchical belief-rule-base (BRB) for APP. The hierarchical belief inference
framework, which determines Pt from Dft, It�1, Wt�1 and Pt�1, and sequentially determines Wt from Pt and Wt�1, is
based on the philosophy of modelling human judgements and decision patterns. Human knowledge is coded as
Figure 1. Hierarchical BRBI framework for APP.
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belief-rule-based if–then rules. For example, if all the variables in BRB have identical linguistic evaluation grades of
low, medium, and high, then the following if–then rules will be included in the two sub-rule bases respectively:
IF demand forecast for the coming month is low, workforce level in this month is medium and production rate in this month ishigh, THEN production rate for the coming month is fðlow, 0:8Þ, ðmedium, 0:2Þ, ðhigh, 0Þg, WITH belief rule weight 1;IF production rate for the coming month is medium and workforce level in this month is medium, THEN workforce level for thecoming month is fðlow, 0Þ, ðmedium, 1Þ, ðhigh, 0Þg, WITH belief rule weight 1.
4.2 Inference engine algorithm
Having represented the antecedent and consequent attributes in the ER framework and determined the formulation
of the BRB, the inference engine algorithm can be directly applied to combine activated belief rules and generate
final decisions for production rate and workforce level. In RIMER, the inference engine algorithm with
deterministic inputs has been proposed (Yang et al. 2006a, 2007). Based on RIMER and interval ER (IER)
approaches (Wang et al. 2006, Guo et al. 2007), an inference engine algorithm for APP with both deterministic and
interval inputs is developed herein. The BRBI method, whose output decision will be uncertain if the input is
uncertain, empowers the decision-maker to make ultimate decision by taking into account external conditions as
well as their experience, preference, intuition, and judgement, while results generated from the model can be adopted
as references.First, the Sub-rule-base 1 is activated to infer Pt. Let STATE
1t ðD
ft, It�1,Wt�1,Pt�1Þ be the state of inputs for
Sub-rule-base 1, i.e. Dft, It�1, Wt�1 and Pt�1, in which Df
t (Dft 2 ½ y
�t , y
þt �) takes interval values, and the others take
deterministic values. Using the information transformation techniques given in Appendix A,
STATE1t ðD
ft, It�1,Wt�1,Pt�1Þ can represented as follows:
Dft : Df
t , H1,1, �1,1� �
, . . . , , H1, j1 , �1, j1� �
, . . . , , H1,J1 , �1,J1� �� �
ð18Þ
It�1 : It�1 , H2,1, �2,1� �
, . . . , , H2, j2 , �2, j2� �
, . . . , , H2,J2 , �2,J2� �� �
ð19Þ
Wt�1 : Wt�1 , H3,1, �3,1� �
, . . . , , H3, j3 , �3, j3� �
, . . . , , H3,J3 , �3,J3� �� �
ð20Þ
Pt�1 : Pt�1 , H4,1, �4,1� �
, . . . , , H4, j4 , �4, j4� �
, . . . , , H4,J4 , �4,J4� �� �
, ð21Þ
where �1, j1 2 ½��1, j1
, �þ1, j1 � (j1 ¼ 1, 2, . . . , J1; ��1, j1� �þ1, j1 ).
The inference for belief degrees of Pt can be generated using the following nonlinear Min/Max optimisation
model:
Min=Max �j4 for j4 ¼ 1, . . . , J4 ð22Þ
s:t: �r1 ¼Y4i¼1
�i, ji , r1 ¼ 1, 2, . . . ,R1ð Þ ð23Þ
!r1 ¼�r1�r1PR1
j¼1 �j�j� , ðr1 ¼ 1, 2, . . . ,R1Þ, ð24Þ
�j4 ¼�QR1
r1¼1!r1�r1, j4 þ 1� !r1
PJ4j4¼1
�r1, j4
� ��QR1
r1¼11� !r1
PJ4j4¼1
�r1, j4
� �h i1� �
QR1
r1¼1ð1� !r1 Þ
h i , j4 ¼ 1, 2, . . . , J4ð Þ ð25Þ
� ¼XJ4j4¼1
YR1
r1¼1
!r1�r1, j4 þ 1� !r1
XJ4j4¼1
�r1, j4
!� ðJ4 � 1Þ
YR1
r1¼1
1� !r1
XJ4j4¼1
�r1, j4
!" #�1, ð26Þ
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where �r1 is the matching degree of the input state STATE1t to the r1th belief rule. !r1 is the activation weight of
STATE1t for the r1th belief rule. �j4 is the inferred belief degrees of Pt. � is the normalising factor. The first, second,
third, and fourth inputs refer to Dft, It�1, Wt�1, and Pt�1 respectively.
