Multiproduct aggregate production planning with fuzzy demands and ...
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Multiproduct Aggregate Production Planning With Fuzzy Demands and Fuzzy Capacities Richard Y. K. Fung, Jiafu Tang, and Dingwei Wang
Abstract—Given the uncertain market demands and capacities in production environment, this paper discusses some practical approaches to modeling multiproduct aggregate production planning problems with fuzzy demands, fuzzy capacities, and financial constraints. By formulating the fuzzy demand, fuzzy equation, and fuzzy capacities, a fuzzy production-inventory balance equation for single period and a dynamic balance equation are formulated as fuzzy/soft equations and they represent the possibility levels of meeting the market demands. Using this formulation and interpretation, a fuzzy multiproduct aggregate production planning model is developed, and its solutions using parametric programming, best balance and interactive techniques are introduced to cater to different scenarios under various decision making preferences. Using the proposed models and techniques, first, the decision maker can select a preferred production plan with a common satisfaction level or different combinations of preferred possibility level and satisfaction levels, according to the market demands and available production capacities, and second, the obtained structure of the optimal solution can help decision maker in aggregate production planning. The decision maker can also make a preferred and reasonable production plan corresponding to one’s most concerned criteria. Hence, decision makers not only can come up with a reasonable aggregate production plan with minimum efforts, but also have more choices of making a preferred aggregate plan based on his most concerned criteria. These models can effectively enhance the capability of an aggregate plan to give feasible family disaggregation plans under different scenarios with fuzzy demands and capacities. Simulation and the results of analysis on the proposed techniques are also given in detail in this paper. Index Terms—Aggregate production planning, fuzzy demands, fuzzy equation, mathematical programming, interactive procedure.
I. INTRODUCTION
A
GGREGATE PRODUCTION PLANNING (APP) [1] is concerned with the determination of production, inventory, and workforce levels required to meet aggregate market demands at the level of product type. Product families with similar product costs and demand seasonality are grouped into a product type. APP is an important tactical level planning activity in a Manuscript received September 12, 2000; revised April 30, 2003. This paper was supported by a Strategic Research Grant (SRG) from City University of Hong Kong, Project Number 7001227, the National Natural Science Foundation, NSFC No. 70002009 of P.R.C., the Excellent Youth Teacher Program of Ministry of Education of China, and Liaoning Provincial Natural Science Foundation, No. 20022019. This paper was recommended by Associate Editor W. Gruver. R. Y. K. Fung is with the Department of Manufacturing Engineering & Engineering Management, City University of Hong Kong, Kow loon, Hong Kong, P.R.C. (e-mail: [email protected]). J. Tang and D. Wang are with the Department of Systems Engineering, School of Information Science and Engineering, Northeastern University, Shenyang 110004, P.R.C. Digital Object Identifier 10.1109/TSMCA.2003.817032
production management system. Other forms of family disaggregation plans, such as master production schedule, capacity plan, and material requirements plan all depend on APP in a hierarchical way [2]. Holt [3] suggested an optimal staffing, production, and inventory policy for a single manufacturer under the assumption of a given sales forecast, and they proposed a continuous time-varied model (HMMS) [3]. HMMS model is one of the best known classical models for tackling aggregated production and inventory planning problems. It aims at minimizing the total cost of regular payroll, overtime and layoffs, inventory, stock-outs, and machine set up. The linear decision rules for production level and workforce level are derived to deal with problems in a single product category. Since then, much attention has been directed toward aggregate production planning, and different models and approaches have been developed. The approaches to APP vary from mathematically optimal procedures [3] and [4], simulation and search methods [5] to heuristic decision rules [6]. Among the optimal models, linear programming has received the widest acceptance. These models can be categorized into deterministic optimization models [3], [7], stochastic programming models [7], [8], and fuzzy optimization models [6], [9]–[11]. Bergstrom and Smith [7] generalized the HMMS approach to a multiproduct formulation, which was further extended by Hausman [8] into a stochastic programming model to cater to the randomness in product demand. Bitran and Yanassee [12] considered the problem of determining production plans over a number of time periods under stochastic demands with known distribution functions. A distribution bound for the problem could be obtained by a deterministic approximation to the original stochastic problem. These stochastic programming models and methods are usually based on the concept of randomness and probability, and they are limited to tackling uncertainties with some probability distributions. The decisions achieved by stochastic programming can only take the form of a distribution function, which can do little to help decision making in practical scenarios. Some other forms of uncertainties, such as fuzzy demands, capacities with tolerance, imprecise process times, etc., are common in aggregate production planning. These types of uncertainties cannot be fully described by frequency-based probability distributions. Therefore, there is a need to formulate APP using fuzzy set theory [13], [14] and fuzzy optimization methods [6], [9], [10], [15]–[18]. Rinks [6] tackled aggregate production planning problems using fuzzy logic and fuzzy linguistics, and developed a production and a workforce algorithm using a series of approximately 40 relational assignment rules. Lee [9] discussed
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fuzzy aggregate production planning problems for single product types, under fuzzy objective, fuzzy workforce levels and fuzzy demands in different time periods. The workforce level and product demands were expressed in fuzzy numbers with tolerance. In this paper [9], a linear programming model with fuzzy objective and fuzzy constraints was developed, and fuzzy solutions under different levels could be attained using parametric programming technology. Wang and Fang [11] developed a genetic-based approach for linear APP with fuzzy objectives and resources. Tang [10], [19] developed a fuzzy approach for formulating integrated production and inventory problems with triangular fuzzy demands. It aims to minimize the total costs of quadratic production cost and linear inventory holding cost. A fuzzy solution is developed by which the solutions corresponding to different satisfaction levels are obtained. These fuzzy optimization models and approaches mainly consider a single product type [3], [6], [9] and they tend to overlook the financial constraint in a practical business environment [2], [3], [6], [7], [9]–[11], [19]. In general, a production plan is not only constrained by production capacities (e.g., machines and workforce), but also by financial resources, i.e., capital level, and all these constraints should be considered in APP. At present, there are no appropriate formulation and interpretation approaches to production-inventory balance equation under a fuzzy environment suitable for APP. In practical production planning systems, the aggregate market demands of a product in each period are not always clear, but uncertain in nature. These demands can be expressed as
tail in Section II. Section III discusses the formulation of fuzzy demands, fuzzy capacities, fuzzy production-inventory balance equation, and the development of the FMAPP model. The parametric programming-based fuzzy solution and the interactive procedures for the model are discussed in Sections IV and V, respectively. An illustrative example and the results of simulation are presented in Section VI, and the conclusion is given in Section VII. II. NOTATION AND FORMULATION Assume that a company manufactures types of products to meet the market demand over a planning horizon . APP is to determine the production, workforce, inventory, and backorder levels for product in period at minimum total cost. A multiproduct aggregate production planning (MAPP) problem under financial constraints with the aim to minimize the total can be formulated by crisp mathematics into cost a MAPP model as follows: (1) (2) (3) (4) (5)
1) random number with probability distribution; 2) fuzzy triangular number; 3) interval number, while the production capacities can be represented as fuzzy numbers with tolerance. Taking into account these factors and the financial constraints in the production environments, this paper proposes a practical approach to modeling multiproduct aggregate production planning problems with fuzzy demands and fuzzy capacities. With the use of fuzzy equation and fuzzy addition, the fuzzy production-inventory balance equation for single period and the dynamic balance equation are formulated as soft equations in terms of degrees of truth, which can be interpreted as possibility levels for meeting the market demands. With this formulation and interpretation, a fuzzy multiproduct aggregate production planning (FMAPP) model is developed. This model can take the form of a parametric programming model, a best balance model or an interactive model, to cater to different scenarios and decision maker’s preference. A parametric programming-based fuzzy solution and an interactive procedure are proposed to solve the problem. With the proposed models and solution, decision makers can make a reasonable APP economically, and more readily under his preferred criteria. This approach effectively allows an aggregate plan to give a feasible family disaggregation plan under different circumstances, especially under fuzzy demands and fuzzy capacities. The remainder of the paper is organized as follows: Notation and formulation of the problem will be presented in de-
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(6) (7) (8) where decision variable, inventory level of product at the end of period ; decision variable, production level of product in period ; decision variable, assigned workforce level for product in period , decision variable, back-order level of product in period ; demands for product in period , unit production costs including materials and overhead for product in period , unit workforce costs for product in period ; unit inventory costs for product in each period; unit back-order costs for product in each period; unit production capacity (man-hours) for product ; available workforce level in period , available financial resources in period . production capacity allowance percentage; the capacity of the warehouse; the initial inventory level for product ; the initial back-order level for product .
