Continuous review inventory models with a mixture of backorders and lost sales under fuzzy demand and different decision situations

Expert Systems with Applications 39 (2012) 4181–4189

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Continuous review inventory models with a mixture of backorders and lost sales under fuzzy demand and different decision situations Lin Wang a,⇑, Qing-Liang Fu a, Yu-Rong Zeng b a b

School of Management, Huazhong University of Science & Technology, Wuhan, Hubei 430074, China School of Information Management, Hubei University of Economics, Wuhan, Hubei 430205, China

a r t i c l e

i n f o

Keywords: Continuous review inventory model Defuzzification Possibilistic mean value Fuzzy simulation Differential evolution algorithm

a b s t r a c t In this paper, continuous review inventory models in which a fraction of demand is backordered and the remaining fraction is lost during the stock out period are considered under fuzzy demands. In order to find the optimal decision under different situations, two decision methods are proposed. The first one is finding a minimum value of the expected annual total cost, and the second one is maximizing the credibility of an event that the total cost in the planning periods does not exceed a certain budget level. For the first decision method, an approach of ranking fuzzy numbers by their possibilistic mean value is adopted to achieve the optimal solution. For the second one, the technique of fuzzy simulation and differential evolution algorithms are integrated to design hybrid intelligent algorithms to solve the fuzzy models. Subsequently, the two decision models are compared and some advices about inventory cash flow management are given. Further, sensitivity analysis is conducted to give more general situations to illustrate the rationality of the management advices. Crown Copyright Ó 2011 Published by Elsevier Ltd. All rights reserved.

1. Introduction In an inventory management, a major concern problem is to decide the time when a replenishment order is to be placed and further what quantity of such replenishment is to be ordered. As is well known continuous review inventory model is an appropriate mathematical model for such problem (Darwish, 2008; Dutta, Chakraborty, & Roy, 2007a). In traditional continuous review inventory systems, the lead-time and the annual demand is assessed by a crisp value (Salameh, Abbound, El-kassar, & Ghattas, 2003; Zhang, Patuwo, & Chu, 2003). But in practical situations, precise values of the cost characteristics are not always known exactly as they may be vague and imprecise. Fuzzy theory is one of good choices to deal with this situation and many scholars have already made some achievements. Gen, Tsujimura, and Zheng (1997) presented mean value of fuzzy number for continuous review inventory model with fuzzy input data. Yao and Chiang (2003) considered the total cost of inventory without backorder and compared the results obtained by two defuzzification methods. Dutta, Chakraborty, and Roy (2007b) analyzed a single period inventory model with fuzzy random demand, and the objective was to determine the optimal order quantity in maximizing the expected profit. Tutuncu, Akoz, Apaydin, and Petrovic (2008) presented new models of continuous review inventory control with or without models in the presence of uncertainty, ⇑ Corresponding author. E-mail address: [email protected] (L. Wang).

and defined fuzzy total annual cost functions involving fuzzy arithmetic operations. Recently, Handfield, Warsing, and Wu (2009) developed a (Q, r) model using fuzzy set representations of various uncertain sources including demand, lead time and penalty cost, then the total cost is computed using defuzzification methods. Moreover, continuous review inventory models should be studied with a mixture of backorder and lost sales. Obviously, when dealing with continuous review inventory problems, most of researchers prefer to assume that the demand during the stock-out period is either completely backorder or completely lost. However, these are always not true. It can be observed, in real market, that when the inventory system is out of stock, a fraction of the customers are willing to fill their demand immediately from another source, while others may wait till the next arrival of stock. Hence some continuous review inventory models were extended to include backorder and lost sales. Vijayan and Kumaran (2008) considered continuous review inventory models in which a fraction of demand was backordered and the remaining fraction was lost during the stock-out period under fuzzy environment. They assumed fixed ordering cost, inventory holding cost, fixed shortage cost and shortage cost of lost sales are fuzzy. In fact, compared with demand, they are easier to obtain accurate data for companies. In the present day scenario, it is really very difficult to forecast the demand for a decision make. On this view, fuzzy inventory models in situations where the customer demand is described imprecisely should be studied in detail. In this paper, we consider a continuous review inventory model under fuzzy environment by assuming average annual demand

0957-4174/$ - see front matter Crown Copyright Ó 2011 Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2011.09.116