The inferred production rate Pt can be represented using the following distributed formulation:
Pt ¼ �RIMERðSTATE1t Þ ¼ fðH4, j4 ,�j4 Þ, j4 ¼ 1, . . . , J4g ¼ fðH4, j4 , ½�
�j4,�þj4 �Þ, j4 ¼ 1, . . . , J4g, ð27Þ
where ��j4 and �þj4 are the optimal solutions of the j4th Min/Max problem for �j4 (j4 ¼ 1, . . . , J4), andPJ4
j4¼1�j4 ¼ 1.
Second, the Sub-rule-base 2 is activated to infer Wt. Let STATE2t ðPt,Wt�1Þ be the state of Pt and Wt�1, in which
Pt is derived from the above part. Using the information transformation techniques given in Appendix A,
STATE2t ðPt,Wt�1Þ can represented as follows:
Pt : Pt , H4,1,�4,1� �
, . . . , H4, j4 ,�4, j4� �
, . . . , H4,J4 ,�4,J4� �� �
ð28Þ
Wt�1 : Wt�1 , H3,1, �3,1� �
, . . . , H3, j3 , �3, j3� �
, . . . , H3,J3 , �3,J3� �� �
, ð29Þ
where �4, j4 2 ½��4, j4
,�þ4, j4 � (j4 ¼ 1, 2, . . . , J4; ��4, j4� �þ4, j4 ).
The inference for belief degrees of Wt can be generated using the following nonlinear Min/Max optimisation
model, which is similar to that of Pt:
Min=Max �j3 for j3 ¼ 1, . . . , J3 ð30Þ
s:t: �r2 ¼ �4, j4 � �3, j3 , ðr2 ¼ 1, 2, . . . ,R2Þ, ð31Þ
!r2 ¼�r2�r2PR2
j¼1 �j�j� , ðr2 ¼ 1, 2, . . . ,R2Þ, ð32Þ
�j3 ¼�QR2
r2¼1!r2�r2, j3 þ 1� !r2
PJ3j3¼1
�r2, j3
� ��QR2
r2¼11� !r2
PJ3j3¼1
�r2, j3
� �h i1� �
QR2
r2¼1ð1� !r2Þ
h i , ð j3 ¼ 1, 2, . . . , J3Þ ð33Þ
� ¼XJ3j3¼1
YR2
r2¼1
!r2�r2, j3 þ 1� !r2
XJ3j3¼1
�r2, j3
!� ðJ3 � 1Þ
YR2
r2¼1
1� !r2
XJ3j3¼1
�r2, j3
!" #�1, ð34Þ
where �r2 is the matching degree of the input state STATE2t to the r2th belief rule, !r2 is the activation weight of
STATE2t for the r2th belief rule. �j3 is the inferred belief degrees of Wt. � is the normalising factor.
The inferred workforce level Wt can be represented using the following distributed formulation:
Wt ¼ �RIMERðSTATE2t Þ ¼ fðH3, j3 ,�j3Þ, j3 ¼ 1, . . . , J3g ¼ fðH3, j3 , ½�
�j3,�þj3 �Þ, j3 ¼ 1, . . . , J3g, ð35Þ
where ��j3 and �þj3 are the optimal solutions of the j3th min–max problem for �j3 (j4 ¼ 1, . . . , J4), andPJ3
j3¼1�j3 ¼ 1.
4.3 Continuous mode and switching mode
Most traditional methods for APP assume that only continuous production rate and workforce level can be used.
However, in some industries, such as the food and chemical industries, the two decision variables should not be
rescheduled frequently, but be carried out at discrete levels, for example, high, normal, and low (Mellichamp and
Love 1978). PSH is an innovative switching method for discrete environment. However, none of the methods from
our literature review offers a unified framework for both continuous and discrete production settings. So, it is
necessary to propose a method for APP to fit in different settings with proper transformations. The BRB model with
a discrete linguistic distribution structure makes this possible with the following different transformation techniques
proposed in this paper.
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4.3.1 Continuous mode
The inferred results for production rate and workforce level in the above subsection are given in distributed formswith evaluation grades and corresponding belief degrees. For the continuous mode, the distributed formulations forPt and Wt, given in Equations (27) and (35), can be further converted into the following form:
Pt 2 P�t ,Pþt
� ¼
XJ4j4¼1
U4, j4��4, j4
,XJ4j4¼1
U4, j4�þ4, j4
" #ð36Þ
Wt 2 W�t ,Wþt
� ¼
XJ3j3¼1
U3, j3��3, j3
,XJ3j3¼1
U3, j3�þ3, j3
" #, ð37Þ
where U4, j4 ¼ �ðH4, j4 Þ, U3, j3 ¼ �ðH3, j3 Þ, ��4, j4¼ minpt �j4 , �
þ4, j4¼ maxpt max
Pt
�j4 , ��3, j3¼ minWt
�j3 , �þ3, j3¼ maxWt
�j3 ,�j4 2 ½�
�j4,�þj4 �, �j3 2 ½�
�j3,�þj3 �, j4 ¼ 1, 2, . . . , J4, and j3 ¼ 1, 2, . . . , J3.