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In the MAPP model, (1) represents the objective of minimizing the total cost of production, inventory and back-order, (2) is the production-inventory balance equation, (3) and (4) denote the constraints of production capacity and financial resources in each period, (5) gives the warehouse capacity constraint, (6) reflects the relation between production level and production capacity, (7) indicates that at least one of the backorder and inventory level in a period is zero, and (8) implies the initial and nonnegative constraints of the decision variables. III. FUZZY MULTIPRODUCT AGGREGATE PRODUCTION PLANNING
B. Fuzzy Demands and Their Addition Let the fuzzy demands be a triangular fuzzy number, i.e., , where , and are the most possible, most pessimistic and most optimistic values of fuzzy demands, respectively. The membership function expresses the possibility measurement of fuzzy demands according to the decision maker’s view. For example, it may be with highest possibility, and with the and and between lowest possibility, and between and with a possibility level between zero and one. can be given as Hence the membership function
A. Formulating the Fuzzy Multiproduct Aggregate Production Planning Model In real-world scenarios, it is difficult to determine product dein advance in each period precisely, especially over mands a long planning horizon. These demands are not a certainty, but an estimate or imprecise/interval number. It is assumed in this is a triangular fuzzy number, denoted paper that the demand . The workforce level in each period by is assumed to take the form of a fuzzy number, denoted by , where the available capacity level is , and the largest acceptable tolerance is . Hence, MAPP problems can be formulated into a fuzzy mathematical model. Owing to the fuzziness of demand, the production-inventory balance (2) is not a crisp, but a fuzzy equation, namely, a soft equation in this paper. It implies that the equation is met in terms of a degree of truth. As a result, an MAPP problem with fuzzy demands and fuzzy production capacities can be formulated into a fuzzy multiproduct aggregate production planning (FMAPP) model as follows:
(9) (10) (11) (12)
Let and be two triangular fuzzy numbers, the addition of two fuzzy numbers remains a triangular fuzzy number, which . Similarly, can be denoted as this concept can be extended to the addition of any number of triangular fuzzy numbers. be the total demands of product from Let the period 1 to a certain period . According to the extension is defined as principle [14], the membership function (17) where and are the maximum and minimum operators respectively. represents the possibility measurement of total fuzzy demands of product from period 1 to period according to the decision maker. can also be deObviously, the total demands scribed by a triangular fuzzy number, i.e., . Triangular fuzzy demands and total demands can be illustrated as shown in Fig. 1(a) and (b), respectively. C. Fuzzy Equations and Their Addition Production-inventory balance (10) in the FMAPP model is not a crisp equation, but a fuzzy one. It implies that the equation is met in terms of a degree of truth, and it can be expressed as depending on the fuzzy demands a membership function . Therefore
(13) (18) (14) (15) (16) where denotes a soft equation. It implies that the productioninventory balance equation is achieved in terms of a degree of truth. The formulations and interpretation of fuzzy demands, fuzzy capacity and fuzzy production-inventory balance equation are explained in the following subsections.
With this formulation, the production-inventory balance (10) indicates that the sum of production level in this period and the difference between inventory level and back-order level in the previous period minus the difference between inventory and back-order level in this period should not differ from the demand too much.The larger the difference, the smaller the degree of truth that the equation holds, and the less likely that reflects the market demands can be met. In this sense, possibility level at which the market demands are met. For ex, the difample, if
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Fig. 1. Formulation of fuzzy demands and fuzzy production-inventory balance equation.
ference is zero, and the production-inventory balance equation is strictly satisfied, and the market demand is most possibly met, i.e., the possibility level equals one. On the contrary, if , the difference is greater than the tolerance, and the production inventory balance equation holds with a degree of truth being zero, and the market demand is unlikely met. Of course, when the right-hand of the , the market demand will be likely equation becomes met, however it will bring higher inventory levels and holding costs, thus, it is not always practical, and therefore the equation has a pessimistic degree of truth in this case. This formulation is illustrated as shown in Fig. 1, and the interpretation coincides is crisp, and the with the crisp situation when the demand production-inventory balance equation holds absolutely over the entire planning horizon. With this interpretation, given the decision maker’s request that the APP is to meet the market demands for product in a period with possibility level , the soft equation can be expressed as follows: (19) as illustrated in Fig. 1. The addition of two soft equations is also a soft equation and its degree of truth is defined as the logic addition of the degrees of truth for each member of soft equations according to the extension principle [14]. It can be explained in the following example. and be two soft equations, then their Let in which the degree of addition is denoted as satisfies truth
according to max-min theory. Similarly, this operation can be extended to the addition of any number of soft equations. By applying the addition of soft equations to (10), we can have (21) more general, for (22) D. Formulation of the Fuzzy Production-Inventory Balance Equations With the formulation of fuzzy addition and fuzzy equation, the fuzzy production-inventory balance equation in a single period can be interpreted as the possibility level of the APP meeting the fuzzy demands. It can be formulated as in (19). Under the same types of fuzzy demands , the production-inventory balance equation in a single period is equivalent to the dynamic balance equation for product from period 1 to , (19) certain period , i.e., for is equivalent to (23) which can be illustrated as shown in Fig. 1(d). Therefore, the production-inventory balance equation under a fuzzy environment can be formulated as in (19) or (23). E. Formulation of the Fuzzy Production Capacity
(20)
in period Assuming the production capacity number with tolerance, its membership function
is a fuzzy re-
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flects the decision maker’s level of satisfaction with the consumption of production capacity, and is defined as follows: (27) (24) (28) For example, if the consumption of production capacity is lower than the available capacity , decision maker is of course fully satisfied. As the consumption of capacity increases, decision maker’s satisfaction level decreases, and the decision maker is becoming dissatisfied when the consumption reaches the toler. ance point For a given threshold level , the decision maker’s satisfaction level with the consumption of production capacity can be , i.e., formulated as
(29) (30) (31) (32) (33)
(25) (34) In general, the fuzzy demand may be extended to the cases of general L-R type and trapezoidal numbers. In those cases, the APP can be formulated into the above model only if the fuzzy demands in each period are of the same type. IV. PARAMETRIC PROGRAMMING MODEL FMAPP model is a model of fuzzy nonlinear programming with fuzzy inequality and equality constraints. Unlike a crisp model, the optimal solution here is not crisp but a fuzzy set “decision” which has different meanings depending on its definition and formulation. APP is an important upstream level decision activity, and the decision is the source of family disaggregation plan at downstream levels. Therefore, the sole optimal solution to APP cannot guarantee that the feasible disaggregation plan will always be achieved. The diversity of the solutions and multiple solutions to APP can offer more opportunities to achieve the feasible disaggregation plan particularly under a fuzzy environment. From this point of view, the solution of the model is “optimal” subject to some preferred possibility levels for meeting the market demands and satisfaction levels with consumption of production capacities. In this case, a fuzzy solution [8], [14], [18], [20]–[22] can be obtained using parametric programming techniques. is given to denote decision A preferred level maker’s request for possibility level at which the APP meets the market demands, the soft equation can be defuzzified such that the cumulative demand of product type from period 1 to a future period should be met with a possibility level higher than . At the same time, another threshold level can be given to express the decision maker’s satisfaction level with the consumption of production capacity which should not be less than . With this understanding, the FMAPP model can be transformed into the following equivalent parametric aggregate production planning (PAPP) model (26)
. The production-inventory balance equation where in the original fuzzy optimization model FMAPP is expressed in terms of inequality constraints (27) and (28) as per the decision maker’s request for possibility level at which APP can meet market demands. and are the preferred parameters, which reflect the constraints of the possibility level of meeting the market demands and satisfaction level with the consumption of production capacity, respectively. For given preferred satisfaction levels and , the PAPP model becomes a parametric nonlinear programming model. Let be an optimal solution to the parametric nonlinear programming model, the following properties on the solution to the PAPP model can be easily proved. increases with the Property 1: In the case of common value of and . increases with the . Property 2: Given a specified increase with the . Property 3: Given a specified For given preferred satisfaction levels and , the optimal sois a fuzzy solution to the origlution inal fuzzy optimization problem FMAPP. That is to say, it’s an optimal solution in a fuzzy sense, because it depends on the decision maker’s preference on the possibility level and the satisfaction level . In particular, the optimal solution is crisp when . Therefore, the fuzzy solution to FMAPP reflects the decision maker’s preference and subjectivity, and it can increase the chances and ability of constructing an APP to guarantee the feasibility of a family disaggregation plan, especially under a fuzzy environment. Of course, in the PAPP model, the possibility level and the satisfaction level may have common or different values. Under some practical situations, the decision maker may expect a higher or lower possibility level of meeting the market demands for a certain product type in a certain period, hence different possibility levels can be applied. Similarly, different satisfaction levels may also be selected in different periods to suit practical situation and the decision maker’s preference. In summary, the above model is flexible and can reflect the decision maker’s preference in a fuzzy environment.
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V. BEST BALANCE MODEL AND INTERACTIVE PROCEDURES As mentioned in the previous section, the total cost increases proportionally with the possibility level and the satisfaction level . In this case, decision maker can make a reasonable plan in light of his preferred possibility level and satisfaction level. However, in some cases, decision maker not only concerns about total cost, but also prefers to make a balance among total cost, possibility level of meeting market demands and satisfaction level on the consumption of production capacities. denote the decision maker’s required possiLet bility level of meeting market demands from period 1 to a speand denote the decision maker’s satcific period isfaction levels with production capacity consumption in period and the total cost in planning horizon respectively. Therefore, the APP problem concerns with not total cost alone any more, but a best balance among the total cost, the possibility of meeting market demands and the satisfaction with consumption of production capacity, and it can be expressed in a balanced aggregate production planning (BAPP) model as follows: (35) (36) (37)
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unique solution. However, this optimal solution may not be desirable or acceptable to decision maker in practice, especially under a fuzzy environment. On the contrary, multiple solutions under different criteria may be preferred. The decision maker’s most important criteria may include minimizing the total cost, maximizing the possibility level of meeting market demands on certain products in a specified period, and maximizing the satisfaction level with consumption of production capacities over specified periods. To cater for this situation, an interactive procedure for APP is proposed to help the decision maker make a preferred plan from a pool of optimal solutions under different criteria. The interactive procedure can be described as follows: A. Interactive Procedure for APP Step 1) Initialization: Given an initialized threshold reflecting decision maker’s overall acceptable satisfaction level with total cost, consumption of production capacities, and possibility level required for meeting market demands. Step 2) Input the decision maker’s most concerned criteria indexes (CI) set
where
represents the criterion for total cost.
