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Construction of a two-stage fuzzy management

planning model for a computer integrated

production management system

SUHUA HSIEH and SIMON HSU

Abstract. A production management system contains many qualitative descriptions and imprecise natures. The conven-tional crisp and/or stochastic model constructed in the computer integrated production management system

(CIPMS) cannot describe these qualitative descriptions and imprecise natures. Therefore, it is difficult to mimic the way managers think, which is conceptual and comprehensive, and to absorb uncertainties such as order cancelled, unstable material supply etc, in a production system. This frequently accounts for why the CIPMS yields a poor performance. This paper presents a fuzzy approach to the CIPMS in order to model qualitative descriptions and imprecise natures. This approach includes two stages. In stage one, a management strategy can be determined in a way that is similar to the way humans think, in which ideas, pictures, images and value systems are formed. In stage two, a fuzzy linear programming model is developed to absorb these imprecise natures in a production system. In doing this, CIPMS can adapt a variety of non-crisp problems in an actual system, thereby improving the performance of CIPMS.

1. Introduction

To ensure the effectiveness of communication within a production system and its ability to respond quickly to the fluctuating changes of market demand, computer integrated production management systems

(CIPMS) have gradually replaced the manual

produc-tion management method. Although it is a highly effective tool for production management and control, CIPMS requires further development before it will find use in practical applications. Derks (1988) cited ten

traps to avoid when planning CIPMS. Willis (1990)

established five rules on how to implement CIPMS

successfully. A production system is usually in a fuzzy and/or imprecise environment within which confusions occasionally occur. Many qualitative descriptions and imprecise natures need to be modelled in the CIPMS to mimic the way managers think and to absorb uncer-tainties, such as order cancelled, unstable material supply etc. This frequently accounts for why the CIPMS yields a poor performance despite performing many tasks and devoting material resources and time.

Owing to the ingenious and efficient nature of human thinking, experienced high-level managers normally adopt qualitative and equivocal approaches to determine a corporation’s management strategies. Managers can normally reach a decision in a con-ceptual or comprehensive way when encountering problems with thousands of information bits. Once CIPMS is adopted, crisp, deterministic, and stochastic models are constructed and installed in the computer. Computers, in terms of analysing and simulating all possible events and, ultimately, obtaining the optimal solutions, can replace managers. However, real situa-tions are frequently not crisp, deterministic or stochas-tic and can also not be described adequately in the same manner. If these models were used to describe an actual system, the nearly complete description would require far more detailed data than a human could recognize. Thus, CIPMS do not work well in general.

The fuzzy theory bridges the gaps between the actual system and the model describing the actual

system. Zadeh (1965) proposed the fuzzy subset and

membership function. Fuzzy set theory provides a strict mathematical framework in which vague conceptual phenomena can be precisely and rigorously studied. Fuzzy theory can also be considered as a modelling language, which can handle a large fraction of semantic contents or uncertainties of actual life situations. Due to International Journal of Computer Integrated Manufacturing

ISSN 0951-192X print/ISSN 1362-3052 online # 2002 Taylor & Francis Ltdhttp://www.tandf.co.uk/journals DOI: 10.1080/09911920110118795

Authors: Suhua Hsieh and Simon Hsu, Department of Mechanical Engineering,

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its generality, this theory can be applied to different circumstances and contexts. However, in many cases this entails context-dependent modification and speci-fication of the original concepts of the formal fuzzy set theory. Unfortunately, this adaptation has not yet progressed to a satisfactory level.

Once a high-level manager determines the corpora-tion’s management strategies, the middle-level execu-tors must implement the aggregate production planning, and then perform the production in the same manner as the planning. In any planning setting, a decision appears to be either a good or poor decision, depending on the market demand after the product is made. A decision is not good or bad in itself, but only relative to the circumstance in which the influence of the decision is felt. Obviously, the future status of the global economy cannot be precisely known, necessitat-ing the decision to be made under imprecise condi-tions.

Since the late 1950s, the imprecise nature of a production management system has been thoroughly investigated using a stochastic approach for decision models where input data have been given probability distributions(Beale 1955, Dantzig 1955, Charnes et al.

1958, and Charnes and Cooper 1959). Zadeh (1978)

examined the feasibility of using fuzzy sets as the basis for a theory of possibility. Subsequent research on the possibility theory has been extensive (Yazenin 1987, Buckley 1990, Tanaka and Asai 1984, Rommelfanger et

al. 1989, Luhandjula 1987, Lai and Hwang 1992, 1993). The notion of a fuzzy set provides the basis for constructing of a conceptual framework that parallels, in many aspects, the framework used in the case of ordinary sets, but is more general than the latter and, potentially, has a much wider scope of applicability. Such a framework provides a natural means of dealing with problems in which impreciseness is largely attributed to the absence of sharply defined criteria of class membership rather than the presence of random variables.