4.3.2 Switching mode
For the switching mode, the inferred Pt and Wt, given in Equations (27) and (35), can be reformulated in thefollowing way:
Pt ¼
H4,1,�1H4,2,�2. . . . . .H4,J4 ,�J4
8>><>>: ¼
H4,1, ½��1 ,�
þ1 �
H4,2, ½��2 ,�
þ2 �
. . . . . .H4,J4 , ½�
�J4,�þJ4 �
8>><>>: ð38Þ
Wt ¼
H3,1,�1H3,2,�2. . . . . .H3,J3 ,�J3
8>><>>: ¼
H3,1, ½��1 ,�
þ1 �
H3,2, ½��2 ,�
þ2 �
. . . . . .H3,J3 , ½�
�J3,�þJ3 �
8>><>>: , ð39Þ
where H4, j4 (j4 ¼ 1, . . . , J4) and H3, j3 (j3 ¼ 1, . . . , J3) are given discrete levels of production and workforce. ½��j4 ,�þj4�
and ½��j3 ,�þj3� are inferred interval belief degrees attached to H4, j4 and H3, j3 .
Usually more than one set of interval belief degrees are inferred to be non-zero, so the most favourable levels forPt andWt can be selected using a suitable interval comparison algorithm. The approach for interval comparison andranking, proposed by Wang et al. (2005), can be used here. The degree of preference �i ¼ ½�
�i ,�
þi � over �j ¼ ½�
�j ,�
þj �
(or �i 4�j) is defined as
Pð�i 4�j Þ ¼maxð0, �þi � �
�j Þ �maxð0, ��i � �
þj Þ
ð�þi � ��i Þ þ ð�
þj � �
�j Þ
, ð40Þ
where i, j 2 ½1, . . . , J4� or i, j 2 ½1, . . . , J3�. The production rate H4, j4 or workforce level H3, j3 with the greatest beliefdegree will be selected as the terminal decision.
4.4 Structure and parameter identifications
The belief-rule expression for APP provides a compact framework for representing expert knowledge. The beliefrules can be determined a priori by experts’ knowledge. However, it is difficult to accurately determine the structuresand parameters of BRBs merely relying on manager’s knowledge, in particular for large-scale BRBs with a largenumber of belief rules and dramatically changing patterns. In order to tackle non-stationary demand with trendsand seasonal variations, the inference structure should be adapted dynamically. As such, it is necessary to develop asupporting mechanism to train both the structure and parameter of BRB using historical information. Structureidentification for BRB aims at determining the number and position of belief rules, i.e. the evaluation grades of theantecedents. The parameter identification for BRB is to approximate the BRB structures to the hidden patterns ofthe planning situation. Most existing applications of RIMER are implemented with predetermined constantstructures, except that Zhou et al. (2010) proposed a sequential learning algorithm for adding and pruning belief
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rules based on the definition of statistical utility. This algorithm is applicable for systems with input–output pair-style sampling data but not for systems with unlabelled sampling data, as is the case in this paper. The structure andparameter identifications can be conducted in two different ways, simultaneous identification and two-stepidentification.
4.4.1 Simultaneous identification
Structure and parameter are two phases for the identification of BRB models, but the preliminary thinking is toidentify them simultaneously. With regard to the BRBI method for APP under uncertainty, we propose thefollowing optimisation model for simultaneous identification of BRB’s structure and parameter.
Min TCt ¼Xt
n¼t�Tþ1
ðCrðtÞ þ CcðtÞ þ CoðtÞ þ CiðtÞÞ ð41Þ
s:t: UmX 6¼ Un
X, ðm, n ¼ 1, 2, . . . ,K; m 6¼ nÞ ð42Þ
x � UmX � x, ðm ¼ 1, 2, . . . ,KÞ ð43Þ
LSX � jjU
mX �Umþ1
X jj � LGX, ðm ¼ 1, 2, . . . ,K� 1Þ, ð44Þ
0 � �1r1, j4 � 1, ðr1 ¼ 1, 2, . . . ,R1; j4 ¼ 1, 2, . . . , J4Þ ð45Þ
XJ4j4¼1
�1r1, j4 � 1, ðr1 ¼ 1, 2, . . . ,R1Þ ð46Þ
0 � �1r1 � 1, ðr1 ¼ 1, 2, . . . ,R1Þ ð47Þ
0 � �2r2, j3 � 1, ðr2 ¼ 1, 2, . . . ,R2; j3 ¼ 1, 2, . . . , J3Þ ð48Þ
XJ3j3¼1
�2r2, j3 � 1, ðr2 ¼ 1, 2, . . . ,R2Þ ð49Þ
0 � �2r2 � 1, ðr2 ¼ 1, 2, . . . ,R2Þ, ð50Þ
where Equation (41) is calculated from Equations (4)–(37) for the continuous mode, or Equations (4)–(35) and(38)–(40) for the switching mode. The cost components can take any form depending on practical conditions, whichdoes not affect the belief-rule inference engine algorithms. K is the number of evaluation grades, Ck
X is the kth(k ¼ 1, . . . ,K) evaluation grade for variable X (i.e. Dt, It, Wt or Pt), x and x are lower and upper bounds of X, LS
X
and LGX are the smallest and greatest distances between two adjacent evaluation grades, �1r1, j4 and �1r1 are the
consequent belief degree and belief rule weight of belief rules in Sub-rule-base 1, and �2r2, j3 and �2r2are the consequent
belief degree and belief rule weight of belief rules in Sub-rule-base 2. Constraints (42)–(44) and (45)–(50) arerespectively on the structure and parameter of the BRB model for APP.