(38) (39) (40) (41) (42)
represent the criteria for meeting market demands in period ; of product represent the and criteria for the consumption of production capacities . in period Step 3) Construct an interactive aggregate production planning (IAPP) model as follows:
(43) (49) (50)
where the original production-inventory balance equation is is the costs incurred in period given in (36), and have been described in (17) and (24) respectively, is defined as while
(51)
(44) (52) is an aspiration level of the total cost prewhere is an acceptable tolerance, ferred by decision maker, and say 5%–15% of . The BAPP model can be transformed into the following nonlinear programming model: (45)
(46) (47) (27), (28), (30), (31), (32), (33), (34)
(48)
and it can be solved by traditional nonlinear optimization techis a crisp and niques. The optimal solution
(53) (54) (30), (31), (32), (33), (34)
(55)
denotes the possibility level of where meeting the market demands on product in period and behaves in the same way as in (18), and is an aggregated satisfaction level with the criteria are the of concerns, and indicators of criteria index, and only one of them equals 1 at a time. and optimal solution Step 4) Set initial criteria index pool to be empty.
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TABLE I BASIC DATA AND MODEL PARAMETERS OF THE SIMPLE EXAMPLE
Step 5) Reset indicators of criteria indexes as follows:
Step 6) Solve the IAPP model by traditional constrained optimization techniques, and reset the optimal , where solution pool is the optimal solution with the th criterion. , go to Step 8; else, , go to Step 7) if Step 5. Step 8) Input decision maker’s most preferred criteria and display the corresponding optimal solutions, stop. Of course, in this interactive procedure, multiple indicators for criteria indices can equal 1 simultaneously, and in this case the indicator vector represents a combination of multiple criteria at a given time. The values of indicator can be further extended to represent the weights of the criteria when it assumes a value in [0, 1], and the multiple criteria can be also considered simultaneously. With this interactive procedure, an optimal solution pool [23] consisting of a set of potential optimal solutions with various criteria will be obtained [24], from which decision maker can interactively establish a production plan to suit particular circumstances. The interactive procedure and the optimal solution pool reflect decision maker’s preference, and can offer decision maker more choices to facilitate the establishment of downstream family disaggregation plans under a fuzzy environment. VI. ILLUSTRATIVE EXAMPLE AND ANALYSIS To illustrate the models proposed in this paper and assess the effect of the possibility level and the satisfaction level on the total cost and production plan in the parametric programming model, a simple simulation example using optimization tool box of software Matlab 5.0 is presented in this section. The basic datum of the example and model parameters are shown in Table I. For this simple example, two types of products denoted by product 1 and product 2, are produced in periods 1–4 to meet the customer demands, of which the value is an approximate
one and it can be formulated as a triangular fuzzy number. For example, the demands of product 1 in period 1 is 85 units most possibly, at least 80 units (pessimistically), and not more than 90 (optimistically). The unit production costs of product may be varied with product types and periods, for example product 1 has identical unit costs in different periods and it for product 2. Other input data has different unit costs and parameters, e.g., workforce costs, capacity, initial inventory level, inventory costs, back-order costs capacity coefficients can be explained in similar way. The simulation is conducted in three parts. The first part aims to analyze the effect of the parameters on the aggregate plan in the parametric programming model. The second and last parts focus on performance of the best balance model (BAPP) and the interactive procedure respectively. The simulation results and analysis are presented as follows. A. Simulation of the Parametric Programming Model and the Results and on the Total 1) Effect of Common Values of Cost: To analyze the effect of the common values on the total cost, common level and are scaled by discretesizing [0, 1] in equal steps of 0.05. Table II presents the production, workforce, inventory and back-order levels for product 1 and 2 in each period and the corresponding total costs when the value of both and varies in steps from 0.60 to 1.0. The effect of the common values on the total cost are depicted in Fig. 2. It can be seen from Table II that the total cost increases with and linearly as shown in Fig. 2. The simulation results suggest the optimal production policy that the inventory levels ) and of product 1 in each period (except when product 2 in the final period are zero, and the back-order in each period is absolutely not permitted, and the workforce level is proportional to the production level. These results coincide with the optimal production policies as shown in Table III. 2) Effect of Level on the Total Cost Given Specified Values of : In this part of the simulation, two specified values of the and , and varies possibility level are used, i.e., from 0.0 to 1.0 discretely in equal steps of 0.10. The results shown in Fig. 4 show that with both of the specified values of ,
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TABLE II AGGREGATE PRODUCTION PLAN WITH COMMON VALUES OF
the total cost increases linearly with , i.e., and . As indicated in the formula, however, there are no great changes in the total cost for different levels of for a specified . Hence, has little effect on the total cost for a specified . That means for a given possibility level of meeting the market demands, the satisfaction with the consumption of capacities has little effect on the total cost. Results of the simulation also conclude that for any specified , the inventory costs increase with , while both the production cost and workforce cost maintain at certain levels except for the . The average inventory cost of workforce cost when increases proportionally to . It implies that both the production cost and the workforce cost are only dependent on the possibility level of meeting the market demands. In fact, in order to maintain the possibility levels of meeting the market demand, the tight constraints of production capacity in each period can only bring more inventory levels and warehouse holding cost, while the production and workforce costs do not decrease as a
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Fig. 2. Total cost varies with common values of theta and gamma.
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TABLE III OPTIMAL STRUCTURE OF AGGREGATE PRODUCTION PLAN AND TOTAL COST
Fig. 3.
Fig. 4.
Total cost varies with gamma given specified theta.
Fig. 5.
Total cost varies with theta given specified gamma.
Total cost varies with theta and gamma simultaneously.
whole. This result reveals a hint that the relaxation of tolerance of the production capacity in each period may decrease the total costs. The simulation also implies that as increases, the production level of product 2 increases in the first two periods and decreases in the last two periods; and the production levels for product 1 are almost unchanged with except for the last period, and it is produced in the optimal way that the inventory and backorder levels in each period remain zero. The optimal production policies shown in Table III help explain this situation. It can be concluded that the results of the simulation not only reflect the effect of on total cost given a specified , but also imply that has substantial impacts on the total cost and hence the production plan, while has relatively little effect. 3) Effect of on the Total Cost Given Specified Values of : In this part of the simulation, two specified values of , i.e., 0.7 and 1.0 are selected, and varies from 0.0 to 1.0 discretely in equal step size of 0.10. The results are shown in Fig. 5. It can be seen that for any given , the total cost increases , linearly with , i.e., . The production and level of product 2 increases rapidly in the first two periods and decreases in the last period with , whereas the production level
for product 1 in each period satisfies that the inventory level in each period is almost zero for any value of . As shown in Fig. 5, there is significant difference in the total cost between neighboring levels of with the same values of , while there are little difference between neighboring levels of with the same . It can be also concluded that has great impacts on the total cost and APP, while has relatively little effect. 4) Combined Effects of and on the Total Cost and Production Plan: In this part of the simulation, 100 sets of values of and are used, where 20 sets are with common values, 40 sets are with either or specified, and the remaining are random numbers in [0 1]. The resulting production, workforce, inventory and back-order levels, as well as the total cost behave linearly as shown in Table III. The total costs with varying and change as shown in Fig. 3. It can be seen from the optimal structure depicted in Table III that the production level for product 1 decreases slightly in the first three periods, and then increases slightly with and rapidly with in the last period. On the other hand, the production level for product 2 increases significantly in the first two periods and then decreases rapidly with and slightly with respectively
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TABLE IV AGGREGATE PRODUCTION PLAN BY WAY OF INTERACTIVE PROCEDURE
in the last two periods. The workforce level varies proportionally to the production level. The inventory level for product 1 is slightly affected by both and , and mostly around zero over the entire planning period, nevertheless the inventory level for product 2 increase rapidly with both and in the planning periods except in the last one. The back-order level for each product over the entire planning period remains zero owing to the high back-order penalty imposed in this example. The optimal production policies coincide with the results in the aforementioned three parts of the simulation. As shown in Table III and Fig. 3, the total cost increases significantly with the level and only slightly with . So far, it can be concluded that the
parameter plays a more influential role than in the model, hence the possibility level has more significant impacts on the production plan. B. Simulation Results on the Best Balance Model The best balance model is simulated using the Minimax function in the optimization tool box of the software Matlab 5.0. The resulting best balance degree, as well as the production, workforce, inventory and back-order levels and the total cost are shown in the first row in Table IV. It can be seen that the best balance among them is at a satisfaction level of 0.5455 with a cost of 170 954 pounds. That means when a higher possibility level