Zadeh pioneered the fuzzy theory in 1965. While

commenting on the fuzzy set theory, Goguen (1967,

1969)attempted to describe the fuzzy properties of the real world efficiently. Subsequent observations on the fuzzy theory or related applications were made. In the early period, Bellman and Zadeh (1970)proposed the

fuzzy decision model. Tanaka et al. (1974, 1976)

proposed fuzzy mathematical programming and the fuzzy decision, and then applied them to investment problems. In a related work, Negoita and Sularia(1976)

discussed the fuzzy mathematical optimization

prob-lem. Dubois and Prade (1978) extended the usual

algebraic operations on real numbers to fuzzy numbers by adopting a fuzzification principle. Zimmermann

(1976) applied fuzzy theory to describe and optimize the fuzzy system. That same investigator proposed fuzzy linear programming with several objective functions

(Zimmermann 1978). Fuzzy-related research has

re-ceived increasing attention since the 1980s. Kacprzyk and Staniewski(1982)applied the fuzzy decision theory to formulate a long-term inventory policy. Karwowski and Evans (1986)extended fuzzy concepts to

produc-tion management research. Lehtimaki (1987) used

fuzzy set theory to determine a master production

schedule. Later, Hintz and Zimmermann (1989)

applied the fuzzy theory to control a flexible manufac-turing system. Werners(1987)described how to use the membership function to represent a fuzzy goal. Trappey et al. (1988) further constructed the fuzzy nonlinear programming model to represent a non-stochastic and very complex manufacturing environ-ment. Fuzzy theory applications in a production system have grown considerably in recent years. Masnata and Settineri(1997)and Gindy and Ratchev(1997)applied the fuzzy theory to effectively manage manufacturing cells. Noto La Diega et al.(1996)later applied the fuzzy theory to production planning, while Grabot and Geneste (1994), Stanfield and Joines (1996) and

Kuroda and Wang (1996) applied it to production

scheduling. Furthermore, Perrone and Noto La Diega

(1996)applied the fuzzy theory to flexible manufactur-ing systems.

While previous investigations have either applied the fuzzy decision theory to certain production prob-lems or have developed fuzzy linear programming models for production planning, the fuzzy theory has seldom been applied to CIPMS. In addition, to our knowledge, an integration fuzzy model has never been developed for the two-stage — high-level management and middle-level execution — problems mentioned above. Therefore, this study presents a novel two-stage fuzzy planning model for CIPMS. The proposed model can effectively handle the qualitative descriptions and imprecise natures of a production system, thereby improving the performance of CIPMS.

For qualitative descriptions in a high-level manage-ment environmanage-ment, fuzzy set and fuzzy decision theories are used to construct a mimic – manager – decision model. By doing so, the optimal corporation manage-ment strategies can be determined in a manner similar to human thinking. Since human thinking is subjective and instinctive, experienced managers inevitably make incorrect decisions. The main purpose of stage one in this research is to present a model enabled with human thinking and feeling, but this model is not committed to making the correct decision. Therefore, analysis of the results or the performance of the proposed model will not be discussed in this paper.

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For the imprecise nature of the middle-level execution environment, fuzzy linear programming is employed to construct an aggregate production plan-ning model. Fuzzy production planplan-ning can therefore be achieved when the actual demand is uncertain. Finally, a case study demonstrates the effectiveness of the proposed model. An analysis and comparison with the crisp plan are given.

2. Two-stage fuzzy model for CIPMS

The production activity consists of a series of organized and correlated activities. The production system acquires customer demand information from the market. The company manager must then determine management strategies. Next, executors implement optimal aggregate production plans, and prepare materials, labour and equipment. Finally, workers produce and deliver products to the custo-mers. Complaints and the fluctuating requirements of customers are fed back to the system and then used to adjust the company management strategy and aggre-gate production plan. The above steps form a production system cycle. A high-level manager nor-mally determines the management strategy, while the middle-level staff implement the aggregate production plan. In this section, we construct a fuzzy model to resolve the two-stage production problem in a CIPMS environment.

2.1. Qualitative properties of a high-level environment High-level managers pursue the goals of an enterprise while adapting to a changing environment. This is the underlying premise of strategic planning. Planning environments faced by managers contain qualitative properties. The qualitative property is included in nearly all potential impacts. These impacts are generally not clearly related to other impacts. Notable examples of these potential impacts include economic, social, and political problems, technological development as well as current and potential compe-titors. At this level, adopting a qualitative perspective is more prevalent than a quantitative one. Managers accumulate all kinds of information that could be directly or indirectly related to their company. Managers evaluate problems by assigning a weight to each of the problems using their own unique judgement, and then arranging the problems accord-ing to the weight or their preference. Hence, the decision-making strategy contains many qualitative properties.

2.2. Imprecise properties of a middle-level environment Middle-level staff employ concrete methods to implement aggregate production planning in order to achieve a high-level manager’s strategic goal. Once the high-level decision is made, the maximum production resources and capability are determined. Therefore, aggregate production planning has the following features: (1)it originates from a quantitative perspec-tive more than from a qualitaperspec-tive one; (2) it seeks to fulfil multiple objectives; and (3) it is short-term planning. The environment is not as complex as the high-level stage. However, the product demand, exact available workforce, reliability of equipment and materials supply conditions change daily. Therefore, this stage contains many imprecise properties.

2.3. Construction of a high-level fuzzy model

Construction of a high-level fuzzy model to deter-mine the management strategy can be divided in the following steps.