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and demand during lead-time as fuzzy variables. And during the stock-out period a part of demand is backordered and the remaining fraction is lost. Then an elegant methodology to determine the optimal order quantity and reorder point that the expected total cost per year has a minimum value. Sometimes, decision makers are not concern about making the total cost minimum, but hope that the total cost does not exceed the budget level, especially when enterprises have much cash flow pressure. In this situation, a fuzzy dependent-chance programming (DCP) model is always regarded as a good choice. The DCP model can be used to find the optimal solution for maximizing the credibility of an event that the total cost in the planning periods does not exceed a certain budget level. Liu (2000) provided the framework of DCP and illustrated a fuzzy simulation (FS) approach for measuring possibility using genetic algorithm by some numerical examples. Wang, Tang, and Zhao (2007) constructed a fuzzy DCP EOQ model in which the function was about credibility and find that it is very difficult to obtain the value with an analytic method. So, they designed a FS approach to estimate credibility of DCP model for fixed Q and r. When the objective functions to be optimized are multimodal or the search spaces are particularly irregular, hybrid intelligent algorithms using FS and optimization algorithms should be designed to solve the fuzzy models. Moreover, optimization algorithms need to be highly robust in order to avoid getting stuck at local optimal solution. Among these algorithms, genetic algorithm (GA) has been proved to be effective for the FS approach (Ke & Liu, 2010; Liu, 2000). However, the GA is only capable of identifying the high performance region at an affordable time and displays inherent difficulties in performing local search for numerical applications. So, it is necessary to find a novel algorithm to deal with FS more efficiently and effectively. Recently, a novel differential evolution (DE) algorithm was proposed by Storn and Price (1997) for complex continuous non-linear, on-differentiable and multi-modal optimization problem. This technique combines simple arithmetic operators with the classical events of crossover, mutation and selection to evolve from a randomly generated starting population to a final solution. DE algorithm is easy to implement, requires only several parameters and shows fast convergence (Aslantas & Kurban, 2010; Chang, 2010; Wang, He, & Zeng, 2011). It is reliable, accurate, robust and fast optimization that make DE algorithm widely used (Lu, Zhou, Qin, Li, & Zhang, 2010; Wang, He, Wu, & Zeng, 2011). However, according to Krink, Filipic, and Fogel (2004), noise may adversely affect the performance of DE due to its greedy nature. How to improve the performance of DE algorithm is a hot focus. There are two methods to obtain better behaviors than the typical DE algorithm. One is improving the parameters of DE, and the other introduces mechanisms of other algorithms into DE. So in this paper, in order to search the optimum solution, two hybrid DE algorithms are also used besides the basic DE algorithms. One called self-adaptive DE (SDE) is an attempt to dynamically adjust F, a scale factor used to control the amplification of the differential variation. SDE is proposed by Salman, Engelbrech, and Omran (2007), and the performance of SDE is investigated and compared with other well-known approaches. The experiments conducted show that SDE generally outperform DE algorithm in all the benchmark. The other one called modified DE (MDE) is in the framework of DE owning new mutation operator and selection mechanism inspired from GA, particle swarm optimization (PSO) and simulated annealing (SA), respectively. In other words, positive characteristics of DE, GA, PSO and SA are combined to create a new efficient stochastic search technique. The algorithm has been successfully applied to solve the non-convex economic dispatch (Amjady & Sharifzadeh, 2010). It is examined on three economic dispatch test systems and compared with some the most recently published economic dispatch solution methods to show the efficiency and

robustness of SDE. So, three novel approaches using FS and DE/ SDE/MDE are designed to solve DCP model, named FSDE/FSSDE/ FSMDE respectively. The aim of this paper is to find the optimal decision for continuous review inventory models with a mixture of backorders and lost sales under fuzzy demand. So, we propose two decision methods under different situations. The first one is minimizing the expected total cost per year, and the second one is maximizing the credibility of an event such that the total cost in the planning periods does not exceed a certain budget level. Then we will compare the two methods and try to provide some advices about the principals of inventory cash flow management. This paper will also present a DCP model about continuous review inventory system under the practical environment for the first time. Moreover, new hybrid intelligent algorithms will be proposed to solve the DCP model in an efficient and reliable way without considering the shapes of fuzzy membership functions. The rest of this paper is organized as follows: Section 2 is preliminaries. Section 3 contains the proposed models and analysis. In Section 4, we defuzzify the fuzzy model by possibilistic mean value to find the optimum solution. In Section 5, we propose three hybrid intelligent DE algorithms to solve the DCP model. Section 6 contains numerical examples, results and advises. In Section 7, we discuss the result and provide directions for future research. 2. Fuzzy preliminaries e ¼ ðA; A; AÞ on the space Definition 1. A triangular fuzzy number A of real numbers R, can be described with following membership function:

leA ðxÞ ¼



LðxÞ ¼ ðx  AÞ=ðA  AÞ

for A 6 x 6 A

RðxÞ ¼ ðA  xÞ=ðA  AÞ for A 6 x 6 A

ð1Þ

e the alpha-cut set Aa is given by For a given fuzzy set A, þ Aa ¼ fxjleðxÞ P ag and is denoted by interval ½A a ; Aa ; 0 6 a 6 1. A

e the interval value possiDefinition 2. For a given fuzzy number A, e ¼ ½M ð AÞ; e M ð AÞ e where M  ð AÞ e and bilistic mean is defined as Mð AÞ e are the lower and upper possibilistic mean values of A e M ð AÞ (Carlsson & Fullér, 2001) and are defined respectively by

R1  e ¼ 0R aAa da ; M ð AÞ 1 ada 0

R1 þ e ¼ 0R aAa da M  ð AÞ 1 ada 0

ð2Þ

e is defined as The possibilistic mean value of A

e ¼ Mð AÞ

e þ M  ð AÞ e M  ð AÞ 2

ð3Þ

In other words, one can write

e ¼ Mð AÞ

Z 0

1





a Aa þ Aþa da

ð4Þ

e and B e be two fuzzy numbers Definition 3. In applications, let A   þ   þ with Aa ¼ Aa ; Aa and Ba ¼ Ba ; Ba ; a 2 ½0; 1 then for ranking e6B e 6 Mð BÞ. e () Mð AÞ e fuzzy numbers, A Let H be a nonempty set, P(H) be the power set of H, and Pos be a possibility measure. Then the triplet (H, P(H), Pos) is called a possibility space. Let A be a set in P(H). The necessity measure of A can then represented by

NecfAg ¼ 1  PosfAC g

ð5Þ

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where AC is the complement of A. Let (H, P(H), Pos) be a possibility space, and A is a set in P(H). Then the credibility measure of A is defined by

1 CrfAg ¼ ðPosfAg þ NecfAgÞ 2

ð6Þ

Definition 4. A fuzzy variable n is defined as a mapping from a possibility space (H, P(H), Pos) to the set of real numbers, and its membership function is defined by

ln ðxÞ ¼ Posfh 2 HjnðhÞ ¼ xg; x 2 R

ð7Þ

Proposition 1. Suppose that (H, P(H), Pos), i = 1, 2, . . . , n, are possibility spaces. Let

H ¼ H1  H2      Hn ¼

n Y

Hi ; PosfAg

i¼1

¼

sup

Pos1 fh1 g ^ Pos2 fh2 g ^    ^ Posn fhn g

ðh1 ;h2 ;...;hn Þ2A

for each A 2 P(H). Then the set function Pos is a possibility measure on P(H), and (H, P(H), Pos) is a possibility space (called the product possibility space of (H, P(H), Pos), i = 1, 2, . . . , n). Definition 5. The fuzzy variable n1, n2, . . . , nn are said to be independent if and only if

Posfni 2 Bi ; i ¼ 1; 2; . . . ng ¼ min Posfni 2 Bi g 16i6n

ð8Þ

maximize the credibility of the event such that the total cost is less than or equal to TC. So, a fuzzy DCP model can be written as

e Crf CðQ; rÞ 6 TCg 8 > 0 6 Q 6D < s:t: r min 6 r 6 r max > : Q min 6 Q 6 Q max

max

ð10Þ

3.3. Model analysis ~  rÞþ . The safety Now, we will discuss value range of r and Mðd stock is defined as the difference between reorder point r and the ~L Þ during lead-time. Since the minimum expected demand Mðd ~L Þ. If r > d L , it level of safety stock is zero, which implies r P Mðd L  r inventory and extra holding cost. So, we results in extra d L and r 2 ½Mðd ~L Þ; d L . For Q, it is obvious that can see r 6 d Q 2 ½0; D is reasonable. Referring to the study of (Detta et al., 2007a), the exact expres~  rÞþ for a given r in ½Mðd ~L Þ; d L  can be obtained as sion of Mðd follows Situation 1. For dL 6 r 6 dL, we have the alpha level set of leadtime demand in under-stocking situation (when the demand rate is greater than r) as Fig. 1.