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of meeting market demand or a satisfaction level higher than 0.5455 is requested, the total cost will exceed 170 954 pounds, giving a less satisfaction level than 0.5455 with total cost, and vice versa. C. Simulation Results on the Interactive Procedure Given an initial overall acceptable satisfaction level with the total cost,consumption of capacities and possibility level , the interactive procedure is implemented using the constrained optimization function in Matlab 5.0, the preferred solutions under different criteria are obtained as shown in Table IV. Following the best balance criteria, the criteria number 0 indicates the aggregate plan under criteria of minimizing the total costs. The lows with criteria numbered 1–4 correspond to the solutions under criteria of maximizing possibility level of meeting market demands of product 1 in periods 1–4, respectively. Similarly the lows with criteria number 5-8 correspond to the solutions under criteria of maximizing possibility level of meeting market demands of product 2 in periods 1–4, respectively. While the criteria numbered 9–12 correspond to the ones of maximizing satisfaction of consumption of production capacity in periods 1–4, respectively. It can be seen from the Table IV that , the optimum satisfaction given an acceptable level level of the total cost can reach 0.6297 with a total cost of 170 406 pounds. While the possibility levels of meeting the market demands of products 1 and 2 in each period can reach 1.0, and the corresponding satisfaction levels with consumption of capacities can reach 1.0 too. It can be seen from the total cost column in Table IV that the minimum total cost is 170 406 pounds achieved at criteria number 0, and the maximum total cost is 171 575 pounds obtained at criteria number 5 and 12, and there is a difference among other criteria, particularly the differences is distinctive among different types of criteria. For example, there is differences of 387 pounds between the minimum value of total costs under criteria of possibility of meeting market demands and of satisfaction of consumption of production capacities. Moreover, there is distinctive differences in solution under different different types of criteria. Both the possibility levels under criteria and satisfaction levels under criteria numbered numbered are achieved as 100%, which coincides the actual scetype nario. D. Implication for Decision Procedure In summary, on the one hand, decision maker can select a preferred production plan with a common satisfaction level, e.g., according to the market demands and available production caare chosen, the third row in Table II pacities, if shows the corresponding APP. Of course, one may choose different combination of and based on the information and implication given in Figs. 2–5. The structure of the optimal solution presented in Table III can help decision maker in aggregate production planning. On the other hand, one can choose preferred production plan corresponding to one’s most concerned criteria, e.g., if one concerns most with best balance among the total cost, meeting market demands and consumption of capacities, then the first row in Table IV gives the corresponding plan. Similarly, if the total cost is of major concern, then the second
row in Table IV offers the feasible plan. Of course, if the criteria of meeting market demands for product 1 in period 4 is of most concern, then the sixth row in Table IV suggest a feasible aggregate plan. VII. CONCLUSION Taking into account the fuzzy demands and fuzzy capacities in each period, this paper proposes a fuzzy approach to formulating MAPP problems under financial constraints. The fuzzy production-inventory balance equation is formulated as a soft equation with a degree of truth, and it can be interpreted as the possibility level of meeting the market demands. With this formulation and interpretation, a FMAPP model is developed, and it is transformed into a parametric programming model and a best balance model to cater for different scenarios and decision maker’s preferences. A parametric programming based fuzzy solution and an interactive aggregate production planning procedure to solve the problem are also discussed. With the proposed models and approaches, on the one hand, first, the decision maker can select a preferred production plan with a common satisfaction level or different combinations of possibility and satisfaction levels, according to the market demands and available production capacities, secondly, the structure of the optimal solution presented in Table III can help decision maker in APP. On the other hand, the decision maker can also make a preferred and reasonable production plan corresponding to one’s most important criteria. The approach could not guarantee a global optimal solution, however the solution is near optimal and it can effectively enhance the capability that an aggregate plan can have feasible family disaggregation plans under different circumstances, especially under the environment of fuzzy demands and fuzzy capacities. ACKNOWLEDGMENT REFERENCES [1] Handbook of Operations Research: Models and Applications, J. Morder and S. E. Elmaghraby, Eds., Van Nostrand, New York, 1978, pp. 127–169. [2] L. Ozdamar, M. A. Bozyel, and S. I. Birbil, “A hierarchical decision support system for production planning (with case study),” Eur. J. Oper. Res., vol. 104, pp. 403–422, 1998. [3] C. C. Holt et al., Planning Production Inventories and Workforce. Englewood Cliffs, NJ: Prentice-Hall, 1960. [4] U. Akinc and G. M. Roodman, “A new approach to aggregate production planning,” IIE Trans., vol. 18, no. 1, pp. 88–94, 1986. [5] R. G. Schroeder, Operations Management: Decision Making in the Operations Function. New York: McGraw-Hill, 1989. [6] D. B. Rinks, “The performance of fuzzy algorithm models for aggregate planning and differing cost structures,” in Approximate Reasoning in Decision Analysis, M. M. Gupta and E. Sachez, Eds, The Netherlands: North-Holland, 1982, pp. 267–278. [7] G. L. Bergstrom and B. E. Smith, “Multi-item production planning-an extension of the HMMS rules,” Manage. Sci., vol. 16, pp. 614–629, 1970. [8] W. H. Hausman and J. D. McClain, “A note on the Bergstrom-Smith multi-item production planning model,” Manage. Sci., vol. 17, pp. 783–785, 1971. [9] Y. Y. Lee, “Fuzzy set theory approach to aggregate production planning and inventory control,” Ph.D. dissertation, Dept. of Industrial Engineer, Kansas State Univ., Manhatten, 1990. [10] J. Tang, R. Y. K. Fung, and D. Wang, “A fuzzy approach to modeling production and inventory planning,” in Proc. 14th IFAC World Congress, vol. A, Beijing, R.O.C., 1999, pp. 261–266.
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[11] D. Wang and S.-C. Fang, “A genetics-based approach for aggregate production planning in fuzzy environment,” IEEE Trans. Syst., Man, Cybern. A, vol. 27, pp. 636–645, May 1997. [12] G. R. Bitran and H. H. Yanassee, “Deterministic approximations to stochastic production problems,” Oper. Res., vol. 32, pp. 999–1018, 1984. [13] R. Y. K. Fung, K. Popplewell, and J. Xie, “An intelligent hybrid system for customer requirements analysis and product attribute targets determination,” Int. J. Prod. Res., vol. 36, no. 1, pp. 13–34, 1998. [14] R. Y. K. Fung, J. Tang, Y. L. Tu, and D. Wang, “Product design resources optimization using a nonlinear fuzzy quality function deployment model,” Int. J. Prod. Res., vol. 40, no. 3, pp. 585–599, 2002. [15] S. Chanas, “Using parametric programming in fuzzy linear programming,” Fuzzy Sets Syst., vol. 11, pp. 243–251, 1983. [16] J. Ramik and J. L. Rommelfanger, “Fuzzy mathematical programming based on some new inequality relations,” Fuzzy Sets Syst., vol. 81, no. 1, pp. 77–88, 1996. [17] J. L. Verdegay, “Application of fuzzy optimization in operational research,” Control Cybern., vol. 13, pp. 229–239, 1984. [18] D. Werners, “An interactive fuzzy programming systems,” Fuzzy Sets Syst., vol. 23, pp. 131–147, 1987. [19] J. Tang, D. Wang, and R. Y. K. Fung, “Fuzzy formulation for multiproduct aggregate production planning,” Prod. Planning Contr., vol. 11, no. 7, pp. 670–676, 2000. [20] H. Tanaka and K. Asia, “Fuzzy solution in fuzzy linear programming problems,” IEEE Trans. Syst., Man, Cybern., vol. SMC–14, pp. 325–328, Feb. 1984. [21] J. Tang and D. Wang, “A nonsymmetric model for fuzzy nonlinear programming problems with penalty coefficients,” Comput. Oper. Res., vol. 24, no. 8, pp. 717–725, 1997. [22] Y. J. Lai and C. L. Hwang, Fuzzy Mathematical Programming, Berlin, Germany: Springer-Verlag, 1992. [23] J. Tang, D. Wang, and R. Y. K. Fung, “Model and method based on GA for nonlinear programming problems with fuzzy objective and resources,” Int. J. Syst. Sci., vol. 29, no. 8, pp. 907–913, 1998. [24] J. Tang and D. Wang, “An interactive approach based on GA for a type of quadratic programming problems with fuzzy objective and resources,” Comput. Oper. Res., vol. 24, no. 5, pp. 413–422, 1997.
Richard Y. K. Fung received the B.Sc. (Hons.) and M.Phil. degrees from University of Aston, Birmingham, U.K., and the Ph.D. degree from Loughborough University, Loughborough, U.K. He is the Founder and Convenor of the government-funded Forum for Environmental Supply Chain Management (FESCM), and the Forum for Industrial Learning Organizations (FILO). In addition to undertaking various academic and administrative functions with the City University of Hong Kong, Hong Kong, he is also the Principal Investigator of a number of research projects funded by the University and the Hong Kong SAR Government. He has published over 100 research and scientific papers, around 50 of which appear in international referred journals. He is the Director of the Enterprise Knowledge Integration and Transfer Laboratory (E-KIT Lab.), and the Group Coordinator of Design and Manufacturing Informatics Activity Group in the Department of MEEM. His current research interests include ERP, fuzzy systems and applications, product design and management, logistics and supply chain management, enterprise knowledge integration, product life-cycle management.
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Jiafu Tang received the B.Sc. degree in mathematics from Hunan Normal University, Changsha, China, in 1989, and the M.Sc. degree and Ph.D. both in control theory and systems engineering from Northeastern University, Shenyang, R.O.C, in 1992 and 1999 respectively. He is currently Professor in the Research Institute of System Engineering, School of Information Science and Engineering, Northeastern University (NEU), in Shenyang, R.O.C. He was Research Assistant and Senior Research Associate at City University of Hong Kong, in 1998 and 2000, respectively, and at the Hong Kong Polytechnic University as Research Fellow in 2002. He authored two Chinese books in Scientific House of China, Beijing, and China Machine Press, Beijing, in 2000 and 2001, and he has published more than 40 papers in international and local journals. He is currently interested in fuzzy optimization theory and its applications, supply chain planning and logistics management, quality design in new product development, quality function deployment.
Dingwei Wang received the B.E. degree in automatic instrument from Northeastern University (NEU), Shenyang, R.O.C, the M.S. degree in systems engineering from Huazhong University of Science and Technology, Wuhan, P.R.C., and the Ph.D. degree in control theory and applications from NEU. He is currently Professor and Chair with the Research Institute of Systems Engineering, School of Information Science and Engineering, NEU. He has been a Post-Doctoral Fellow at the North Carolina State University, Raleigh. Since 2001, he has been a Panel Member of the Evaluation Committee of Management Science Department of National Natural Science of Foundation of China (NSFC). He is currently interested in modeling and optimization, soft computing, genetic algorithms, agile manufacturing, production planning and scheduling. Dr. Wang is a member of The New York Academy of Sciences and a member of The Economic and Management Systems Committee of The Chinese Association Automation. He is now member of Editorial board of International Journal of Computer and Operations Research, Editorial board of International Journal of Fuzzy Optimization and Decision Making.