Step 1. List the entire possible goals and constraints of

the corporation. The possible items for the corporation’s goals or constraints depend on the type of corporation, and the experience, knowledge, and preference of managers. All common corporations’ goals and constraints are listed below.

(1) Goals. G1: obtaining the maximum profit or lowest production cost,

G2: owning the corporation’s

leadership in a product market, G3: maintaining the best competi-tion policies such as to ensure the product delivery in time, or to accept changes in the purchase order from customers.

(2) Constraints. C1: limited to the level of economic prosperity,

C2: limited to the production capacity,

C3: limited to the ability of research and development.

Step 2. Establish the membership function for each

corporation’s goal or constraint. Once the corporation’s goals and constraints in step 1 are defined, the measurement standards to achieve the corporation’s goals and constraints must be established. The membership functions exactly represent the measurement standards. Figure 1

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illustrates the membership function of maximum profit for a company. When the company makes a profit equal to, or more than, $687 500, one degree of membership is reached, indicating that people think that the company makes maximum profit. The company making a profit of $450 000 indicates that 65.5% people think that the company makes maximum profit.

Step 3. Propose all the possible management strategies

for the corporation. In addition, assume that there are k (k is a limited positive integer)

possible strategies, N1, N2, . . ., Nk, proposed. For

example:

N1: Satisfy the entire customer demands com-pletely, regardless of any problem.

N2: Never manufacture products when they

exceed the maximum production scale.

Step 4. Estimate the degree of membership of each goal or

constraint under different strategies. In addition, estimate a group of membership degrees, mGmn(Ni) or mCmn(Ni), for every strategy Ni.

Step 5. Select the strategy among the strategies of N1,

N2, . . ., Nk. For simulating the human

decision-making process, the rationality of the Max-Min operator is used here. The optimal strategy can be determined by selecting the strategy that has the Max-Min degree of membership, i.e. Max[Min mGmn (Ni)or mCmn (Ni)for a particular

strategy Ni], i ˆ 1, . . ., k.

2.4. Construction of a middle-level fuzzy model

Middle-level staff perform the aggregate production plan by following the corporation’s management strategy. Owing to the many imprecise properties in

this stage, the fuzzy linear programming model is used here. Steps to construct the fuzzy linear programming model are described in the following.

Step 1. Construct a crisp linear programming model.

By assuming that the aggregate production plan includes variables x1, x2, x3, . . ., xn, a variable

vector, X ˆ [x1 x2 x3 . . . xn]T, X 2 Rn£1, can

represent them. In general, the objective func-tion is either to maximize profit or to minimize cost, i.e.

Max(or Min)fa…X † ˆ CatX ; a ˆ 1; 2; 3; . . . ; n where

fa2 R1,

Ca2 Rn£1: the unit cost vector for each variable,

subject to DX µ B, X ¶ 0, where

D 2 Rr£n

B 2 Rr£1.

Step 2. Construct the fuzzy constraints. As described

above, some resources and demands in the production system are imprecise. Therefore, some elements in constraint vector B could be fuzzy numbers. Then the constraints can be modified as follows.

The fuzzy constraint portion AX µ ~B

The non-fuzzy constraint portion NX µ B0,

X ¶ 0 A N 8 > : 9 > ;ˆD and BB~0 8 > > : 9 > > ;ˆB and ‰~B ˆ b1; b2; . . . ; bp]. Figure 2 depicts the assumption of the

member-ship function of these fuzzy resources bj,

j ˆ 1; 2; . . . ; p, where bj ˆ ‰abj;bbj;`bj;rbjŠ is a fuzzy constraint, `bj is the minus tolerance for

bjandrbjis the plus tolerance for bj.

Correspondingly, the membership function mj(X)of bjcan be derived. mj…x†ˆ 0 …AX †j µabj¡`bj …AX †j¡abj‡`bj `bj abj¡`bj5…AX †jµabj 1 abj5…AX †j µbbj bbjrbj¡…AX †j rbj bbj5…AX †jµb bjrbj 0 …AX †j4bbjrb 8 > > > > > > > > > > < > > > > > > > > > > : …1†

Figure 1. The membership function of maximum profit for a company.

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According to a-cut, a variable l, 8mj(x),

mj(x)¶ l ˆ a is introduced. Then, the constraint

function can be written as …AX †j¡ l`bjabj¡`bj; …AX †j‡ lrbjµbbjrbj; j ˆ 1; 2; 3; . . . ; p » …2† NX µ B ’ X ¶ 0.

Step 3. Construct the fuzzy objective function: since the

constraint is fuzzified, the objective function must be fuzzified as well.

fa…X † ˆ CatX ; a ˆ 1; 2; 3; . . . ; n where fa(X)2 [Oa, Pa], Oais the optimistic value

of the purpose function, and Pais the

pessimis-tic value of the purpose function.

When the objective function is to maximize profit, the membership function, ma(X)can be

obtained, mA…X † ˆ 1 Ct aX 4Oa Ct aX ¡ Pa Oa¡ Pa Pa µ C t aX µ Oa 0 Ct aX 5Pa 8 > < > : …3†

and when the objective function is to minimize cost, the membership function, ma(X) can be

obtained, ma…x† ˆ 1 Ct aX 5Oa Pa¡ CatX Pa¡ Oa Oaµ C t aX µ Pa 0 Ct aX 4Pa 8 > < > : …4†

Step 4. Construct the fuzzy linear programming model.