8h i > < r; dþL;a for a 6 LðrÞ i ðdL Þa ¼ h  þ > : dL;a ; dL;a for a > LðrÞ

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

which implies

3. Mathematics modeling and analysis

8h i > < 0; dþL;a  r for a 6 LðrÞ þ ~L  rÞ Þ ¼ h i ððd a > : dL;a  r; dþL;a  r for a > LðrÞ

3.1. Fuzzy continuous review (Q, r) inventory model (Model-I)

Therefore, the possibilistic mean of probable shortage is computed by

Notations are as follows e D ~ d

average annual demand, a triangular fuzzy number,

Q r A h s b

lot-size or order quantity (decision variable), reorder point (decision variable), fixed cost of placing an order, yearly per unit inventory holding cost, fixed shortage cost per unit short, fraction of demand backordered during the stork out period, 0 < b < 1, shortage cost of lost sales including the lost profit.

L

p

~L  rÞþ ¼ Mðd

Z

1

0

demand during lead time, a triangular fuzzy number,

¼

Z

1

0

¼

Z

0

¼

1





a ððd~L  rÞþ Þa þ ððd~L  rÞþ Þþa da 

Z



a dþL ðaÞ  r da þ þ dL ð

a

aÞda þ

Z

1

LðrÞ

1



LðrÞ  dL ð

a



a dL ðaÞ  r da

aÞda  rð1  0:5ðLðrÞÞ2 Þ

L þ dL þ 4dL dL  r  r  dL 2 d þ 3 6 dL  dL  2 ! 1 r  dL r 1 2 dL  dL

ð11Þ

The expression for annual total cost is formulated as

e ðQ ; rÞ ¼ C

!   e D Q ~L Þ Aþh þ r  Mðd Q 2 " # e pð1  bÞ D e sD þ ~ þ Mðd  rÞ hð1  bÞ þ þ Q Q

L , the possibilistic mean of probable Situation 2. For dL 6 r 6 d shortages is computed by

ð9Þ

~L ¼ ðdL ; dL ; d L Þ; Mðd ~L Þ is expected lead-time dee ¼ ðD; D; DÞ; d where D ~  rÞþ is the expected or possible shortages during mand, and Mðd each cycle. 3.2. DCP model (Model-II) Usually, the decision maker hopes that total cost does not exceed the budget level TC. In this situation, a natural idea is to

~L . Fig. 1. Alpha cut of d

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~L  rÞþ ¼ Mðd ¼

Z Z

1 0

Therefore,

aðððd~L  rÞþ Þa þ ððd~L  rÞþ Þþa Þda

RðrÞ 0

adþL ðaÞda  0:5rðRðrÞÞ2

L  r d L  r d ¼  6 dL  dL

!2 ð12Þ

! ~L  rÞþ ½s þ pð1  bÞ D þ D þ 4D @MðQ ; rÞ h A þ Mðd ¼  @Q 2 6 Q2 " !# 2 @MðQ ; rÞ ðRðrÞÞ s þ pð1  bÞ D þ D þ 4D ¼h hð1  bÞ þ @r Q 6 2 Equating

(1) The optimum Q⁄ and r⁄ For r 2 [dL, dL], we have

MðQ ; rÞ ¼

0

Equating

@MðQ ;rÞ @r

ð15Þ

¼ 0, we get

2

ðRðrÞÞ ½s þ pð1  bÞ D þ D þ 4D Q¼ 6 2h  hðRðrÞÞ2 ð1  bÞ

! ð16Þ

By equating the square of Eq. (16) with Eq. (15), r⁄ can be found. The optimum Q⁄ is then solved by substituting r⁄ into Eq. (16). Eqs. (15) and (16) are a necessary condition for existence Q⁄ and r⁄.

4.1. Situation 1



!

2

The possibilistic mean of expected average total cost per year is a function of Q and r. The simultaneous equation for solving Q and r and the necessary conditions to attain the optimality are derived as follows:

1

¼ 0, we get

2 ~L  rÞþ ½s þ pð1  bÞÞ D þ D þ 4D Q ¼ ðA þ Mðd h 6

4. Defuzzifition by possibilistic mean value for Model- I

Z

@MðQ ;rÞ @Q

~  rÞþ 4.3. Sufficient condition to attain the minimum of Mðd

þ

a C a þ C a da

!   Q A D þ D 2D ~ ~L  rÞþ þ Mðd þ r  MðdL Þ þ þ ¼h 2 Q 6 3 " !# s þ pð1  bÞ D þ D 2D  hð1  bÞ þ þ Q 6 3

In both situations mentioned above, the sufficient condition and the range of r must be satisfied at the same time. A sufficient condition for Q⁄ and r⁄ to attain the minimum is the convexity of MðQ ; rÞ. That means Hessian matrix of MðQ ; rÞ must be positive at (Q⁄, r⁄), which requires