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Multiproduct Aggregate Production Planning With Fuzzy Demands and Fuzzy Capacities Richard Y. K. Fung, Jiafu Tang, and Dingwei Wang
Abstract—Given the uncertain market demands and capacities in production environment, this paper discusses some practical approaches to modeling multiproduct aggregate production planning problems with fuzzy demands, fuzzy capacities, and financial constraints. By formulating the fuzzy demand, fuzzy equation, and fuzzy capacities, a fuzzy production-inventory balance equation for single period and a dynamic balance equation are formulated as fuzzy/soft equations and they represent the possibility levels of meeting the market demands. Using this formulation and interpretation, a fuzzy multiproduct aggregate production planning model is developed, and its solutions using parametric programming, best balance and interactive techniques are introduced to cater to different scenarios under various decision making preferences. Using the proposed models and techniques, first, the decision maker can select a preferred production plan with a common satisfaction level or different combinations of preferred possibility level and satisfaction levels, according to the market demands and available production capacities, and second, the obtained structure of the optimal solution can help decision maker in aggregate production planning. The decision maker can also make a preferred and reasonable production plan corresponding to one’s most concerned criteria. Hence, decision makers not only can come up with a reasonable aggregate production plan with minimum efforts, but also have more choices of making a preferred aggregate plan based on his most concerned criteria. These models can effectively enhance the capability of an aggregate plan to give feasible family disaggregation plans under different scenarios with fuzzy demands and capacities. Simulation and the results of analysis on the proposed techniques are also given in detail in this paper. Index Terms—Aggregate production planning, fuzzy demands, fuzzy equation, mathematical programming, interactive procedure.
I. INTRODUCTION
A
GGREGATE PRODUCTION PLANNING (APP) [1] is concerned with the determination of production, inventory, and workforce levels required to meet aggregate market demands at the level of product type. Product families with similar product costs and demand seasonality are grouped into a product type. APP is an important tactical level planning activity in a Manuscript received September 12, 2000; revised April 30, 2003. This paper was supported by a Strategic Research Grant (SRG) from City University of Hong Kong, Project Number 7001227, the National Natural Science Foundation, NSFC No. 70002009 of P.R.C., the Excellent Youth Teacher Program of Ministry of Education of China, and Liaoning Provincial Natural Science Foundation, No. 20022019. This paper was recommended by Associate Editor W. Gruver. R. Y. K. Fung is with the Department of Manufacturing Engineering & Engineering Management, City University of Hong Kong, Kow loon, Hong Kong, P.R.C. (e-mail: [email protected]). J. Tang and D. Wang are with the Department of Systems Engineering, School of Information Science and Engineering, Northeastern University, Shenyang 110004, P.R.C. Digital Object Identifier 10.1109/TSMCA.2003.817032
production management system. Other forms of family disaggregation plans, such as master production schedule, capacity plan, and material requirements plan all depend on APP in a hierarchical way [2]. Holt [3] suggested an optimal staffing, production, and inventory policy for a single manufacturer under the assumption of a given sales forecast, and they proposed a continuous time-varied model (HMMS) [3]. HMMS model is one of the best known classical models for tackling aggregated production and inventory planning problems. It aims at minimizing the total cost of regular payroll, overtime and layoffs, inventory, stock-outs, and machine set up. The linear decision rules for production level and workforce level are derived to deal with problems in a single product category. Since then, much attention has been directed toward aggregate production planning, and different models and approaches have been developed. The approaches to APP vary from mathematically optimal procedures [3] and [4], simulation and search methods [5] to heuristic decision rules [6]. Among the optimal models, linear programming has received the widest acceptance. These models can be categorized into deterministic optimization models [3], [7], stochastic programming models [7], [8], and fuzzy optimization models [6], [9]–[11]. Bergstrom and Smith [7] generalized the HMMS approach to a multiproduct formulation, which was further extended by Hausman [8] into a stochastic programming model to cater to the randomness in product demand. Bitran and Yanassee [12] considered the problem of determining production plans over a number of time periods under stochastic demands with known distribution functions. A distribution bound for the problem could be obtained by a deterministic approximation to the original stochastic problem. These stochastic programming models and methods are usually based on the concept of randomness and probability, and they are limited to tackling uncertainties with some probability distributions. The decisions achieved by stochastic programming can only take the form of a distribution function, which can do little to help decision making in practical scenarios. Some other forms of uncertainties, such as fuzzy demands, capacities with tolerance, imprecise process times, etc., are common in aggregate production planning. These types of uncertainties cannot be fully described by frequency-based probability distributions. Therefore, there is a need to formulate APP using fuzzy set theory [13], [14] and fuzzy optimization methods [6], [9], [10], [15]–[18]. Rinks [6] tackled aggregate production planning problems using fuzzy logic and fuzzy linguistics, and developed a production and a workforce algorithm using a series of approximately 40 relational assignment rules. Lee [9] discussed
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fuzzy aggregate production planning problems for single product types, under fuzzy objective, fuzzy workforce levels and fuzzy demands in different time periods. The workforce level and product demands were expressed in fuzzy numbers with tolerance. In this paper [9], a linear programming model with fuzzy objective and fuzzy constraints was developed, and fuzzy solutions under different levels could be attained using parametric programming technology. Wang and Fang [11] developed a genetic-based approach for linear APP with fuzzy objectives and resources. Tang [10], [19] developed a fuzzy approach for formulating integrated production and inventory problems with triangular fuzzy demands. It aims to minimize the total costs of quadratic production cost and linear inventory holding cost. A fuzzy solution is developed by which the solutions corresponding to different satisfaction levels are obtained. These fuzzy optimization models and approaches mainly consider a single product type [3], [6], [9] and they tend to overlook the financial constraint in a practical business environment [2], [3], [6], [7], [9]–[11], [19]. In general, a production plan is not only constrained by production capacities (e.g., machines and workforce), but also by financial resources, i.e., capital level, and all these constraints should be considered in APP. At present, there are no appropriate formulation and interpretation approaches to production-inventory balance equation under a fuzzy environment suitable for APP. In practical production planning systems, the aggregate market demands of a product in each period are not always clear, but uncertain in nature. These demands can be expressed as
tail in Section II. Section III discusses the formulation of fuzzy demands, fuzzy capacities, fuzzy production-inventory balance equation, and the development of the FMAPP model. The parametric programming-based fuzzy solution and the interactive procedures for the model are discussed in Sections IV and V, respectively. An illustrative example and the results of simulation are presented in Section VI, and the conclusion is given in Section VII. II. NOTATION AND FORMULATION Assume that a company manufactures types of products to meet the market demand over a planning horizon . APP is to determine the production, workforce, inventory, and backorder levels for product in period at minimum total cost. A multiproduct aggregate production planning (MAPP) problem under financial constraints with the aim to minimize the total can be formulated by crisp mathematics into cost a MAPP model as follows: (1) (2) (3) (4) (5)
1) random number with probability distribution; 2) fuzzy triangular number; 3) interval number, while the production capacities can be represented as fuzzy numbers with tolerance. Taking into account these factors and the financial constraints in the production environments, this paper proposes a practical approach to modeling multiproduct aggregate production planning problems with fuzzy demands and fuzzy capacities. With the use of fuzzy equation and fuzzy addition, the fuzzy production-inventory balance equation for single period and the dynamic balance equation are formulated as soft equations in terms of degrees of truth, which can be interpreted as possibility levels for meeting the market demands. With this formulation and interpretation, a fuzzy multiproduct aggregate production planning (FMAPP) model is developed. This model can take the form of a parametric programming model, a best balance model or an interactive model, to cater to different scenarios and decision maker’s preference. A parametric programming-based fuzzy solution and an interactive procedure are proposed to solve the problem. With the proposed models and solution, decision makers can make a reasonable APP economically, and more readily under his preferred criteria. This approach effectively allows an aggregate plan to give a feasible family disaggregation plan under different circumstances, especially under fuzzy demands and fuzzy capacities. The remainder of the paper is organized as follows: Notation and formulation of the problem will be presented in de-
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(6) (7) (8) where decision variable, inventory level of product at the end of period ; decision variable, production level of product in period ; decision variable, assigned workforce level for product in period , decision variable, back-order level of product in period ; demands for product in period , unit production costs including materials and overhead for product in period , unit workforce costs for product in period ; unit inventory costs for product in each period; unit back-order costs for product in each period; unit production capacity (man-hours) for product ; available workforce level in period , available financial resources in period . production capacity allowance percentage; the capacity of the warehouse; the initial inventory level for product ; the initial back-order level for product .