When seeking the maximum profit or minimum cost, the model becomes

Maximize l Subject to

Ct

aX ¡ l…Oa¡ Pa† ¶ Pa; a ˆ 1; 2; . . . ; n; when seeking the maximum profit,

Ct

aX ‡ l…Pa¡ Oa† µ Pa; a ˆ 1; 2; . . . ; n; when seeking the minimum cost,

…AX †j¡ l`bjabj¡`bj;

…AX †j‡ lrbj µbbj¡rbj; j ˆ 1; 2; . . . ; p,

NX µ B0,

X ¶ 0: (5)

Step 5. Search for the optimal solution by using

LINDO. 3. Illustrative example

This study considers a manufacturer of sunglasses as an illustrative example. As assumed herein, the manufacturer produces only one kind of sunglasses, the optimal production capacity is 25 000 pairs monthly and each pair costs $500. If the production quantity exceeds the optimal production capacity, then the cost per pair for the exceeded portion is $650. The monthly fixed production cost is $860 000. The manufacturer has only one major material supplier. The optimal material supply quantity equals the optimal production quantity {Pi-1}+5%. Notably, market prosperity affects the stability of the material supply. The more prosper-ous is the market, the less stable is the supply; otherwise the supply is more stable. The major customers are A and B. Customer A is the first major customer, customer B is the second major customer, and others are not important and are not considered here. The manufacturer provides customer A at $550 each pair and customer B at $570 each pair. Upon receipt of the purchase order from customers, the manufacturer must deliver the product in time. If not, the manufacturer gives 5% and 2% rebate of the delayed product amount to customers A and B, respectively. Different customers have different levels of tolerance to the product delay. Whenever customers are not tolerant of the delay, they cancel their purchase orders. For a delay of less than or equal to 0.1 month, both customers A and B can 100% accept such a delay. However, the tolerance linearly decreases if the delay lasts longer. For customer A, the tolerance is 100% absent when the delay is equal to, or over, two months. For customer B, the tolerance is 100% absent when the delay is equal to, or over, three months. Market prosperity affects the possibilities that customers fulfil the purchase order. The more prosper-ous the market, the higher the possibility that the customer fulfils the purchase order; otherwise the

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possibility is lower. Table 1 lists the next four-month purchase orders that the manufacturer received. 3.1. Construction of a high-level fuzzy model

In this section, we sequentially carry out the five steps described in section 3.

Step 1. List all the possible items of the corporation’s

goals and constraints.

(1) Goals. Assume that the manufacturer has

only one goal, G1, i.e. to reach the

maximum profit. In this example, the maximum profit possible is defined as 5% of the gross income at the optimal production quantity, 25 000 pairs per month, and the lowest sale price, $550 per pair, i.e. maximum profit possi-ble ˆ $550/pair£25 000 pairs/month£ 5% ˆ $687 500/month.

(2) Constraints.

(a) The constraint of the optimal produc-tion quantity, C1. As expected, the production quantity is greater than or equal to the minimum production quantity and less than or equal to the maximum production quantity. Due to the monthly fixed cost of $860 000, the manufacturer’s production quan-tity should exceed a specific quanquan-tity so that the manufacturer can pay for the fixed cost. The specific quantity is exactly the minimum production quantity, and is equal to

Minimum production quantity ˆ $860,000/month/($550 – $500)/pair

ˆ $17,200 pair/month

In addition, the manufacturer never produces products if no profit can be made. The maximum tion quantity is defined as the produc-tion quantity in which the producproduc-tion cost is equal to the gross income. In this example, the maximum produc-tion quantity is equal to

Maximum production quantity ˆ [$(550 – 500)£25 000 pairs/month ¡$860 000/month)] ¥ $(650 – 550)/ pair ‡ 25 000 pairs ˆ 28 900 pairs/ month.

(b) The constraint of the optimal

ma-terial supply quantity, C2. The ma-terial supply quantity can never exceed the maximum production quantity, 28 900 pairs, and never be less than the minimum production quantity, 17 200 pairs. The company would be satisfied if the material supply quantity is equal to 25 000 pairs +5%.

(c) The constraint of the stability of the

material supply, C3. The material

supplier may or may not supply the material as they promised. Market prosperity affects the stability of the material supply.

(d) The constraint of the customer’s

tolerance to the product delay, C4. When no longer tolerating the delay, customers cancel their purchase or-ders. The tolerance linearly decreases with a longer delay.

(e) The constraint of the fulfilment of the

purchase order by customer, C5.

Market prosperity affects the possibi-lity that customers fulfil the purchase orders.

Step 2. Establish the membership function for each

goal or constraint.

(1) Goals. Figure 1 illustrates the membership function of the maximum profit.

mG1ˆ 1, if profit ¶ 687 500,

mG1ˆ profit/687 500, if 0 µ profit

µ 687 500, mG1ˆ 0, if profit µ 0.

(2) Constraints.