4.4.2 Two-step identification
As an alternative, the structure identification and parameter identification can be conducted in two steps. Wepropose a belief k-means (BKM) clustering algorithm for structure identification of BRB based on k-means(MacQueen 1967) and FCM (Bezdek 1981), and a parameter approximation model for parameter identification ofBRB. Unlike k-means, where each datum belongs to only one cluster, and fuzzy c-means, where each datum isassociated with every cluster using a membership function, the new belief k-means approach defines each datum tobelong to two adjacent clusters, because each deterministic datum is assigned to two adjacent evaluation grades with
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belief degrees in the ER framework. The new approach consists of the following model. The objective function for
the variable X (i.e. Dt, It, Wt, or Pt) is
Min �ðCkXjk ¼ 1, 2, . . . ,KÞ ¼
XNi¼1
�ipkxi � CpXk þ �iqkxi � C
qXk
� �ð51Þ
s:t: CpX ¼Max
K
k¼1Ck
XjCkX � xi
� �, ði ¼ 1, 2, . . . ,NÞ ð52Þ
CqX ¼Min
K
k¼1Ck
XjCkX � xi
� �, ði ¼ 1, 2, . . . ,NÞ ð53Þ
�ip ¼kxi � C
qXk
kCqX � C
pXk
, ði ¼ 1, 2, . . . ,NÞ ð54Þ
�iq ¼kxi � C
pXk
kCqX � C
pXk
, ði ¼ 1, 2, . . . ,NÞ ð55Þ
CmX 6¼ Cn
X, ðm, n ¼ 1, 2, . . . ,K; m 6¼ nÞ ð56Þ
x � CmX � x, ðm ¼ 1, 2, . . . ,KÞ ð57Þ
LSX � kC
mX � Cmþ1
X k � LGX, ðm ¼ 1, 2, . . . ,K� 1Þ, ð58Þ
where xi is the kth historical datum for variable X, CkX is the kth cluster centre (i.e. evaluation grade), C
pX and C
qX are
the nearest two adjacent evaluation grades of xi, �ip and �iq are the belief degrees of xi assigned to CpX and C
qX, x and
x are the lower and upper bounds of X, LSX and LG
X are the smallest and greatest distances between two adjacent
evaluation grades, K is the number of clusters, N is the number of historical data, and k*k is the Euclidean distance.From Equations (51), (54), and (55), we can get
�ðCkjk ¼ 1, . . . ,KÞ ¼ 2XNi¼1
kxi � CpXk � kxi � C
qXk
kCqX � C
pXk
: ð59Þ
Equation (59) implies two points: (1) the optimal cluster is proportional to the distance between two adjacent
evaluations, which is consistent with human cognition; and (2) the optimal cluster ensures that the sampling data are
around the evaluation grades with minimum distances, which improves the accuracy of identification and inference.After the structure of BRB is determined, its parameters, i.e. rule weights and consequent belief degrees, can be
identified from Perspective 2 using the following parameter approximation model based on a series of historical
demand data in order to refine the belief rules:
Min TCt ¼Xt
n¼t�Tþ1
ðCrðtÞ þ CcðtÞ þ CoðtÞ þ CiðtÞÞ ð60Þ
s:t: 0 � �1r1, j4 � 1, ðr1 ¼ 1, 2, . . . ,R1; j4 ¼ 1, 2, . . . , J4Þ ð61Þ
XJ4j4¼1
�1r1, j4 � 1, ðr1 ¼ 1, 2, . . . ,R1Þ ð62Þ
0 � �1r1 � 1, ðr1 ¼ 1, 2, . . . ,R1Þ ð63Þ
0 � �2r2, j3 � 1, ðr2 ¼ 1, 2, . . . ,R2; j3 ¼ 1, 2, . . . , J3Þ ð64Þ
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XJ3j3¼1
�2r2, j3 � 1, ðr2 ¼ 1, 2, . . . ,R2Þ ð65Þ
0 � �2r2 � 1, ðr2 ¼ 1, 2, . . . ,R2Þ, ð66Þ
where Equation (60) is calculated from Equations (4)–(37) for the continuous mode, or Equations (4)–(35) and
(38)–(40) for the switching mode. The cost components can take any forms. �1r1, j4 and �1r1are the consequent belief
degree and belief rule weight of belief rules in Sub-rule-base 1, and �2r2, j3 and �2r2are the consequent belief degree and
belief rule weight of belief rules in Sub-rule-base 2.