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In the MAPP model, (1) represents the objective of minimizing the total cost of production, inventory and back-order, (2) is the production-inventory balance equation, (3) and (4) denote the constraints of production capacity and financial resources in each period, (5) gives the warehouse capacity constraint, (6) reflects the relation between production level and production capacity, (7) indicates that at least one of the backorder and inventory level in a period is zero, and (8) implies the initial and nonnegative constraints of the decision variables. III. FUZZY MULTIPRODUCT AGGREGATE PRODUCTION PLANNING
B. Fuzzy Demands and Their Addition Let the fuzzy demands be a triangular fuzzy number, i.e., , where , and are the most possible, most pessimistic and most optimistic values of fuzzy demands, respectively. The membership function expresses the possibility measurement of fuzzy demands according to the decision maker’s view. For example, it may be with highest possibility, and with the and and between lowest possibility, and between and with a possibility level between zero and one. can be given as Hence the membership function
A. Formulating the Fuzzy Multiproduct Aggregate Production Planning Model In real-world scenarios, it is difficult to determine product dein advance in each period precisely, especially over mands a long planning horizon. These demands are not a certainty, but an estimate or imprecise/interval number. It is assumed in this is a triangular fuzzy number, denoted paper that the demand . The workforce level in each period by is assumed to take the form of a fuzzy number, denoted by , where the available capacity level is , and the largest acceptable tolerance is . Hence, MAPP problems can be formulated into a fuzzy mathematical model. Owing to the fuzziness of demand, the production-inventory balance (2) is not a crisp, but a fuzzy equation, namely, a soft equation in this paper. It implies that the equation is met in terms of a degree of truth. As a result, an MAPP problem with fuzzy demands and fuzzy production capacities can be formulated into a fuzzy multiproduct aggregate production planning (FMAPP) model as follows:
(9) (10) (11) (12)
Let and be two triangular fuzzy numbers, the addition of two fuzzy numbers remains a triangular fuzzy number, which . Similarly, can be denoted as this concept can be extended to the addition of any number of triangular fuzzy numbers. be the total demands of product from Let the period 1 to a certain period . According to the extension is defined as principle [14], the membership function (17) where and are the maximum and minimum operators respectively. represents the possibility measurement of total fuzzy demands of product from period 1 to period according to the decision maker. can also be deObviously, the total demands scribed by a triangular fuzzy number, i.e., . Triangular fuzzy demands and total demands can be illustrated as shown in Fig. 1(a) and (b), respectively. C. Fuzzy Equations and Their Addition Production-inventory balance (10) in the FMAPP model is not a crisp equation, but a fuzzy one. It implies that the equation is met in terms of a degree of truth, and it can be expressed as depending on the fuzzy demands a membership function . Therefore
(13) (18) (14) (15) (16) where denotes a soft equation. It implies that the productioninventory balance equation is achieved in terms of a degree of truth. The formulations and interpretation of fuzzy demands, fuzzy capacity and fuzzy production-inventory balance equation are explained in the following subsections.
With this formulation, the production-inventory balance (10) indicates that the sum of production level in this period and the difference between inventory level and back-order level in the previous period minus the difference between inventory and back-order level in this period should not differ from the demand too much.The larger the difference, the smaller the degree of truth that the equation holds, and the less likely that reflects the market demands can be met. In this sense, possibility level at which the market demands are met. For ex, the difample, if
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Fig. 1. Formulation of fuzzy demands and fuzzy production-inventory balance equation.
ference is zero, and the production-inventory balance equation is strictly satisfied, and the market demand is most possibly met, i.e., the possibility level equals one. On the contrary, if , the difference is greater than the tolerance, and the production inventory balance equation holds with a degree of truth being zero, and the market demand is unlikely met. Of course, when the right-hand of the , the market demand will be likely equation becomes met, however it will bring higher inventory levels and holding costs, thus, it is not always practical, and therefore the equation has a pessimistic degree of truth in this case. This formulation is illustrated as shown in Fig. 1, and the interpretation coincides is crisp, and the with the crisp situation when the demand production-inventory balance equation holds absolutely over the entire planning horizon. With this interpretation, given the decision maker’s request that the APP is to meet the market demands for product in a period with possibility level , the soft equation can be expressed as follows: (19) as illustrated in Fig. 1. The addition of two soft equations is also a soft equation and its degree of truth is defined as the logic addition of the degrees of truth for each member of soft equations according to the extension principle [14]. It can be explained in the following example. and be two soft equations, then their Let in which the degree of addition is denoted as satisfies truth
according to max-min theory. Similarly, this operation can be extended to the addition of any number of soft equations. By applying the addition of soft equations to (10), we can have (21) more general, for (22) D. Formulation of the Fuzzy Production-Inventory Balance Equations With the formulation of fuzzy addition and fuzzy equation, the fuzzy production-inventory balance equation in a single period can be interpreted as the possibility level of the APP meeting the fuzzy demands. It can be formulated as in (19). Under the same types of fuzzy demands , the production-inventory balance equation in a single period is equivalent to the dynamic balance equation for product from period 1 to , (19) certain period , i.e., for is equivalent to (23) which can be illustrated as shown in Fig. 1(d). Therefore, the production-inventory balance equation under a fuzzy environment can be formulated as in (19) or (23). E. Formulation of the Fuzzy Production Capacity
(20)
in period Assuming the production capacity number with tolerance, its membership function
is a fuzzy re-
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flects the decision maker’s level of satisfaction with the consumption of production capacity, and is defined as follows: (27) (24) (28) For example, if the consumption of production capacity is lower than the available capacity , decision maker is of course fully satisfied. As the consumption of capacity increases, decision maker’s satisfaction level decreases, and the decision maker is becoming dissatisfied when the consumption reaches the toler. ance point For a given threshold level , the decision maker’s satisfaction level with the consumption of production capacity can be , i.e., formulated as
(29) (30) (31) (32) (33)
(25) (34) In general, the fuzzy demand may be extended to the cases of general L-R type and trapezoidal numbers. In those cases, the APP can be formulated into the above model only if the fuzzy demands in each period are of the same type. IV. PARAMETRIC PROGRAMMING MODEL FMAPP model is a model of fuzzy nonlinear programming with fuzzy inequality and equality constraints. Unlike a crisp model, the optimal solution here is not crisp but a fuzzy set “decision” which has different meanings depending on its definition and formulation. APP is an important upstream level decision activity, and the decision is the source of family disaggregation plan at downstream levels. Therefore, the sole optimal solution to APP cannot guarantee that the feasible disaggregation plan will always be achieved. The diversity of the solutions and multiple solutions to APP can offer more opportunities to achieve the feasible disaggregation plan particularly under a fuzzy environment. From this point of view, the solution of the model is “optimal” subject to some preferred possibility levels for meeting the market demands and satisfaction levels with consumption of production capacities. In this case, a fuzzy solution [8], [14], [18], [20]–[22] can be obtained using parametric programming techniques. is given to denote decision A preferred level maker’s request for possibility level at which the APP meets the market demands, the soft equation can be defuzzified such that the cumulative demand of product type from period 1 to a future period should be met with a possibility level higher than . At the same time, another threshold level can be given to express the decision maker’s satisfaction level with the consumption of production capacity which should not be less than . With this understanding, the FMAPP model can be transformed into the following equivalent parametric aggregate production planning (PAPP) model (26)
. The production-inventory balance equation where in the original fuzzy optimization model FMAPP is expressed in terms of inequality constraints (27) and (28) as per the decision maker’s request for possibility level at which APP can meet market demands. and are the preferred parameters, which reflect the constraints of the possibility level of meeting the market demands and satisfaction level with the consumption of production capacity, respectively. For given preferred satisfaction levels and , the PAPP model becomes a parametric nonlinear programming model. Let be an optimal solution to the parametric nonlinear programming model, the following properties on the solution to the PAPP model can be easily proved. increases with the Property 1: In the case of common value of and . increases with the . Property 2: Given a specified increase with the . Property 3: Given a specified For given preferred satisfaction levels and , the optimal sois a fuzzy solution to the origlution inal fuzzy optimization problem FMAPP. That is to say, it’s an optimal solution in a fuzzy sense, because it depends on the decision maker’s preference on the possibility level and the satisfaction level . In particular, the optimal solution is crisp when . Therefore, the fuzzy solution to FMAPP reflects the decision maker’s preference and subjectivity, and it can increase the chances and ability of constructing an APP to guarantee the feasibility of a family disaggregation plan, especially under a fuzzy environment. Of course, in the PAPP model, the possibility level and the satisfaction level may have common or different values. Under some practical situations, the decision maker may expect a higher or lower possibility level of meeting the market demands for a certain product type in a certain period, hence different possibility levels can be applied. Similarly, different satisfaction levels may also be selected in different periods to suit practical situation and the decision maker’s preference. In summary, the above model is flexible and can reflect the decision maker’s preference in a fuzzy environment.
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V. BEST BALANCE MODEL AND INTERACTIVE PROCEDURES As mentioned in the previous section, the total cost increases proportionally with the possibility level and the satisfaction level . In this case, decision maker can make a reasonable plan in light of his preferred possibility level and satisfaction level. However, in some cases, decision maker not only concerns about total cost, but also prefers to make a balance among total cost, possibility level of meeting market demands and satisfaction level on the consumption of production capacities. denote the decision maker’s required possiLet bility level of meeting market demands from period 1 to a speand denote the decision maker’s satcific period isfaction levels with production capacity consumption in period and the total cost in planning horizon respectively. Therefore, the APP problem concerns with not total cost alone any more, but a best balance among the total cost, the possibility of meeting market demands and the satisfaction with consumption of production capacity, and it can be expressed in a balanced aggregate production planning (BAPP) model as follows: (35) (36) (37)
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unique solution. However, this optimal solution may not be desirable or acceptable to decision maker in practice, especially under a fuzzy environment. On the contrary, multiple solutions under different criteria may be preferred. The decision maker’s most important criteria may include minimizing the total cost, maximizing the possibility level of meeting market demands on certain products in a specified period, and maximizing the satisfaction level with consumption of production capacities over specified periods. To cater for this situation, an interactive procedure for APP is proposed to help the decision maker make a preferred plan from a pool of optimal solutions under different criteria. The interactive procedure can be described as follows: A. Interactive Procedure for APP Step 1) Initialization: Given an initialized threshold reflecting decision maker’s overall acceptable satisfaction level with total cost, consumption of production capacities, and possibility level required for meeting market demands. Step 2) Input the decision maker’s most concerned criteria indexes (CI) set
where
represents the criterion for total cost.