(a) Figure 3 illustrates the membership function of the optimal production quantity, C1.

mC1ˆ 0, if the production quantity

517 200,

Figure 3. The distribution of the membership function of the optimal production quantity.

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mC1ˆ(28 000¡the production

quantity)/(28 900¡25 000), if 25 000 µ the production quantity µ 28 900, and

mC1ˆ 0, if the production quantity4

28 900.

(b) Figure 4 illustrates the membership function of the optimal material supply quantity.

mC2ˆ 0, if the material supply quantity

µ 17 200,

mC2ˆ(supply quantity¡17 200)/

(23 750¡17 200), if the supply quantity417 200 and the supply quantity µ 23 750,

mC2ˆ 1, if the supply quantity 423 750

and the supply quantity µ 26 250, mC2ˆ(28 900¡supply quantity)/

(28 900¡23 750), if the supply quantity 423 750 and the supply quantity µ 28 900, and

mC2ˆ 0, if the supply quantity

428 900.

(c) Figure 5 illustrates the membership function of the stability of the ma-terial supply. The stability of the material supply is linearly decreased when the market prosperity is from fair to good. The stability of the material supply is perfect when the market prosperity is poor.

mC3ˆ 1, if market prosperity is poor;

the average degree of the member-ship, mC3ˆ 0.9, if the market

prosper-ity is fair;

the average degree of the member-ship, mC3ˆ 0.7, if the market

prosper-ity is good.

(d) Figure 6 illustrates the membership function of customer’s tolerance to product delay. The customer’s toler-ance linearly decreases with an in-crease of the delay time.

mC4ˆ 1.0, if the delay time is less than

or equal to 0.1 month for both A and B,

mC4ˆ 0.667 for customer A and

mC4ˆ 0.8 for customer B, if the

delay time is 1.0 month, mC4ˆ 0 for customer A and mC4ˆ 0.4

for customer B, if delay time is 2.0 months,

mC4ˆ 0 if the delay time is greater than

2 months for customer A and

greater than 3 months for cus-tomer B.

(e) Figure 7 illustrates the membership function of customers’ fulfilment of the purchase order. The possibility of a customer’s fulfilment of the pur-chase linearly increases with an

in-Figure 4. The distribution of the membership function of the optimal material supply quantity.

Figure 5. The distribution of the membership function of the stability of the material supply.

Figure 6. The distribution of the membership function of the customer’s tolerance to the product delay.

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crease of the prosperity of the market:

the average degree of the member-ship, mC5ˆ 0.5, if the market

prosper-ity is poor;

the average degree of the member-ship, mC5ˆ 0.8, if the market

prosper-ity is fair;

mC5ˆ 1.0, if the market prosperity is

good.

Step 3. Propose all possible management strategies.

Assume that the company has the following possible strategies, Ni, i ˆ 1,2,3,4.

N1: Satisfy the entire customer demand com-pletely, regardless the profit,

N2: Treat customers equally and never produce more products if the optimal production capacity is reached. Therefore, when the accumulative quantity exceeds the optimal production capacity, each customer re-ceives part of the order and shares the shortage equally.

N3: Deal with the customer who has the most shortage and never produces any product once the optimal production capacity is reached. Therefore, the customer with a maximum delayed product quantity has the highest priority to receive the order. When none of the customers have a shortage, customer A would have higher priority than the others.

N4: Satisfy the maximum profit order prior to others when the accumulative quantity of orders exceeds the optimal production capacity.

Step 4. Estimate the degree of membership of each goal or

constraint under different strategies. Based on the

membership function of each goal or constraint, the degree of membership of each goal or constraint can be estimated under different strate-gies. Therefore, the fuzzy evaluation chart for different strategies can be established.

(1) Fuzzy evaluation of the maximum profit

goal. According to the order received, table 2 summarizes the company profits for different strategies. The degree of the membership for each strategy is calculated as follows:

N1: mG1 (N1)ˆ 0.685, the average monthly profit is $471 250;

N2: mG1 (N2)ˆ 0.670, the average monthly profit is $460 469;

N3: mG1(N3)ˆ 0.660, the average monthly profit is $453 000;

N4: mG1(N4)ˆ 0.651, the average monthly profit is $447 813.

(2) Fuzzy evaluation of the constraints.

(a) Constraint of the optimal production quantity. Table 3 lists the degrees of the membership of the monthly opti-mal production quantity for each strategy. The average degree of the membership can thus be estimated.

N1: the average degree of

member-ship mC1(N1)ˆ 0.600,

N2: the average degree of

member-ship mC1(N2)ˆ 0.920,

N3: the average degree of

member-ship mC1(N3)ˆ 0.920,

N4: the average degree of

member-ship mC1(N4)ˆ 0.920,

(b) Constraint of the optimal quantity of the material supply. Table 4 estimates the degrees of the membership of the optimal material supply quantity per month. Then, the average degree of membership for each strategy is:

N1: the average degree of

member-ship mC2(N1)ˆ 0.716,

N2: the average degree of

member-ship mC2(N2)ˆ 0.952,

N3: the average degree of

member-ship mC2(N3)ˆ 0.952,

N4: the average degree of

member-ship mC2(N4)ˆ 0.952.

