Fuzzy Economic Order Quantity Inventory Models

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nificantly enhance a company's profit. In 1913, the ... examined the economic order quantity model by treat- ing ordering costs and inventory holding costs as trapezoidal fuzzy ... example, when a trapezoidal fuzzy number and a. Gaussian fuzzy ..... ate to two crisp numbers, the model (9) changes to the following form, min. 2.
TSINGHUA SCIENCE AND TECHNOLOGY ISSN 1007-0214 14/17 pp91-96 Volume 12, Number 1, February 2007

Fuzzy Economic Order Quantity Inventory Models Without Backordering* WANG Xiaobin (王小斌)1,2, TANG Wansheng (唐万生)1,**, ZHAO Ruiqing (赵瑞清)1 1. Institute of Systems Engineering, Tianjin University, Tianjin 300072, China; 2. School of Computer and Information Engineering, Shandong University of Finance, Ji’nan 250014, China Abstract: In economic order quantity models without backordering, both the stock cost of each unit quantity and the order cost of each cycle are characterized as independent fuzzy variables rather than fuzzy numbers as in previous studies. Based on an expected value criterion or a credibility criterion, a fuzzy expected value model and a fuzzy dependent chance programming (DCP) model are constructed. The purpose of the fuzzy expected value model is to find the optimal order quantity such that the fuzzy expected value of the total cost is minimal. The fuzzy DCP model is used to find the optimal order quantity for maximizing the credibility of an event such that the total cost in the planning periods does not exceed a certain budget level. Fuzzy simulations are designed to calculate the expected value of the fuzzy objective function and the credibility of each fuzzy event. A particle swarm optimization (PSO) algorithm based on a fuzzy simulation is designed, by integrating the fuzzy simulation and the PSO algorithm. Finally, a numerical example is given to illustrate the feasibility and validity of the proposed algorithm. Key words: inventory; fuzzy variable; dependent chance programming; fuzzy simulation; particle swarm optimization

Introduction Inventory control is an important field in supply chain management. A proper control of inventory can significantly enhance a company’s profit. In 1913, the economic order quantity (EOQ) formula was introduced by Harris[1]. Since then, a large number of academic papers have been published describing numerous variations of the basic EOQ model (for a review, see Brahimi et al.[2]). This body of research assumes that the parameters involved in the EOQ model, such as the demand and the purchasing cost, are crisp values Received: 2005-12-22; revised: 2006-04-12

﹡ Supported by the National Natural Science Foundation of China (Nos.70471049 and 70571056) and the China Postdoctoral Science Foundation (No. 2004035013)

﹡﹡ To whom correspondence should be addressed. E-mail: [email protected]; Tel: 86-22-27406095

or random variables. However, in reality, the demand and the cost of the items often change slightly from one cycle to another. Moreover, it is very hard to estimate the probability distribution of these variables due to a lack of historical data. Instead, the cost parameters are often estimated based on experience and subjective managerial judgment. Thus, the fuzzy set theory, rather than the traditional probability theory, is well suited to the inventory problem. The fuzzy set theory was first introduced by Zadeh[3], and has now been applied in inventory control systems to model behavior more realistically. In 1981, Sommer[4] used fuzzy dynamic programming to solve a real-world inventory and production scheduling problem, where linguistic statements such as “the stock should be at best zero at the end of the planning horizon” and “diminish production capacity as continuously as possible” were used to describe management's