@ 2 MðQ; rÞ @Q 2

Therefore, ! ~L  rÞþ ½s þ pð1  bÞ D þ D þ 4D @MðQ ; rÞ h A þ Mðd ¼  @Q 2 6 Q2 " !# @MðQ ; rÞ s þ pð1  bÞ D þ D þ 4D ¼ h  ½1  0:5ðLðrÞÞ2  hð1  bÞ þ @r Q 6

@ 2 MðQ ; rÞ @ 2 MðQ ; rÞ @ 2 MðQ ; rÞ  2 2 @r @Q @r @Q

and

!2 > 0:

~  rÞþ indicates that Mðd ~  rÞþ > 0 The derivation process of Mðd is established. And hence

@ 2 MðQ; rÞ @Q 2

¼

2 h Q3

i ~  rÞþ ðs þ pð1  bÞÞ D þ D þ 4D A þ Mðd 6

!

is always positive. The sufficient condition reduces to Equating

@MðQ ;rÞ @Q

2 ~L  rÞþ ½s þ pð1  bÞÞ D þ D þ 4D ðA þ Mðd h 6

Q2 ¼ Equating

¼ 0, we get

@MðQ ;rÞ @r

! ð13Þ

The detailed expression can be found in Appendix A.

¼ 0, we get

! ½1  0:5ðLðrÞÞ ½s þ pð1  bÞ D þ D þ 4D Q¼ 6 h  h½1  0:5ðLðrÞÞ2 ð1  bÞ

!2 @ 2 MðQ; rÞ @ 2 MðQ ; rÞ @ 2 MðQ ; rÞ  > 0: @r2 @Q @r @Q 2

5. Credibility calculation approach based on fuzzy simulation and DE/SDE/MDE algorithm

2

ð14Þ 5.1. Fuzzy simulation

By equating the square of Eq. (14) with Eq. (13), r⁄ can be found. The optimum Q⁄ is then solved by substituting r⁄ into Eq. (14). Eqs. (13) and (14) are necessary conditions for the existence of Q⁄ and r⁄.

Usually, it is difficult to solve Eq. (10) with an analytic method. Instead, we use a fuzzy simulation approach to obtain the value of Eq. (10) for a fixed Q and r. According to (Wang et al., 2007), a typical FS method can be adopted as follows:

4.2. Situation 2

Step 1. Set e1 = 0, e2 = 0, and n = 1. Step 2. Uniformly generate a sequence h1n from H such that Pos{h1n} P e where e is sufficiently small number. Thus, we can obtain a real vector D(h1n). Step 3. If r 2 [dL, dL], utilize Eq. (11) to calculate

L , we have For r 2 ½dL ; d

MðQ ; rÞ ¼

Z 0

1





a C a þ C þa da

!   Q ~L Þ þ A D þ D þ 2D þ r  Mðd ¼h 2 Q 6 3 " !# ~L  rÞþ hð1  bÞ þ s þ pð1  bÞ D þ D þ 2D þ Mðd Q 6 3

CðQ ; rÞ ¼

and l = lD(Din).

    Din Q ~L Þ Aþh þ r  Mðd 2 Q  ~  rÞþ hð1  bÞ þ sDin þ pð1  bÞDin þ Mðd Q Q

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in the DE. It means the resulting trial individual will only replace the original if it has a lower objective function value. Otherwise, the parent will remain in the next generation.

L , utilize Eq. (12) to calculate If r 2 ½dL ; d

    Din Q ~Þ þ r  Mðd CðQ ; rÞ ¼ Aþh L 2 Q  ~  rÞþ hð1  bÞ þ sDin þ pð1  bÞDin þ Mðd Q Q

 xi;Gþ1 ¼

and l = lD(Din). Step 4. If C(Q, r) 6 TC, and e1 < l, set e1 = l. Step 5. If C(Q, r) > TC, and e2 < l, set e2 = l. Step 6. Return to step 2 with n + 1 replacing n until a given number of iterations is reached. Step 7. Return e = (1/2)⁄(e1 + 1  e2). Where e1 denotes Pos {C(Q, r) 6 TC} and e2 denotes Pos {C(Q, r) > TC} or 1-Nec {C(Q, r) 6 TC}. 5.2. Differential evolution algorithm and modification DE algorithms are adopted to search a satisfactory solution in the acceptable time. When the objective functions to be optimized are multimodal or the search spaces are particularly irregular, algorithms need to be highly robust in order to avoid getting stuck at local optimal solution. The advantage of DE algorithms is just to obtain the global optimal solution fairly. In the following, we will discuss three different DE algorithms. 5.2.1. Basic DE algorithm The basic DE algorithm consists of three evolution operators: mutation, crossover, and selection. In mutation operator, instead of making use of some probability distribution function in order to introduce variations into the population, DE uses the differences between randomly selected individuals to generate a trial individual. Then crossover operator is used to produce one offspring which is only accepted if it improves on the fitness of parent individual. The process of choosing individuals is called selection. The three evolution operators: mutation, crossover, and selection, are described in more detail below.