(c) Constraint of the stability of the

material supply. As could be known, the market prosperity is fair for the next four months. The degrees of membership of the stability for differ-ent strategies for each month are all

Figure 7. The distribution of the membership function for the customer’s fulfilment of the purchase order.

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0.9. Then the average degree of the membership of each strategy is:

N1: mC3 (N1)ˆ 0.900,

N2: mC3 (N2)ˆ 0.900,

N3: mC3 (N3)ˆ 0.900,

N4: mC3 (N4)ˆ 0.900.

(d) Constraint of customer’s tolerance to the product delay. The degrees of the membership for each strategy are:

N1: the maximum delay time for

customers A and B is 0 month, mC4 (N1)ˆ 1.0,

N2: the maximum delay time for

customers A and B is 3 months, mC4 (N2)ˆ 0,

N3: the maximum delay time for

customers A and B is 1 month, mC4 (N3)ˆ 0.667 for A and mC4

(N3)ˆ 0.800 for B. The average

degree of membership mC4

(N3)ˆ 0.734,

N4: The maximum delay time for

customer A is 3 months, and the

degree of membership mC4

(N4)ˆ 0. The maximum delay

time for customer B is 0 month, and the degree of membership mC4 (N4)ˆ 1.000. The average

degree of membership mC4

(N4)ˆ 0.500.

(e) The constraint of the possibility of customers’ fulfilment of the purchase order. As could be known, the market prosperity is fair in the next four months. The degrees of membership of the possibility that customers would carry out their orders are all 0.8. Then the average degree of membership of each strategy is:

Table 1. The next four-month purchase orders that the manufacturer received.

Month

Customer Month 1 Month 2 Month 3 Month 4

A 20 000 17 500 12 500 17 500

B 2500 10 000 15 000 7500

Table 2. The estimated company profit according to the customer orders received.

Month Month 1 Month 2 Month 3 Month 4

Company A B A B A B A B

Ordered quantity 20 000 2500 17 500 10 000 12 500 15 000 17 500 7500

Policy Production quantity 20 000 2500 17 500 10 000 12 500 15 000 17 500 7500

N1 Shortage 0 0 0 0 0 0 0 0

Gross income 11 000 000 1 425 000 9 625 000 5 700 000 6 875 000 8 550 000 9 625 000 4 275 000

Rebate 0 0 0 0 0 0 0 0

Cost 12 110 000 14 860 000 14 860 000 13 360 000

Profit 315 000 465 000 565 000 540 000

Policy Production quantity 20 000 2500 16 250 8750 11 250 13 750 17 500 7500

N2 Shortage 0 0 1250 1250 2500 2500 2500 2500

Gross income 11 000 000 1 425 000 8 937 500 4 987 500 6 187 500 7 837 500 9 625 000 4 275 000

Rebate 0 0 34 375 14 250 68 750 28 500 68 750 28 500

Cost 12 110 000 13 360 000 13 360 000 13 360 000

Profit 315 000 516 375 567 750 442 750

Policy Production quantity 20 000 2500 17 500 7500 7500 17 500 22 500 2500

N3 Shortage 0 0 0 2500 5000 0 0 5000

Gross income 11 000 000 1 425 000 9 625 000 4 275 000 4 125 000 9 975 000 1 275 000 1 425 000

Rebate 0 0 0 28 500 137 500 0 0 57 000

Cost 12 110 000 13 360 000 13 360 000 13 360 000

Profit 315 000 511 500 602 500 383 000

Policy Production quantity 20 000 2500 15 000 10 000 10 000 15 000 17 500 7500

N4 Shortage 0 0 2500 0 5000 0 5000 0

Gross income 11 000 000 1 425 000 8 250 000 5 700 000 5 500 000 8 550 000 9 625 000 4 275 000

Rebate 0 0 68 750 0 137 500 0 137 500 0

Cost 12 110 000 13 360 000 13 360 000 13 360 000

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N1: mC5 (N1)ˆ 0.800,

N2: mC5 (N2)ˆ 0.800,

N3: mC5 (N3)ˆ 0.800,

N4: mC5 (N4)ˆ 0.800.

The proposed high-level management strategy evaluation is performed and summarized in table 5.

Step 5. Select the management strategy:

the minimum degree of membership ˆ 0.600 for strategy N1,

the minimum degree of membership ˆ 0 of strategy N2,

the minimum degree of membership ˆ 0.660 of strategy N3, and

the minimum degree of membership ˆ 0.500 of strategy N4.

The optimal strategy is the one with the largest minimum degree of membership:

i

Max

m

Min…mG

1…Ni†; mCm…Ni†† ˆ 0:660

the optimal management strategy D ˆ N3. As mentioned above, this part of the research is intended to present a model enabled with human thinking and feeling. Although it is selected, strategy 3 is not committed to be the best one.

3.2. Construction of a middle-level fuzzy model

High-level managers determine the optimal com-pany management strategy, and middle-level staff

execute the production in an imprecise/uncertain environment. Since the resources and demands in production system are imprecise, fuzzy numbers can represent some of the elements in the constraints. The given conditions, cost coefficients and variables of the production system are described as follows.