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fuzzy aspirations for inventory and production capacity reduction in a planned withdrawal from a market. In 1982, Kacprzyk and Staniewski[5] applied the fuzzy set theory to the inventory problem and considered longterm inventory policy-making through fuzzy decisionmaking models. An algorithm was also presented to find the optimal time-invariant strategy for determining the replenishment to current inventory levels that maximized the membership function for the decision. After that, several scholars developed the EOQ inventory problems in the fuzzy sense. For example, Park[6] examined the economic order quantity model by treating ordering costs and inventory holding costs as trapezoidal fuzzy numbers. The mode and median rules were suggested for transforming the fuzzy cost information into a scalar for input to the EOQ model. Vujošević et al.[7] investigated a fuzzy EOQ model by introducing fuzzy inventory costs and fuzzy order costs. They then obtained the fuzzy total cost and defuzzified the fuzzy total cost with the moments method. Gen et al.[8] expressed their input data as fuzzy numbers, and then the interval mean value concept was introduced to solve the inventory problem. Lee and Yao[9] depicted the order quantity as a triangular fuzzy number, then obtained the fuzzy total cost, and defuzzified it with the centroid method. Yao et al.[10] depicted the order quantity and the total demand as triangular fuzzy numbers, and then solved the problem following the same method as Lee and Yao[9]. Yao and Chiang[11] treated the total demand and the cost of storing one unit per day as triangular fuzzy numbers, then obtained the fuzzy total cost, and defuzzified it with the signed distance method and the centroid method. In all these EOQ problems, the parameters were assumed to be triangular fuzzy numbers or trapezoidal fuzzy numbers. Therefore, the membership function of the total cost can be calculated easily. However, if the membership function of fuzzy variable is complex, for example, when a trapezoidal fuzzy number and a Gaussian fuzzy number coexist in a model, it is hard to obtain the membership function of the total cost by the methods applied in Refs. [6-11]. Moreover, the moments method, the centroid method, and the signed distance method can only be considered as the ranking function for fuzzy numbers in some special problems. For more general cases, such as where a trapezoidal fuzzy number is divided by a Gaussian fuzzy number, the membership function of the quotient is hard to

Tsinghua Science and Technology, February 2007, 12(1): 91-96

obtain. Hence, the moments method and centroid method are both difficult to apply. In addition, when the membership function figure is multi-ridged, the αlevel cut may include several different intervals, but the signed distance only considers the problem in one interval. Thus, all these methods are unsatisfactory. In this paper, the EOQ problem is investigated by introducing the fuzzy theory to an inventory system. As a general extension of the classical EOQ model, a fuzzy expected value model (EVM) and a fuzzy dependent chance programming (DCP) model are constructed, then an intelligent algorithm is suggested for solving these models.

1

Preliminaries

Let Θ be a nonempty set, P(Θ ) be the power set of

Θ , and Pos be a possibility measure. Then the triplet (Θ , P(Θ ), Pos) is called a possibility space. Let A be a set in P (Θ ) . The necessity measure of A can then be represented by Nec{ A} = 1 − Pos{ Ac }

(1)

c

where A is the complement of A. Definition 1[12] Let (Θ , P (Θ ), Pos) be a possibility space, and A a set in P (Θ ). Then the credibility measure of A is defined by 1 Cr{ A} = (Pos{ A} + Nec{ A}) (2) 2 Definition 2 A fuzzy variable ξ is defined as a mapping from a possibility space (Θ , P (Θ ), Pos) to

the set of real numbers, and its membership function is defined by µξ ( x) = Pos{θ ∈Θ |ξ (θ ) = x}, x ∈ R (3) Proposition 1[13] Suppose that (Θi , P(Θi ), Posi ), are possibility spaces. Let i = 1, 2, ..., n,

Θ = Θ1 × Θ 2 × Pos{ A}=

sup

n

× Θ n = ∏Θi , and i =1

(θ1 ,θ 2 ,..., θ n )∈ A

Pos1{θ1} ∧ Pos 2 {θ 2 } ∧

∧ Pos n {θn}

for each A ∈ P(Θ ). Then the set function Pos is a possibility measure on P(Θ ), and (Θ , P(Θ ), Pos) is a possibility space (called the product possibility space of (Θi , P(Θi ), Posi ), i = 1, 2, ..., n ). Definition 3[13]

The fuzzy variables ξ1 , ξ 2 , ..., ξ n are

said to be independent if and only if Pos{ξi ∈ Bi , i = 1, 2, ..., n} = min Pos{ξi ∈ Bi } 1≤i≤n

for any sets B1 , B2 , ..., Bn of R.

(4)

WANG Xiaobin (王小斌) et al:Fuzzy Economic Order Quantity Inventory Models …

Definition 4[12] Let ξ be a fuzzy variable on the possibility space (Θ , P(Θ ), Pos) . Then the expected value E[ξ ] is defined as E[ξ ] = ∫

+∞ 0

0

Cr{ξ ≥ r}dr − ∫ Cr{ξ ≤ r}dr −∞

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spaces (Θi , P(Θi ), Posi ), i = 1, 2, respectively. A typical behavior of the EOQ model without backordering is depicted in Fig. 1.