ui;Gþ1 ; if f ðui;Gþ1 Þ P f ðxi;G Þ if f ðui;Gþ1 Þ < f ðxi;G Þ xi;G ;

ð19Þ

This is the same for all variants of the DE. Although the selection pressure is only one, the best individual of the next generation will be at least as fit as the best individual of the current generation. 5.2.2. Self-adaptive differential evolution (SDE) Recently, there were several attempts to dynamically adjust the parameters of DE. Abbass (2002) proposed the self-adaptive pareto DE (SPDE), a self-adaptive approach to DE for multi-objective optimization problems. In SPDE, the parameter F is generated for each variable from a normal distribution, N(0, 1). Each individual, i, has its own crossover rate, CRi. The parameter CRi is first initialized for each individual in the population from a uniform distribution between 0 and 1. Then a self-adaptive strategy is adopted. Due to the success achieved in SPDE by self-adapting CRi, Salman et al. (2007) proposed that the same mechanism be applied to self-adapt the value of F. Also, in SDE, CRi is generated for each individual from normal distribution. 5.2.2.1. Mutation. Eq. (17) changes as follows

v i;Gþ1 ¼ xr1;G þ F i;G ðxr2;G  xr3;G Þ;

r 1 –r2 –r 3

ð20Þ

where

F i;G ¼ F r4;G þ Nð0; 0:5Þ ðF r5;G  F r6;G Þ

ð21Þ

with r4 – r5 – r6 and r4, r5, r6  U(1, . . . , D) and Fi,G is initialized for each individual from a normal distribution, N(0.5, 0.15), generating values which fits well within the range (0, 1]. 5.2.2.2. Crossover. Eq. (18) changes as follows

5.2.1.1. Mutation. For each target individual xi,G (i = 1, 2, . . . , NP), a mutant individual vi,G+1 is generated according to

v i;Gþ1 ¼ xr1;G þ Fðxr2;G  xr3;G Þ;

r1 – r2 – r3

5.2.1.2. Crossover. It is also called recombination operation. The basic DE crossover operator implements a discrete recombination of the trial individual vi,G+1 and the parent individual xi,G to produce offspring ui,G+1. The crossover is implemented as follows

uji;Gþ1 ¼

v ji;Gþ1

if rand ðjÞ 6 CR or j ¼ rnbðiÞ

xji;G

otherwise

v ji;Gþ1

if rand ðjÞ 6 CRi;G

xji;G

otherwise

or j ¼ rnbðiÞ

ð17Þ

with randomly chosen integer indexes r1, r2, r3 2 {1, 2, . . . , NP}. Note that indexes have to be different from each other and from the running index. F is called mutation factor between [0, 1] which controls the amplification of the differential variation (xr2,G, xr3,G).



 uji;Gþ1 ¼

j ¼ 1; 2; . . . ; D ð18Þ

where xji,G refers to the jth element of the individual xi,G.uji,G+1 and vji,G+1 are similarly defined. rand (j) is the jth evaluation of a uniform random number generator between [0, 1]. rnb (i) is a randomly chosen index from 1, 2, . . . , D which ensures that ui,G+1 gets at least one parameter from vi,G+1. Otherwise, no new parent individual would be produced and the population would not alter. CR is the crossover or recombination rate between [0, 1] which has to be determined by the user. 5.2.1.3. Selection. The selection in DE is deterministic and simple. The evaluation function of an offspring is one-to-one competition

j ¼ 1; 2; . . . ; D ð22Þ

where

CRi;G ¼ CRr7;G þ Nð0; 1Þ ðCRr8;G  CRr9;G Þ

ð23Þ

with r7 – r8 – r9 and r7, r8, r9  U(1, . . . , D) and CRi,G is initialized for each individual from a normal distribution, N(0.5, 0.15), generating values which fits well within the range (0, 1]. The other procedure of SDE is the same with basic DE. 5.2.3. Modified differential evolution (MDE) In MDE, the new mutation operation and selection mechanism inspired from PSO, GA and SA are described as follows (Amjady & Sharifzadeh, 2010). 5.2.3.1. Mutation. The information of the best individuals of current population is used in the proposed mutation operation. Eq. (17) of basic DE is replaced by Eq. (24) as follows

v i;Gþ1 ¼ xr1;G þ F

k X i¼1

"

f ðxib;G Þ

Pk

i¼1 f ðxib;G Þ

!