4.4.3 Comparative characteristics and search solutions
We summarise the following comparative characteristics of simultaneous identification and two-step identification
from the points of optimality, pertinence, simplification, and computational speed.
(1) Optimality. The simultaneous identification does not discriminate and classify the structure identification
and parameter identification and tries to solve different levels of matters all at once. Theoretically it can
guarantee the optimal structure and parameter given sufficient domain information, but technically the
simultaneous identification is a very difficult problem and hardly achieves the global optimum because
the global optimisation problem belongs to the complexity class of NP-hard problems. Besides, in practice,
the information available is often inadequate, and decisions must be made with limited information and
time. In the light of solution complexity, information limitation, and decision timeliness, the two-step
identification abandons the sole irrational pursuit of unreachable optimality and pursue reasonable
near-optimal with full utilisation of limited information and time constraints. Because of the underdevel-
opment of extant global optimisation techniques and inadequate utilisation of known information, the
simultaneous identification would generate inferior solutions compared with the two-step identification
especially for complex problems.(2) Pertinency. The data produced in the operational process contain the special working pattern of the system.
With the BKM clustering algorithm and parameter approximation model, the two-step identification
exploits the historical information about demand, production, workforce, and inventory to mine the
potential patterns embodied in structure and parameter. This pertinent manipulation makes the two-step
identification more suitable for the specific problem under consideration. However, the simultaneous
identification does not make full use and seek a thorough understanding of known information, so the
solution deduced would be superficial and impertinent.(3) Simplification. Because the training of BRB is a very complex problem, a sensible disposition is to
decompose the difficult problem into a set of simple subproblems to reduce the degree of difficulty. The
two-step identification is one such divide-and-conquer approach and especially well suited to handling
large-scale, highly complex, nonlinear, and uncertain problems. In contrast, the simultaneous identification
does not simplify the solution for the problem under consideration and is only feasible for small-scale and
low-complexity problems.(4) Computational speed. With the simplification operation, the two-step identification could accelerate the
computational speed of the BRB training process. This is critical to make real-time decisions in the dynamic
data-driven system, including but not limited to the APP problem. Constrained by the underdevelopment of
extant global optimisation techniques, the simultaneous identification has not only a lower precision but also
a lower speed especially for highly complex problems.
Yang et al. (2007) proposed the use of the FMINCON function in MATLAB to train BRB. Chang et al. (2007)
improved the work and developed a gradient and dichotomy algorithm (GDA) to improve the local search
capability to train BRB parameters. For the following numerical case studies, GDA, genetic algorithm and
FMINCON functions are utilised to solve the nonlinear optimisation models of simultaneous identification and
two-step identification.
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5. Application of the BRBI method for two production cases
5.1 Case 1: HMMS paint-factory example
The HMMS paint-factory example (Holt et al. 1955), which has become a benchmark for comparison of APPmethods, is employed to illustrate the BRBI method under deterministic demand forecast. A 5-year range ofdemands from 1949 to 1953 is depicted in Figure B.1 in Appendix B. It contains periods of severe fluctuations,which imply that the planning decisions will be subjected to substantial severity. Four essential categories ofproduction related costs are considered herein, i.e. regular payroll cost Crp, hiring and firing cost Chf, overtime andunder-time cost Cou and inventory-related cost (including holding and shortage/backorder penalty cost) Cir. Thesecost components and the total cost functions are listed as follows:
CrpðtÞ ¼ C1Wt ð67Þ
ChfðtÞ ¼ C2ðWt �Wt�1Þ2
ð68Þ
CouðtÞ ¼ C3ðPt � C4WtÞ2þ C5Pt � C6Wt ð69Þ
CirðtÞ ¼ C7ðIt � ðC8 þ C9QtÞÞ2
ð70Þ
TC1t ¼
XtþT�1n¼t
ðCrpðtÞ þ ChfðtÞ þ CouðtÞ þ CirðtÞÞ ð71Þ
TC2t ¼
Xtn¼t�Tþ1
ðCrpðtÞ þ ChfðtÞ þ CouðtÞ þ CirðtÞÞ, ð72Þ
where Cnðn ¼ 1, . . . , 9Þ are constant coefficients, Qt ¼ Dft in Perspective 1 and Qt ¼ Dt in Perspective 2. The
coefficients Cnðn ¼ 1, . . . , 9Þ are defined as 340, 64.3, 0.2, 5.67, 51.2, 281, 0.0825, 320, and 0 in turn. The planninghorizon is T ¼ 12 months. Because no explicit forecasts have been recorded by the factory, a set of deterministicdemand forecasts, given in Table B.1 in Appendix B, are produced as substitutes to simulate the operation processwith comparison and sensitivity analyses. The absolute forecast error for the coming nth month is �ð1þ Þn�1