(5)

provided that at least one of the two integrals is finite. Especially, if ξ is a positive fuzzy variable then E[ξ ] = ∫

+∞ 0

Cr{ξ ≥ r}dr.

Proposition 2[14]

Let ξ and η be independent

fuzzy variables with finite expected values. Then for any numbers a and b, we have E[aξ + bη ] = aE[ξ ] + bE[η ] (6)

2

Model Formulation

The purpose of the EOQ model is to find the optimal order quantity of inventory items at each time such that the combination of the order cost and the stock cost is minimal. In the classical EOQ model without backordering, an instantaneous replenishment is assumed to take place when the inventory level drops to zero, and the stock items are exhausted with a fixed demand rate. Moreover, the stock cost and the order cost in each replenishment are assumed to be deterministic. But in real situations, the order cost and the stock cost are usually affected by various uncontrollable factors and often show some fluctuation. In most cases, these parameters are described as “approximately equal to some certain amounts” or “located in some intervals with a membership degree”. It is more reasonable, therefore, to characterize these parameters as fuzzy variables. The purpose of this section is to discuss the EOQ model in the fuzzy sense. For the sake of clarity, the following notations and assumptions are used throughout this paper. t, length of one cycle; h, stock cost per unit quantity per unit time; T, the plan period; K, order cost for each cycle; D, total demand quantity during the plan period; Q, order quantity per cycle; F(Q), total cost in the plan period. Assumptions (1) No shortage is allowed. (2) The parameters h and K are assumed to be independent fuzzy variables defined on possibility

Fig. 1 EOQ model without backordering

It is obvious that the order quantity Q is less than or equal to the total demand D, i.e., Q ≤ D. The total demand D in plan period [0, T ] can be expressed as T D = Q. t Hence, T D = (7) t Q htQ and the Note that the stock cost in one cycle is 2 T number of cycles in the plan period [0, T ] is . It t follows from Eq. (7) that the total cost F (Q) in the plan period [0, T ] can be expressed by ⎛ htQ ⎞ T hQT KD F (Q) = ⎜ +K⎟ = + (8) 2 Q ⎝ 2 ⎠t Since the parameters h and K are fuzzy variables defined on the possibility spaces (Θi , P (Θ i ), Posi ), i = 1, 2, the total cost F (Q) is a fuzzy variable defined on the product possibility space (Θ , P (Θ ), Pos) , where Θ = Θ1 × Θ 2 and Pos = Pos1 ∧ Pos 2 .

2.1

Fuzzy expected and value EOQ model

If the decision maker wants to find an order quantity Q such that the fuzzy expected value of the total cost is minimal, a fuzzy EVM can be constructed as follows, ⎡ hQT KD ⎤ min E ⎢ + (9) Q ⎥⎦ ⎣ 2

s.t. 0 ε , k = 1, 2, ..., N , where ε is a sufficiently small number. Then set xk = h(θ k ) and rearrange the order of xk such that x1 < x2 < … < xN . Additionally, calculate µk = µh ( xk ) by the membership function of h for k = 1, 2, ..., N . Step 2

Employ

N

∑x w k =1

k

k

to estimate E[h], where

1 w1 = ( µ1 + max µ j − max µ j ), 1≤ j 1< j 2 1 wk = ( max µ j − max µ j + max µ j − max µ j ), 1≤ j < k k≤ j k< j 2 1≤ j≤k 2 ≤ k ≤ N − 1, 1 wN = ( max µ j − max µ j + µ N ). 1≤ j < N 2 1≤ j≤N Example 1 Let h be a drum-shape fuzzy variable with membership function ⎧1 − ( x − 3)2 , 2 ≤ x < 3; ⎪ 3 ≤ x ≤ 4; ⎪1, µh ( x) = ⎨ 2 ⎪1 − ( x − 4) , 4 < x ≤ 5; ⎪⎩ 0, otherwise. Then running the fuzzy simulation described above with N = 1000, we can obtain E[h] = 3.4960.

Remark 1 If the fuzzy parameters h and K degenerate to two crisp numbers, the model (9) changes to the following form, ⎧ hQT KD ⎫ min ⎨ + (12) ⎬ Q ⎭ ⎩ 2 s.t. 0 ≤ Q ≤ D.