(38) (39) (40) (41) (42)
represent the criteria for meeting market demands in period ; of product represent the and criteria for the consumption of production capacities . in period Step 3) Construct an interactive aggregate production planning (IAPP) model as follows:
(43) (49) (50)
where the original production-inventory balance equation is is the costs incurred in period given in (36), and have been described in (17) and (24) respectively, is defined as while
(51)
(44) (52) is an aspiration level of the total cost prewhere is an acceptable tolerance, ferred by decision maker, and say 5%–15% of . The BAPP model can be transformed into the following nonlinear programming model: (45)
(46) (47) (27), (28), (30), (31), (32), (33), (34)
(48)
and it can be solved by traditional nonlinear optimization techis a crisp and niques. The optimal solution
(53) (54) (30), (31), (32), (33), (34)
(55)
denotes the possibility level of where meeting the market demands on product in period and behaves in the same way as in (18), and is an aggregated satisfaction level with the criteria are the of concerns, and indicators of criteria index, and only one of them equals 1 at a time. and optimal solution Step 4) Set initial criteria index pool to be empty.
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TABLE I BASIC DATA AND MODEL PARAMETERS OF THE SIMPLE EXAMPLE
Step 5) Reset indicators of criteria indexes as follows:
Step 6) Solve the IAPP model by traditional constrained optimization techniques, and reset the optimal , where solution pool is the optimal solution with the th criterion. , go to Step 8; else, , go to Step 7) if Step 5. Step 8) Input decision maker’s most preferred criteria and display the corresponding optimal solutions, stop. Of course, in this interactive procedure, multiple indicators for criteria indices can equal 1 simultaneously, and in this case the indicator vector represents a combination of multiple criteria at a given time. The values of indicator can be further extended to represent the weights of the criteria when it assumes a value in [0, 1], and the multiple criteria can be also considered simultaneously. With this interactive procedure, an optimal solution pool [23] consisting of a set of potential optimal solutions with various criteria will be obtained [24], from which decision maker can interactively establish a production plan to suit particular circumstances. The interactive procedure and the optimal solution pool reflect decision maker’s preference, and can offer decision maker more choices to facilitate the establishment of downstream family disaggregation plans under a fuzzy environment. VI. ILLUSTRATIVE EXAMPLE AND ANALYSIS To illustrate the models proposed in this paper and assess the effect of the possibility level and the satisfaction level on the total cost and production plan in the parametric programming model, a simple simulation example using optimization tool box of software Matlab 5.0 is presented in this section. The basic datum of the example and model parameters are shown in Table I. For this simple example, two types of products denoted by product 1 and product 2, are produced in periods 1–4 to meet the customer demands, of which the value is an approximate
one and it can be formulated as a triangular fuzzy number. For example, the demands of product 1 in period 1 is 85 units most possibly, at least 80 units (pessimistically), and not more than 90 (optimistically). The unit production costs of product may be varied with product types and periods, for example product 1 has identical unit costs in different periods and it for product 2. Other input data has different unit costs and parameters, e.g., workforce costs, capacity, initial inventory level, inventory costs, back-order costs capacity coefficients can be explained in similar way. The simulation is conducted in three parts. The first part aims to analyze the effect of the parameters on the aggregate plan in the parametric programming model. The second and last parts focus on performance of the best balance model (BAPP) and the interactive procedure respectively. The simulation results and analysis are presented as follows. A. Simulation of the Parametric Programming Model and the Results and on the Total 1) Effect of Common Values of Cost: To analyze the effect of the common values on the total cost, common level and are scaled by discretesizing [0, 1] in equal steps of 0.05. Table II presents the production, workforce, inventory and back-order levels for product 1 and 2 in each period and the corresponding total costs when the value of both and varies in steps from 0.60 to 1.0. The effect of the common values on the total cost are depicted in Fig. 2. It can be seen from Table II that the total cost increases with and linearly as shown in Fig. 2. The simulation results suggest the optimal production policy that the inventory levels ) and of product 1 in each period (except when product 2 in the final period are zero, and the back-order in each period is absolutely not permitted, and the workforce level is proportional to the production level. These results coincide with the optimal production policies as shown in Table III. 2) Effect of Level on the Total Cost Given Specified Values of : In this part of the simulation, two specified values of the and , and varies possibility level are used, i.e., from 0.0 to 1.0 discretely in equal steps of 0.10. The results shown in Fig. 4 show that with both of the specified values of ,
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TABLE II AGGREGATE PRODUCTION PLAN WITH COMMON VALUES OF
the total cost increases linearly with , i.e., and . As indicated in the formula, however, there are no great changes in the total cost for different levels of for a specified . Hence, has little effect on the total cost for a specified . That means for a given possibility level of meeting the market demands, the satisfaction with the consumption of capacities has little effect on the total cost. Results of the simulation also conclude that for any specified , the inventory costs increase with , while both the production cost and workforce cost maintain at certain levels except for the . The average inventory cost of workforce cost when increases proportionally to . It implies that both the production cost and the workforce cost are only dependent on the possibility level of meeting the market demands. In fact, in order to maintain the possibility levels of meeting the market demand, the tight constraints of production capacity in each period can only bring more inventory levels and warehouse holding cost, while the production and workforce costs do not decrease as a
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PARAMETRIC PROGRAMMING
Fig. 2. Total cost varies with common values of theta and gamma.
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TABLE III OPTIMAL STRUCTURE OF AGGREGATE PRODUCTION PLAN AND TOTAL COST
Fig. 3.
Fig. 4.
Total cost varies with gamma given specified theta.
Fig. 5.
Total cost varies with theta given specified gamma.
Total cost varies with theta and gamma simultaneously.
whole. This result reveals a hint that the relaxation of tolerance of the production capacity in each period may decrease the total costs. The simulation also implies that as increases, the production level of product 2 increases in the first two periods and decreases in the last two periods; and the production levels for product 1 are almost unchanged with except for the last period, and it is produced in the optimal way that the inventory and backorder levels in each period remain zero. The optimal production policies shown in Table III help explain this situation. It can be concluded that the results of the simulation not only reflect the effect of on total cost given a specified , but also imply that has substantial impacts on the total cost and hence the production plan, while has relatively little effect. 3) Effect of on the Total Cost Given Specified Values of : In this part of the simulation, two specified values of , i.e., 0.7 and 1.0 are selected, and varies from 0.0 to 1.0 discretely in equal step size of 0.10. The results are shown in Fig. 5. It can be seen that for any given , the total cost increases , linearly with , i.e., . The production and level of product 2 increases rapidly in the first two periods and decreases in the last period with , whereas the production level
for product 1 in each period satisfies that the inventory level in each period is almost zero for any value of . As shown in Fig. 5, there is significant difference in the total cost between neighboring levels of with the same values of , while there are little difference between neighboring levels of with the same . It can be also concluded that has great impacts on the total cost and APP, while has relatively little effect. 4) Combined Effects of and on the Total Cost and Production Plan: In this part of the simulation, 100 sets of values of and are used, where 20 sets are with common values, 40 sets are with either or specified, and the remaining are random numbers in [0 1]. The resulting production, workforce, inventory and back-order levels, as well as the total cost behave linearly as shown in Table III. The total costs with varying and change as shown in Fig. 3. It can be seen from the optimal structure depicted in Table III that the production level for product 1 decreases slightly in the first three periods, and then increases slightly with and rapidly with in the last period. On the other hand, the production level for product 2 increases significantly in the first two periods and then decreases rapidly with and slightly with respectively
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TABLE IV AGGREGATE PRODUCTION PLAN BY WAY OF INTERACTIVE PROCEDURE
in the last two periods. The workforce level varies proportionally to the production level. The inventory level for product 1 is slightly affected by both and , and mostly around zero over the entire planning period, nevertheless the inventory level for product 2 increase rapidly with both and in the planning periods except in the last one. The back-order level for each product over the entire planning period remains zero owing to the high back-order penalty imposed in this example. The optimal production policies coincide with the results in the aforementioned three parts of the simulation. As shown in Table III and Fig. 3, the total cost increases significantly with the level and only slightly with . So far, it can be concluded that the
parameter plays a more influential role than in the model, hence the possibility level has more significant impacts on the production plan. B. Simulation Results on the Best Balance Model The best balance model is simulated using the Minimax function in the optimization tool box of the software Matlab 5.0. The resulting best balance degree, as well as the production, workforce, inventory and back-order levels and the total cost are shown in the first row in Table IV. It can be seen that the best balance among them is at a satisfaction level of 0.5455 with a cost of 170 954 pounds. That means when a higher possibility level