(1) Given conditions

(a) The planning production period is 4 months.

(b) The planned production quantity, Ri, under

strategy N3 is R1ˆ 22 500 pairs, and R

R3ˆ R4ˆ 25 000 pairs.

(c) The initial manpower M0is 40 men/month.

(d) The initial inventory stock, I0, is 0 pairs.

(e) The initial backorder, B0, is 0 pairs.

(f) The working hours per person per day are 8

hours.

(g) The ratio of overtime and regular working

hours cannot be greater than 20%.

(h) The manufacturing time for a pair of glasses is 0.32 hours/pair.

(i) There are 25 working days in a month.

(2) Cost coefficient

CP: material cost ˆ $420/pair.

CR: regular labour cost ˆ $15 000/person, month. CH: hiring cost ˆ $5000/person, month.

CL: layoff cost ˆ $15 000/person, month. CO: overtime labour cost ˆ $125/hour. CI: inventory cost ˆ $3/pair, month. CB: backorder cost ˆ $20/pair, month.

(3) Variables

Mj: month j manpower,

Hj: month j hiring work force,

Lj: month j layoff work force,

Pnj: month j production quantity in regular

work-ing hours,

Poj: month j production quantity in overtime,

Ij: month j inventory level,

Bj: month j backorder level.

Step 1. Construct a crisp linear programming model:

Assume that the execution achievement goal

Table 3. The degree of the membership of the monthly optimal production quantity.

Degree of

membership Month1 Month2 Month3 Month4

Strategy N1 0.679 0.360 0.360 1.0

Strategy N2 0.679 1.0 1.0 1.0

Strategy N3 0.679 1.0 1.0 1.0

Strategy N4 0.679 1.0 1.0 1.0

Table 4. The degree of the membership of the stability of the component supply.

Membership

degree Month1 Month2 Month3 Month4

N1 0.809 0.528 0.528 1.0

N2 0.809 1.0 1.0 1.0

N3 0.809 1.0 1.0 1.0

N4 0.809 1.0 1.0 1.0

Table 5. The proposed high-level management strategy evaluation results. Strategy N1 N2 N3 N4 mG1(Ni) 0.685 0.670 0.660 0.651 mC1(Ni) 0.600 0.920 0.920 0.920 mC2(Ni) 0.716 0.952 0.952 0.952 mC3(Ni) 0.900 0.900 0.900 0.900 mC4(Ni) 1.000 0.000 0.734 0.500 mC5(Ni) 0.800 0.800 0.800 0.800

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minimizes the total production cost that in-cludes (1) the material cost, (2) the inventory and backlog cost,(3)the personnel cost. Based on the above information, a crisp linear programming model is established as follows:

(1) The minimum production cost objects: MinfC ˆ

X4

jˆ1

Cp…Pnj‡ Poj†‡…CIIj‡ CBBj†‡ …CRMj‡CHHj‡CLLj‡CO…0:32POj††

It includes material cost, inventory cost, backlog cost, regular labour cost, hiring cost, layoff cost and overtime cost.

(2) Constraints:

(a) The planned production quantity is

equal to the regular production quan-tity plus overtime production quanquan-tity plus the inventory difference between month j and month j¡1 minus the backorder difference between month

j and month j¡1, i.e.

Rj ˆPnj‡ Poj‡ Ij¡1¡ Ij¡ …Bj¡1¡ Bj†;

j ˆ 1; 2; 3; 4

(b) The manpower level in month j is

equal to the manpower level in month

j¡1 plus the hiring workforce in

month j minus the layoff workforce in month j, i.e.

Mjˆ Mj¡1‡ Hj¡ Lj; j ˆ 1; 2; 3; 4

(c) The regular working hours in a

month must be less than or equal to the working day hours, i.e.

0:32Pnjµ 8 £ 25Mj; j ˆ 1; 2; 3; 4

(d) The overtime work hours must be less than or equal to 20% of the working day hours, i.e.

0:32Pojµ 0:2 £ 8 £ 25Mj; j ˆ 1; 2; 3; 4

(e) The total planned production

quan-tity must be less than or equal to the total production quantity in the plan-ning production period, i.e.

XN jˆ1 Rj µ XN jˆ1 Pnj‡ Poj; j ˆ 1; 2; 3; 4

(f) All the variables are not less than 0:

Mj; Hj; Lj; Pnj; Poj; IjBj¶ 0;

j ˆ 1; 2; 3; 4

(g) Use a linear programming software to search the optimum crisp plan. A crisp plan is shown in table 6.