This is just the classical EOQ model, and the optimal order quantity Q* in Eq. (11) changes to Q* =

2 KD hT

with

the

minimal

total

cost

F (Q* ) = 2hKDT .

2.2

Fuzzy dependent chance programming EOQ model

Usually, the decision maker hopes that the total cost does not exceed the budget level r. In this situation, a natural idea is to maximize the credibility of the event such that the total cost is less than or equal to r. Thus a fuzzy DCP model can be constructed as follows: ⎧ hQT KD ⎫ max Cr ⎨ ≤ r⎬ (13) + Q 2 ⎩ ⎭ s.t. 0 < Q ≤ D. The key for solving the model described by Function (13) is to calculate the value of ⎧ hQT KD ⎫ Cr ⎨ ≤ r ⎬ for each given Q. Usually, it is + Q ⎩ 2 ⎭ very difficult to obtain this value with an analytic method. Instead, we design a fuzzy simulation to esti⎧ hQT KD ⎫ ≤ r ⎬ for a fixed Q as follows. mate Cr ⎨ + Q ⎩ 2 ⎭ Step 1

Set e1 = 0, e2 = 0, and n = 1.

Step 2

Uniformly generate a sequence (θ1n , θ 2 n )

from Θ = Θ1 × Θ 2 such that Pos{θin } > ε ,

i = 1, 2,

where ε is a sufficiently small number. Thus, we can obtain a real vector (h(θ1n ), K (θ 2 n )). h(θ1n )QT K (θ 2 n ) D Step 3 Calculate + and 2 Q µ = min {µh (h(θ1n )), µ K ( K (θ 2 n ))}. h(θ1n )QT K (θ 2 n ) D Step 4 If + ≤ r and e1 < µ , 2 Q set e1 = µ. h(θ1n )QT K (θ 2 n ) D Step 5 If + > r and e2 < µ , 2 Q set e2 = µ . Step 6 Return to Step 2 with n + 1 replacing n until a given number of iterations is reached. 1 Step 7 Return e = (e1 + 1 − e2 ). 2

WANG Xiaobin (王小斌) et al:Fuzzy Economic Order Quantity Inventory Models …

3

PSO Algorithm Based on the Fuzzy Simulation

The particle swarm optimization (PSO) algorithm, which is a population-based algorithm, was originally designed by Kennedy and Eberhart[15]. Compared with other evolutionary algorithms, such as the genetic algorithm, the PSO algorithm has a faster convergence rate and very few adjustable parameters. The algorithm has been developed by many researchers, such as Shi and Eberhart[16,17], and has already been used for a wide range of applications. In order to solve the proposed DCP model, we design an algorithm by embedding the fuzzy simulation into the PSO algorithm, where the fuzzy simulation is employed to estimate the credibility of each fuzzy event and the PSO algorithm is used to find the optimal solution. The steps of the newly designed algorithm are described as follows. Step 1 Set k = 1. Step 2 Randomly generate an initial position Qik from the interval (0, D] for the particle i, where i = 1, 2, ..., N , and N is the population size of the swarm. Then randomly sample a velocity Vi k from [0, v] for the particle i, where v is the largest available velocity, i = 1, 2, ..., N . Step 3

Let Pi k be a position of the particle i such

that ⎧ hP T KD ⎫ ⎧ hQ T KD ⎫ Cr ⎨ i + k ≤ r ⎬ = max Cr ⎨ + l ≤ r ⎬, l k 1 ≤ ≤ Pi Qi ⎩ 2 ⎭ ⎩ 2 ⎭ l ⎧ hQi T KD ⎫ + l ≤ r ⎬ can be calwhere the value of Cr ⎨ Qi ⎩ 2 ⎭ culated by the fuzzy simulation designed in Section 2.2, and where i = 1, 2, ..., N . We call Pi k the p-best of k

l i

the particle i. Further, let Pgk be a position such that ⎧⎪ hPgk T KD ⎫⎪ Cr ⎨ + k ≤ r⎬ = Pg ⎪⎩ 2 ⎪⎭ ⎧ hQ l T KD ⎫ Cr ⎨ i + l ≤ r ⎬ . 1≤l ≤k , 1≤i≤ N 2 Q i ⎩ ⎭ k We call Pg the g-best of all particles. max

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Step 4 Set Vi k +1 = ωVi k + c1r1 ( Pi k − Qik ) + c2 r2 ( Pgk − Qik ), where ω is the inertia weight, c1 and c2 are acceleration constants, and r1 and r2 are two random numbers in the interval [0, 1].