(� ¼ 0:05, ¼ 0:1), where � is the value for the next month, and is the growth rate. This represents the fact that animmediate forecast is more accurate than a distant forecast.In this case, the structure and parameter of BRB areadjusted by the two-step identification technique, while in the next case, both simultaneous identification and two-step identification techniques are used for comparison. The BRBI method in both continuous and discrete modes iscompared with LDR and PSH in a continuous setting and discrete setting respectively. The curves of production,workforce, over-/under-time hours and inventory using LDR and BRBI-CM (BRBI in continuous mode) in acontinuous setting are depicted in Figure 2. The curves using PSH and BRBI-SM (BRBI in switching mode) in adiscrete setting are depicted in Figure 3. The cost components using the four methods are listed in Table 2. In thecontinuous setting, BRBI-CM and LDR yield similar production and workforce decisions, but the cost componentsshow that BRBI-CM is better than LDR. LDR is an optimal strategy when the average forecast error is zero, butnot optimal when the average forecast error is not zero. Holt et al. (1955) analysed the performance of LDR withdifferent degrees of forecast accuracy. However, they did not elaborate how to improve their rules when the forecasterror is not zero. BRBI provides an advantageous strategy to fit in conditions where the forecast error is biased andmagnifying. In the discrete setting, the production rate is restricted to 400, 550, and 700, and workforce level to 75,95, and 110. As for production rate, PSH switches 11 times, while BRBI-SM switches nine times. As for workforcelevel, PSH still switches 11 times, because in PSH production, rate and workforce level have the same rule-antecedent, while BRBI-SM switches five times. BRBI-SM attains much less production change cost and total costthan PSH.
A group of trained belief rules for production and workforce under continuous mode at month 1 is listed inTables B.2 and B.3 in Appendix B for illustration. The belief-rule structure is concise and enables the decision-makerto improve it by tuning the rule weights and consequent belief degrees directly. BRBI can store favourable decisionexperiences into the belief rules to make new decisions, and dynamically train the belief rules using real-time demanddata. In making new decisions, workforce and inventory levels in the last month are considered in LDR, and only
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the inventory level is considered in PSH, while all the conditions of production, workforce, and inventory areconsidered in BRBI simultaneously. This mechanism ensures that BRBI can assist decision-making with a morecomprehensive consideration.
To test whether the BRBI method for APP is sensitive and reasonable, the cost coefficients in HMMS are variedto investigate three kinds of costs: hiring/firing cost C2, over-time/under-time cost C3 and inventory-related cost C7.Each kind of cost is studied at three levels: low (one-third of original value); medium (original value); and high(triple original value). Figure 4 depicts the sensitivity analyses of workforce, over-/under-time and inventory forBRBI-CM with differing cost structures. When the hiring/firing cost decreases to the low level, there is an increasingtendency to change workforce in response to the demand fluctuations. When the hiring/firing cost rises to the highlevel, there is a decreasing tendency to change workforce. The curves of over-/under-time and inventory present thesame sensitive phenomenon. These results illustrate that the BRBI method is sensitive to different cost structures bymaking reasonable planning strategies.
5.2 Case 2: an automotive production example
The automotive industry is challenged by fierce competition, high product variety, and shortened product life cycles.Automotive manufacturers should establish strategic, tactical, and operational decisions to keep their businessespowering ahead, in which the tactical APP is of paramount importance (Omar 2011). An automotive supplier inWuhan, China manufactures a mechanical component with three different variation models that can bedistinguished from the size and the amount of steel used. Related managers must plan the component’s production
Figure 2. Curves of production, workforce, over-/under-time, and inventory using LDR and BRBI-CM.
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on an aggregated basis for the coming year based on customers’ demands from the three models. The sales for the
aggregated unit in a 36-month period from 2008 to 2010 are depicted in Figure B.2 in Appendix B, and the
corresponding rolling interval style forecasts are given in Table B.4 in Appendix B. The forecast error has an
increasing tendency from immediate to distant periods. The first 12-month horizon is only used for initialisation,
and the other 24-month horizon is used for comparison. The cost components and the total cost functions are listed
as follows:
CrpðtÞ ¼ C1Wt ð73Þ
Figure 3. Curves of production, workforce, over-/under-time, and inventory using PSH and BRBI-SM.
Table 2. Cost components for the paint factory using LDR, BRBI-CM, PSH, and BRBI-SM.