Step 2. Construct the fuzzy constraints. Owing to

imprecise factors such as unstable material supply, customer order cancellation or change, poor products, and machine break-down, the produced quantity is usually less than the planned one. Therefore, some of the planned resources, workforce or materials are wasted. Here, the planned production quan-tity is represented by a fuzzy number that gives a minus tolerance of 10% on the original planned production quantity. As ex-pected, the fuzzy planning can provide less waste than the crisp planning. The fuzzy planned production quantity can be repre-sented as follows: ~ R1ˆ [22500, 22500, 2250, 0] ~ R2ˆ [25000, 25000, 2500, 0] ~ R3ˆ [25000, 25000, 2500, 0] ~ R4ˆ [25000, 25000, 2500, 0]

According to equations(1)and(2)the original constraint

Rj ˆPnj‡ Poj‡ Ij¡1¡ Ij¡ …Bj¡1¡ Bj†;

j ˆ 1; 2; 3; 4

can be transferred into

Pn1‡Po1‡Io¡I1¡…B0¡B1†¡2250l ¶ 20250

Pnj‡Poj‡Ij¡1¡Ij¡…Bj¡1¡Bj†¡2500l ¶ 22500;

j ˆ 2; 3; 4

Step 3. Construct the fuzzy objective function: Since the

constraint is fuzzified, the objective function must be fuzzified as well. According to equation

(4), we obtain mG…X † ˆ 1 fC…X †5Oa Pa¡ fC…X † Pa¡ Oa Oa µ fC…X † µ Pa 0 fC…X †4Pa 8 > < > :

where Oaˆ min fC(X~), and Paˆ max fC(X~).

Table 6. The crisp production plan.

Month 1 2 3 4

Planned regular

production quantity 24 375 24 375 24 375 24 375 Planned inventory level 1875 1250 625 0

Planned manpower level 39 39 39 39

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Step 4. Construct the fuzzy linear programming model.

According to steps 1 to 3 and equation(5), the result is summarized as follows:

Maximise l Subject to fC…X † ‡ l…Pa¡ Oa† µ Pa; Pn1‡Po1‡I0¡I1¡…B0¡B1†¡2250l ¶ 20250 Pn1‡ Po1‡ I0¡ I1¡ …B0¡ B1† µ 22500 Pnj‡Poj‡Ij¡1¡Ij¡…Bj¡1¡Bj†¡2500l ¶ 22500, j ˆ 2,3,4 Pnj‡ Poj ‡ Ij¡1¡ Ij¡ …Bj¡1¡ Bj† µ 25000, j ˆ 2,3,4 Mjˆ Mj¡1‡ Hj¡ Lj; j ˆ 1; 2; 3; 4 0:32Pnjµ 8 £ 25Mj; j ˆ 1; 2; 3; 4 0:32Poj µ 0:2 £ 8 £ 25Mj; j ˆ 1; 2; 3; 4 XN jˆ1 Rjµ XN jˆ1 Pnj‡ Poj; j ˆ 1; 2; 3; 4 Mj; Hj; Lj; Pnj; Poj; Ij; Bj ¶ 0; j ˆ 1; 2; 3; 4

Step 5. Obtain the optimal solution. A linear

program-ming package LINDO (Linear INteractive and

Discrete Optimizer; see Schrage 1989)is used to search for Oa, Pa and the optimum solution.

After the compromise among objectives and constraints, we obtain:

l ˆ 0.7446, and a fuzzy production plan shown in table 7.

Environmental conditions and resources are time-variant. To verify the effectiveness of the proposed fuzzy plan, several numerical experiments are conducted. Perturbing the planned production quantity by 1% to 10% to generate the forecast errors can simulate ten actual production quantities. If the actual production quantity is greater than that planned, the shortage quantity can be produced over time. If the actual production quantity is less than that planned, the excess quantity can be stored in the inventory. The actual production cost can thus be estimated. Figure 8 illustrates the comparison between the fuzzy plan and the crisp plan for the ten cases.

3.3. Summary

This section has demonstrated the two-stage fuzzy planning model in the CIPMS for the sunglasses manufacturer. This case study confirms that high-level strategic decision and middle-level production plan-ning can be determined in a fuzzy way.

4. Conclusions

CIPMS largely focuses on gaining maximum profits through efficient management/administra-tion. However, the production environment is full of qualitative descriptions and imprecise natures. Thus, using the conventional approach yields an unsatisfactory performance of CIPMS. This study focuses mainly on the qualitative properties in a high-level decision stage. A high-level fuzzy model is also proposed. The management strategy can there-fore be determined through the computer in a qualitative way that closely resembles human think-ing. Also addressed here are the imprecise proper-ties in the production planning period, in which a middle-level fuzzy model is proposed. The optimal production planning can therefore be obtained at the moment that factors such as the actual demand, resources and manpower are uncertain. The pro-posed two-stage fuzzy model enables CIPMS to adapt to the complex and changeable environment. Therefore, the model proposed here improves CIPMS.

Acknowledgements

The authors would like to thank the National Science Council of the Republic of China for financially supporting this research under Contract No. NSC87-2213-E002-032.

Table 7. The fuzzy production plan.

Month 1 2 3 4

Planned regular

production quantity 23 752 23 752 23 752 23 752 Planned inventory level 1827 1218 609 0

Planned manpower level 38 38 38 38

Planned layoff workforce 2 0 0 0

Figure 8. The comparison between the fuzzy plan and the crisp plan on different forecast errors.

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數據

Figure 2 depicts the assumption of the member- member-ship function of these fuzzy resources b j , j ˆ 1; 2;
Figure 3. The distribution of the membership function of the optimal production quantity.
Figure 5. The distribution of the membership function of the stability of the material supply.
Table 2. The estimated company profit according to the customer orders received.
+3

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