#

xib;G  xr3;G

! ð24Þ

where xib,G (i = 1, . . . , k are k best individuals owning the highest fitness values in the current population. To enhance the diversity of the search process, xr3,G, inspired from GA, is the result of variable weight arithmetic crossover

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between randomly selected parents xr31,G and xr32,G from current population:

xr3;G ¼ ð1  xa Þxr31;G þ xa xr32;G

ð25Þ

where the variable weight xa is randomly chosen in the range of [0, 1]. Large value of k may cause that poor individuals are involved in the mutation and decelerates the convergence speed. On the other hand, small value of k could lead to premature termination and trapping in the local minimum. To enhance exploitation of proposed mutation, k is adaptively changed along the evolution process.

kðGÞ ¼ k0  Roundððk0  1Þ G=MiterÞ

ð26Þ

where k0 is the initial value of k and Round () is a function that rounds its argument to nearest integer value. G is the counter of iteration of algorithm and Miter is the maximum iteration of algorithm. In the mutation, the mechanism of PSO where a distance between the positions of each particle and the best one is used to modify the positions of the particle is applied.

5.2.3.2. Selection. In the MDE, a new probabilistic selection mechanism inspired from Simulated Annealing (SA) is used instead of the deterministic selection of the basic DE. The proposed selection mechanism of MDE can be described as follows

xi;Gþ1

8 > < ui;Gþ1 ; if f ðxi;G Þ 6 f ðui;Gþ1 Þ ¼ ui;Gþ1 ; if f ðxi;G Þ > f ðui;Gþ1 Þ and hðxi;G ; ui;Gþ1 Þ > randðÞ > : otherwise xi;G ; ð27Þ

hðxi;G ; ui;Gþ1 Þ ¼ exp½ððf ðxi;G Þ  f ðui;Gþ1 ÞÞ=f ðxi;G ÞÞ=T

where exp () is exponential function, and T is temperature like that defined in SA technique. Here, the parameter T is adaptively changed in the evolution process as follows

TðG þ 1Þ ¼ aTðGÞ

ð29Þ

Tð0Þ ¼ T 0

where, the parameter a is the rate of reducing the temperature (a < 1) and T0 is the initial temperature. The proposed selection mechanism begins with a large value for the initial temperature.

START

Initialization:Population NP, Individual (Q,r)0,G, Parameter F CR,G=1, iteration times Miter

Mutation: to produce vi,G+1, Basic DE uses Eq.(17), SDE uses Eq.(20) and (21), and MDE uses Eq.(24),(25) and (26).

Crossover: to produce ui,G+1, Basic DE and MDE use Eq.(18), and SDE uses Eq.(22) and (23).

Adjustment: It's possible some of ui,G+1 exceeding the range of value [Qmin,Qmax] and [rmin,rmax].

Given (Q, r) START

Step 1:e1=0,e2=0, n=1 simulation times M

Step 2:Generate Dn and calculate µn

No

Yes >dL?

Step 4:Utilize Eq.(12) to calculate C(Q,r), if C
Step 3:Utilize Eq.(11) to calculate C(Q,r), if C
Evaluation: Calculate Cr{C(ui,G+1)M? Selection: to produce an offspring, Basic DE and SDE use Eq.(19) and MDE uses Eq.(27), (28) and (29).

No, G=G+1 Is G>Miter?

ð28Þ

Yes Step 5:Cr{C(Q,r)
Yes END

Fig. 2. The flowchart of FSDE/FSSDE/FSMDE.

END

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L. Wang et al. / Expert Systems with Applications 39 (2012) 4181–4189 Table 1 Results obtained by FSDE/FSSDE/FSMDE.