Cost components
Continuous mode Switching mode
LDR BRBI-CM PSH BRBI-SM
Regular payroll 1,929,353 1,894,135 1,881,900 1,909,100Hiring and firing 20,539 25,797 255,593 85,197Over-/under-time 84,295 118,710 114,044 126,062Inventory 30,879 25,906 236,258 336,702Total cost 2,065,066 2,064,548 2,487,795 2,457,061
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CrmðtÞ ¼ C2Pt ð74Þ
ChfðtÞ ¼ C3 Wt �Wt�1½ �þþC4 Wt �Wt�1½ �
�ð75Þ
CouðtÞ ¼ C6 Pt � C5Wt½ �þþC7 Pt � C5Wt½ �
�ð76Þ
CirðtÞ ¼ C8 It½ �þþC9 It½ �
�ð77Þ
TC1t ¼
XtþT�1n¼t
ðCrpðtÞ þ CrmðtÞ þ ChfðtÞ þ CouðtÞ þ CirðtÞÞ ð78Þ
TC2t ¼
Xtn¼t�Tþ1
ðCrpðtÞ þ CrmðtÞ þ ChfðtÞ þ CouðtÞ þ CirðtÞÞ, ð79Þ
where Crp, Crm, Chf, Cou, and Cir are regular payroll cost, raw material and manufacturing cost, hiring and firing
cost, overtime and under-time cost, and inventory related cost respectively. The constant coefficients
Cnðn ¼ 1, . . . , 9Þ are defined as 3500, 500, 2800, 4900, 11.5, 6500, 5400, 3700, and 8100. The planning horizon is
T ¼ 12 months. ½x�þ ¼ maxðx, 0Þ and ½x�� ¼ maxð�x, 0Þ.This case study is designed to exhibit BRBI’s performance on dealing with interval demand forecast under both
simultaneous identification and two-step identification of BRBI’s structure and parameter. The NINP model for
generalised production planning (Jiang et al. 2008, Abass et al. 2010) is employed to conduct comparative analysis.
As a general guidance, three distinct preferences, i.e. pessimistic, neutral, and optimistic, are defined. The pessimistic
lower bound, centre, and upper bound of the inference output intervals in BRBI or the possibility degree intervals in
NINP are paralleled to pessimistic, neutral, and optimistic respectively. The curves of production, workforce, and
inventory using NINP and BRBI in three preferences are depicted in Figure 5, and the cost components using them
are listed in Table 3. BRBI-SI means the BRBI method under simultaneous identification of structure and
parameter, and BRBI-TI means the BRBI method under two-step identification of structure and parameter.
Figure 4. Sensitivity analysis for BRBI-CM with differing cost structures.
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A group of trained belief rules for production and workforce in neutral preference at Month 1 is listed in Tables B.5and B.6 in Appendix B for illustration.
The production, workforce, and inventory curves in pessimistic preference are illustrated in (a), (d), and (g),those in neutral preference are illustrated in (b), (e), and (h), and those in optimistic preference are illustrated in (c),(f), and (i) in Figure 5. The curves of the three decision variables using BRBI-SI and BRBI-TI keep almost the sameshape in three different preferences. This is because the BRBI method is a member of the rule-based inferencemethods owning logical antecedents and consequents. The matching and inference mechanism can deduce consistentresults with the same demand patterns. This feature verifies that the BRBI method can ensure credible and rationaldecisions are made in an uncertain environment. In contrast, the shapes of the curves obtained by NINP changegreatly and irregularly, since the transformed programming models must be reprogrammed under differentpossibility degree levels in parallel with different preferences. Without consistent condition-action rules as in the
Figure 5. Curves of production, workforce, and inventory using NINP and BRBI in three preferences.
Table 3. Cost components for the automotive manufacturer using NINP, BRBI-SI, and BRBI-TI.