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of meeting market demand or a satisfaction level higher than 0.5455 is requested, the total cost will exceed 170 954 pounds, giving a less satisfaction level than 0.5455 with total cost, and vice versa. C. Simulation Results on the Interactive Procedure Given an initial overall acceptable satisfaction level with the total cost,consumption of capacities and possibility level , the interactive procedure is implemented using the constrained optimization function in Matlab 5.0, the preferred solutions under different criteria are obtained as shown in Table IV. Following the best balance criteria, the criteria number 0 indicates the aggregate plan under criteria of minimizing the total costs. The lows with criteria numbered 1–4 correspond to the solutions under criteria of maximizing possibility level of meeting market demands of product 1 in periods 1–4, respectively. Similarly the lows with criteria number 5-8 correspond to the solutions under criteria of maximizing possibility level of meeting market demands of product 2 in periods 1–4, respectively. While the criteria numbered 9–12 correspond to the ones of maximizing satisfaction of consumption of production capacity in periods 1–4, respectively. It can be seen from the Table IV that , the optimum satisfaction given an acceptable level level of the total cost can reach 0.6297 with a total cost of 170 406 pounds. While the possibility levels of meeting the market demands of products 1 and 2 in each period can reach 1.0, and the corresponding satisfaction levels with consumption of capacities can reach 1.0 too. It can be seen from the total cost column in Table IV that the minimum total cost is 170 406 pounds achieved at criteria number 0, and the maximum total cost is 171 575 pounds obtained at criteria number 5 and 12, and there is a difference among other criteria, particularly the differences is distinctive among different types of criteria. For example, there is differences of 387 pounds between the minimum value of total costs under criteria of possibility of meeting market demands and of satisfaction of consumption of production capacities. Moreover, there is distinctive differences in solution under different different types of criteria. Both the possibility levels under criteria and satisfaction levels under criteria numbered numbered are achieved as 100%, which coincides the actual scetype nario. D. Implication for Decision Procedure In summary, on the one hand, decision maker can select a preferred production plan with a common satisfaction level, e.g., according to the market demands and available production caare chosen, the third row in Table II pacities, if shows the corresponding APP. Of course, one may choose different combination of and based on the information and implication given in Figs. 2–5. The structure of the optimal solution presented in Table III can help decision maker in aggregate production planning. On the other hand, one can choose preferred production plan corresponding to one’s most concerned criteria, e.g., if one concerns most with best balance among the total cost, meeting market demands and consumption of capacities, then the first row in Table IV gives the corresponding plan. Similarly, if the total cost is of major concern, then the second
row in Table IV offers the feasible plan. Of course, if the criteria of meeting market demands for product 1 in period 4 is of most concern, then the sixth row in Table IV suggest a feasible aggregate plan. VII. CONCLUSION Taking into account the fuzzy demands and fuzzy capacities in each period, this paper proposes a fuzzy approach to formulating MAPP problems under financial constraints. The fuzzy production-inventory balance equation is formulated as a soft equation with a degree of truth, and it can be interpreted as the possibility level of meeting the market demands. With this formulation and interpretation, a FMAPP model is developed, and it is transformed into a parametric programming model and a best balance model to cater for different scenarios and decision maker’s preferences. A parametric programming based fuzzy solution and an interactive aggregate production planning procedure to solve the problem are also discussed. With the proposed models and approaches, on the one hand, first, the decision maker can select a preferred production plan with a common satisfaction level or different combinations of possibility and satisfaction levels, according to the market demands and available production capacities, secondly, the structure of the optimal solution presented in Table III can help decision maker in APP. On the other hand, the decision maker can also make a preferred and reasonable production plan corresponding to one’s most important criteria. The approach could not guarantee a global optimal solution, however the solution is near optimal and it can effectively enhance the capability that an aggregate plan can have feasible family disaggregation plans under different circumstances, especially under the environment of fuzzy demands and fuzzy capacities. ACKNOWLEDGMENT REFERENCES [1] Handbook of Operations Research: Models and Applications, J. Morder and S. E. Elmaghraby, Eds., Van Nostrand, New York, 1978, pp. 127–169. [2] L. Ozdamar, M. A. Bozyel, and S. I. Birbil, “A hierarchical decision support system for production planning (with case study),” Eur. J. Oper. Res., vol. 104, pp. 403–422, 1998. [3] C. C. Holt et al., Planning Production Inventories and Workforce. Englewood Cliffs, NJ: Prentice-Hall, 1960. [4] U. Akinc and G. M. Roodman, “A new approach to aggregate production planning,” IIE Trans., vol. 18, no. 1, pp. 88–94, 1986. [5] R. G. Schroeder, Operations Management: Decision Making in the Operations Function. New York: McGraw-Hill, 1989. [6] D. B. Rinks, “The performance of fuzzy algorithm models for aggregate planning and differing cost structures,” in Approximate Reasoning in Decision Analysis, M. M. Gupta and E. Sachez, Eds, The Netherlands: North-Holland, 1982, pp. 267–278. [7] G. L. Bergstrom and B. E. Smith, “Multi-item production planning-an extension of the HMMS rules,” Manage. Sci., vol. 16, pp. 614–629, 1970. [8] W. H. Hausman and J. D. McClain, “A note on the Bergstrom-Smith multi-item production planning model,” Manage. Sci., vol. 17, pp. 783–785, 1971. [9] Y. Y. Lee, “Fuzzy set theory approach to aggregate production planning and inventory control,” Ph.D. dissertation, Dept. of Industrial Engineer, Kansas State Univ., Manhatten, 1990. [10] J. Tang, R. Y. K. Fung, and D. Wang, “A fuzzy approach to modeling production and inventory planning,” in Proc. 14th IFAC World Congress, vol. A, Beijing, R.O.C., 1999, pp. 261–266.
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[11] D. Wang and S.-C. Fang, “A genetics-based approach for aggregate production planning in fuzzy environment,” IEEE Trans. Syst., Man, Cybern. A, vol. 27, pp. 636–645, May 1997. [12] G. R. Bitran and H. H. Yanassee, “Deterministic approximations to stochastic production problems,” Oper. Res., vol. 32, pp. 999–1018, 1984. [13] R. Y. K. Fung, K. Popplewell, and J. Xie, “An intelligent hybrid system for customer requirements analysis and product attribute targets determination,” Int. J. Prod. Res., vol. 36, no. 1, pp. 13–34, 1998. [14] R. Y. K. Fung, J. Tang, Y. L. Tu, and D. Wang, “Product design resources optimization using a nonlinear fuzzy quality function deployment model,” Int. J. Prod. Res., vol. 40, no. 3, pp. 585–599, 2002. [15] S. Chanas, “Using parametric programming in fuzzy linear programming,” Fuzzy Sets Syst., vol. 11, pp. 243–251, 1983. [16] J. Ramik and J. L. Rommelfanger, “Fuzzy mathematical programming based on some new inequality relations,” Fuzzy Sets Syst., vol. 81, no. 1, pp. 77–88, 1996. [17] J. L. Verdegay, “Application of fuzzy optimization in operational research,” Control Cybern., vol. 13, pp. 229–239, 1984. [18] D. Werners, “An interactive fuzzy programming systems,” Fuzzy Sets Syst., vol. 23, pp. 131–147, 1987. [19] J. Tang, D. Wang, and R. Y. K. Fung, “Fuzzy formulation for multiproduct aggregate production planning,” Prod. Planning Contr., vol. 11, no. 7, pp. 670–676, 2000. [20] H. Tanaka and K. Asia, “Fuzzy solution in fuzzy linear programming problems,” IEEE Trans. Syst., Man, Cybern., vol. SMC–14, pp. 325–328, Feb. 1984. [21] J. Tang and D. Wang, “A nonsymmetric model for fuzzy nonlinear programming problems with penalty coefficients,” Comput. Oper. Res., vol. 24, no. 8, pp. 717–725, 1997. [22] Y. J. Lai and C. L. Hwang, Fuzzy Mathematical Programming, Berlin, Germany: Springer-Verlag, 1992. [23] J. Tang, D. Wang, and R. Y. K. Fung, “Model and method based on GA for nonlinear programming problems with fuzzy objective and resources,” Int. J. Syst. Sci., vol. 29, no. 8, pp. 907–913, 1998. [24] J. Tang and D. Wang, “An interactive approach based on GA for a type of quadratic programming problems with fuzzy objective and resources,” Comput. Oper. Res., vol. 24, no. 5, pp. 413–422, 1997.
Richard Y. K. Fung received the B.Sc. (Hons.) and M.Phil. degrees from University of Aston, Birmingham, U.K., and the Ph.D. degree from Loughborough University, Loughborough, U.K. He is the Founder and Convenor of the government-funded Forum for Environmental Supply Chain Management (FESCM), and the Forum for Industrial Learning Organizations (FILO). In addition to undertaking various academic and administrative functions with the City University of Hong Kong, Hong Kong, he is also the Principal Investigator of a number of research projects funded by the University and the Hong Kong SAR Government. He has published over 100 research and scientific papers, around 50 of which appear in international referred journals. He is the Director of the Enterprise Knowledge Integration and Transfer Laboratory (E-KIT Lab.), and the Group Coordinator of Design and Manufacturing Informatics Activity Group in the Department of MEEM. His current research interests include ERP, fuzzy systems and applications, product design and management, logistics and supply chain management, enterprise knowledge integration, product life-cycle management.
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Jiafu Tang received the B.Sc. degree in mathematics from Hunan Normal University, Changsha, China, in 1989, and the M.Sc. degree and Ph.D. both in control theory and systems engineering from Northeastern University, Shenyang, R.O.C, in 1992 and 1999 respectively. He is currently Professor in the Research Institute of System Engineering, School of Information Science and Engineering, Northeastern University (NEU), in Shenyang, R.O.C. He was Research Assistant and Senior Research Associate at City University of Hong Kong, in 1998 and 2000, respectively, and at the Hong Kong Polytechnic University as Research Fellow in 2002. He authored two Chinese books in Scientific House of China, Beijing, and China Machine Press, Beijing, in 2000 and 2001, and he has published more than 40 papers in international and local journals. He is currently interested in fuzzy optimization theory and its applications, supply chain planning and logistics management, quality design in new product development, quality function deployment.
Dingwei Wang received the B.E. degree in automatic instrument from Northeastern University (NEU), Shenyang, R.O.C, the M.S. degree in systems engineering from Huazhong University of Science and Technology, Wuhan, P.R.C., and the Ph.D. degree in control theory and applications from NEU. He is currently Professor and Chair with the Research Institute of Systems Engineering, School of Information Science and Engineering, NEU. He has been a Post-Doctoral Fellow at the North Carolina State University, Raleigh. Since 2001, he has been a Panel Member of the Evaluation Committee of Management Science Department of National Natural Science of Foundation of China (NSFC). He is currently interested in modeling and optimization, soft computing, genetic algorithms, agile manufacturing, production planning and scheduling. Dr. Wang is a member of The New York Academy of Sciences and a member of The Economic and Management Systems Committee of The Chinese Association Automation. He is now member of Editorial board of International Journal of Computer and Operations Research, Editorial board of International Journal of Fuzzy Optimization and Decision Making.