Step 5

Set

Qik +1 = Qik + Vi k +1 . Step 6 Set k ← k + 1, and return to Step 3 until a given number of cycles is reached. Step 7 Return the value of Pgk as the optimal

solution.

4

A Numerical Example

Consider a fuzzy EOQ inventory system without backordering in which the plan period T = 5, the total demand quantity during the plan period D = 32, the stock cost h = (2.8, 3, 3.1), and the order cost K = (3.6, 4, 4.1, 4.5). In order to maximize the expected total cost, a fuzzy EVM can be constructed as ⎧ 5 E[h]Q 32 E[ K ] ⎫ min ⎨ (14) + ⎬ Q ⎭ ⎩ 2 s.t. 0< Q ≤ 32. It follows from Eq. (11) that the optimal solution is 5 E[h]Q 32 E[ K ] Q* = 4.1744 with + = 62.0935. 2 Q If instead we want to maximize the credibility that the total cost does not exceed 65, a fuzzy DCP model can be constructed as ⎧ 5hQ 32 K ⎫ max Cr ⎨ ≤ 65⎬ (15) + Q ⎩ 2 ⎭ s.t. 0 < Q ≤ 32. Use the PSO algorithm based on the fuzzy simulation designed in Section 3 to solve the model (15). Set the largest available velocity v = 2, the inertia weight ω = 1, and the acceleration constants c1 = c2 = 2. By running the PSO algorithm based on the fuzzy simulation (3000 cycles for the fuzzy simulation and 1000 iterations in the PSO), we obtain the optimal order size and the maximal credibility as follows, ⎧ 5hQ* 32 K ⎫ Q* = 4.2228 and Cr ⎨ + * ≤ 65⎬ = 0.8140. Q ⎩ 2 ⎭ The graph in Fig. 2 shows the variation of the maximal credibility with the number of iterations N .

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From Fig. 2, it can be observed that the maximal value ⎧ 5hQ 32 K ⎫ of Cr ⎨ ≤ 65⎬ tends to remain steady + Q ⎩ 2 ⎭ after about 500 iterations.

Tsinghua Science and Technology, February 2007, 12(1): 91-96 338-356. [4] Sommer G. Fuzzy inventory scheduling. In: Lasker G, ed. Applied Systems and Cybernetics. New York: Pergamon Press, 1981, VI: 3052-3060. [5] Kacprzyk J, Staniewski P. Long-term inventory policymaking through fuzzy decision-making models. Fuzzy Sets and Systems, 1982, 8: 117-132. [6] Park K. Fuzzy-set theoretic interpretation of economic order quantity. IEEE Transactions on Systems, Man, and Cybernets, 1987, 17: 1082-1084. [7] Vujošević M, Petrović D, Petrović R. EOQ formula when

Fig. 2 The variation of the maximal credibility with the number of iterations N

In order to verify the feasibility of our proposed algorithm, we combine a GA and a fuzzy simulation, and design a GA based on the fuzzy simulation to solve the same numerical example. In the model, the population size is 30, the probability of crossover is 0.3, and the probability of mutation is 0.2. After 2000 cycles, the Q* = 4.1916 with result obtained is ⎧ 5hQ* 32 K ⎫ Cr ⎨ + * ≤ 65⎬ = 0.8108. Accordingly, the 2 Q ⎩ ⎭ performance of the PSO algorithm based on the fuzzy simulation is acceptable.

5

Conclusions

This paper studied the fuzzy EOQ inventory problems without backordering. In contrast to previous studies, we characterize the order cost and the stock cost in an inventory system as fuzzy variables, and construct a fuzzy EVM and a fuzzy DCP model. In order to solve these complex models, we have designed fuzzy simulations and a PSO algorithm based on the fuzzy simulation. The algorithm has been tested using a numerical example. The results show the algorithms described in this paper perform well.

References

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