Cost components
Pessimistic preference Neutral preference Optimistic preference
NINP BRBI-SI BRBI-TI NINP BRBI-SI BRBI-TI NINP BRBI-SI BRBI-TI
Regular payroll 20,692,000 20,625,500 20,237,000 21,416,500 21,259,000 20,930,000 22,988,000 21,822,500 21,598,500Manufacturing 33,991,500 34,871,500 34,296,000 33,530,000 34,954,000 34,392,000 35,267,500 35,045,000 34,487,000Hiring and firing 8,791,2,00 6,651,6,00 6,619,7,00 5,492,5,00 6,647,6,00 6,566,3,00 7,814,7,00 6,562,6,00 6,701,9,00Over-/under-time 49,049,050 12,827,750 13,643,500 24,970,200 781,050 590,800 57,644,150 10,772,150 10,789,250Inventory 18,659,180 22,865,340 13,630,290 15,505,430 23,769,270 14,637,750 28,458,280 24,752,950 15,653,620Total cost 131,182,930 97,841,690 88,426,490 100,914,630 87,410,920 77,116,850 152,172,630 98,955,200 89,230,270
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BRBI method, the programming results from NINP would be quite different with different model coefficients. Their
priority relationship can also be clarified from Table 3 in which the total costs generated by BRBI-SI and BRBI-TI
are rather lower than those generated by NINP in all three preferences. Besides, BRBI as an artificial intelligence
method can compensate the demand forecasting error and requires little computational effort through a belief-rule
structure with related inference and identification algorithms (Li et al. 2011), while NINP as a mathematical
programming method does not consider demand forecasting error and is computationally expensive and inefficient,
especially for large-scale, complex, and uncertain problems.The production and workforce curves acquired from BRBI-SI and BRBI-TI in three preferences appear to be
very similar in profile shape and variation range, but distinctions using these two alternative identification
techniques can be distinguished from the inventory curves. Under the fluctuating demand given in Figure B.2, the
inventory curves using BRBI-SI drift remarkably, whereas those using BRBI-TI increase approximately linearly.The regular modality of the inventory curves obtained by the latter demonstrates that BRBI-TI can identify the
fluctuating demand pattern precisely to provide appropriate production and workforce decisions. This contrast
verifies that BRBI-TI is more pertinent than BRBI-SI. In addition, the computational speed of BRBI-TI is faster
than that of BRBI-SI in the simulation process. From Table 3, BRBI-TI can cut total costs significantly in
comparison with BRBI-SI.
6. Conclusion and discussion
This paper proposed a BRBI method for aggregate production planning under uncertainty, particularly when
demand is forecast in uncertain form, and forecast error is magnifying and biased from immediate to distant
periods. Various forms of forecast can be represented and transacted in this method, but only deterministic and
interval styles are considered in this paper. A hierarchical BRBI framework was constructed that contains two sub-
rule bases for inference and determination of production and workforce respectively. The belief-rule structure isconcise and enables the decision-maker to improve it by tuning the rule weights and consequent belief degrees
directly. Based on RIMER and IER, an inference algorithm for BRBI with both deterministic and interval inputs
was developed. In order to make the method applicable to both continuous and discrete production settings, a
continuous mode and switching mode were proposed with different transformation techniques for the distributed
formulations. Because demands usually fluctuate with trends and seasonal variations, the structure and parameter
of BRB should be adapted dynamically to approximate the hidden patterns of planning situations. Two
identification methods, simultaneous identification and two-step identification, are put forward and analysed for
comparison. The two-step identification contains a belief k-means (BKM) clustering algorithm extended from k-
means and fuzzy c-means. BKM ensures that an optimal cluster can both facilitate human cognition and improve
inference accuracy.Most mathematical programming methods for APP, such as LDR and NINP, are constructed on Perspective 1
with the tacit assumption that demand forecasts are accurate. In real cases, however, demand forecast error always
exists and increases, so methods based on Perspective 2, such as PSH and BRBI, are better choices in such
situations. LDR is suitable for continuous settings with deterministic forecasts, PSH for discrete settings with
deterministic forecasts, and NINP for continuous settings with interval forecasts. BRBI can be applied to both
continuous and discrete settings with both deterministic and interval forecasts. From the paint-factory example,
LDR will not engender optimal solutions with magnifying and biased forecast errors, and PSH has limitations in
smoothing production rate and workforce level because of insufficient consideration of rule antecedents. In
contrast, BRBI could gain from a better performance from a flexible belief-rule structure and rational inference and
identification algorithms. By varying the appropriate cost coefficients, BRBI shows the ability to make reasonable
planning strategies for different cost structures sensitively. When the forecast is given in interval forms as in the
automotive production example, BRBI exhibits a superiority over mathematical NINP method in differentpreferences. In comparison with the simultaneous identification, the pertinent two-step identification can identify
the fluctuating demand pattern efficiently and precisely to provide appropriate production and workforce decisions,
and improve the computational speed by virtue of its divide-and-conquer strategy.In this paper, only deterministic and interval forecasts are considered, but other forms of forecasts, such as fuzzy,
are also always involved in APP problems. So, it is beneficial to propose an approach for BRBI to handle fuzzy
forecasts based on the fuzzy ER (FER) algorithm (Yang et al. 2006b).
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Acknowledgements
This work was supported by the Natural Science Foundation of China (No. 60674085, No. 70971046, No. 60736026, No.90924301, No. 71125001, and No. 61174147), and the International Science and Technology Cooperation Program of theMinistry of Science and Technology of China (No. 2008DFA12150). The third author was also supported by the UKEngineering and Physical Science Research Council under Grant No. EP/F024606/1. We are also grateful to the anonymousreviewers for their valuable comments and constructive criticism.
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Appendices
Please see Appendix A and Appendix B at the following link: https://skydrive.live.com/view.aspx/.Public/Appendices.doc?cid=c805202c5ba2cc47
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