FSDE FSSDE FSMDE

Q⁄⁄

r⁄⁄

Cr{C(Q, r) < TC}

Iteration numbers

Convergence times

MðQ  ; r  Þ

352.966 351.878 349.929

96.527 96.658 96.419

0.8550 0.8550 0.8550

19.63 29.98 10.28

100 91 100

1727.38 1727.16 1726.60

In other words, at the beginning of the evolution process, many new worse solutions have chance to be selected to increase the exploration. By evolving the individuals, the temperature T decreases along the iterations and so the probability of selecting the worse solutions is decreased. The other procedure of SDE is the same with basic DE. 5.3. FSDE/FSSDE/FSMDE procedures for the proposed DCP model The main flowchart of FSDE/FSSDE/FSMDE is shown in Fig. 2. In the adjustment, if r > rmax or r < rmin, then r = rmin + rand (1)⁄(rmax  rmin); if Q > Qmax or Q < Qmin, then Q = Qmin + rand (1)⁄(Qmax  Qmin), where rand (1) 2 (0, 1). 6. Numerical example and sensitivity analysis 6.1. Numerical example In order to verify the above models in fuzzy environment, an inventory system is illustrated where average annual demand e and demand during lead time ð e ð DÞ d L Þ are both described as triangular fuzzy numbers. The basic data are as follows. ~L ¼ ð70; 85; 100Þ, A = 80, h = 5, s = 10, e ¼ ð2400; 3600; 3900Þ, d D p = 15, b = 0.5, and TC = 1810. The result obtained with defuzzifition by possibilistic mean value are Q⁄ = 333.44 and r⁄ = 96.499. Some necessary results should be shown. The range of r is 2

2 2 ;rÞ @ 2 MðQ ;rÞ MðQ ;rÞ [85, 100], the sufficient condition @ MðQ  @ @Q ¼ @r @r 2 @Q 2 0:0426 > 0 is satisfied and the total cost is 1724.665. For (Q, r) = (333.44, 96.499), Cr{C(Q, r) < TC} is 0.8400. Before we use FSDE, FSSDE and FSMDE to compute the credibility, the parameters of three DE algorithms must be confirmed. The value of NP may affect the speed and robustness of the search. Amin & Hong (2008) suggested that the guideline for the number of population is 2Nd 6 NP 6 20Nd(Nd is the dimension of each individual), and in general the larger NP the more robust will be the search. However, the choice of NP is mainly based on empirical evidence and practical experience. It’s hard to be proven in theory. We set NP = 50 for the problems since it has shown a good performance during experimentation. Likewise, the maximum generation was set as 200 since it has also shown a good performance during experimentation. And CR and F were set as 0.8 and 1.2 respectively according to the recommendation of the inventor of DE algorithm (Storn & Price, 1997). Since k and T are adaptively changed along the evolution process, it is not very important to find out optimum values for k0 and T0 and that will be not economical. So, we set k0 = 5, T0 = 1 and a = 0.7 respectively in this paper according to (Amjady & Sharifzadeh, 2010). Using the language programming of MATLAB and parameters mentioned above, we obtained the results summarized in Table 1. Iteration numbers is the average number of iterations to get the final credibility for algorithms executed for 100 times. Convergence times are denoted as the number getting the optimum solution in 100 times algorithms executed. From Table 1, we can see that FSDE and FSMDE are better than FSSDE for convergence efficiency. In the design of algorithms, the only objective is to gain the maximum credibility. And the performances of FSDE and FSMDE are equal. So we adopt FSDE, because FSDE has fewer parameters to control than FSMDE. Then,

two methods have been applied for continuous review inventory model. The detailed results are shown in Table 2(I) and the total cost gained by two decision methods under different D and dL are shown in Table 2(II). Then some advices can be concluded. First, Model-I using defuzzification approach by possobilistic mean value is to find the minimum cost in the fuzzy environment for a long time, but in some special case, for example (D, dL) = (3900, 100), the total costs is bigger. The solution found by solving DCP model with FSDE is to maximize the credibility of event such that the total cost in the planning periods does not exceed a certain budget level. That means in almost every period the total cost can not exceed the certain budge. But in a long term, the total cost does not need to be minimum one. For companies, if they have much cash flow pressure, choosing the DCP model is a better decision that makes companies run smoothly. It can assure that companies will not face cash flow pressure in almost every period. But for a long-term running, it is not the most economic method. If capital is enough, enterprises should adopt defuzzification by possibilistic mean value that can minimize the total cost in long run even if in some period more money is spent. 6.2. Sensitivity analysis Sensitivity analysis is also conducted to demonstrate the advice ~L e and d is right not only for the designed example. In this section, D are both triangular fuzzy numbers. Moreover, for triangular fuzzy numbers, for example (a, b, c), there are three situations that are b  a = c  b, b  a > c  b, and b  a < c  b. Therefore, in the sensitivity analysis, nine situations are considered and the results are summarized in Table 3. where Cr {  } denotes Cr {C(Q, r) < TC}. From Table 3, we can see that sometimes MðQ  ; r Þ and Cr {C(Q, r) < TC} obtained by two optimization methods have no distinct difference. But when companies are in high cash flow pressure, every coin should be spent on where it should do. Even if a little money is spent in a wrong way, companies may have to face adverse situation. So, they should select the right model carefully. 7. Conclusion and future research This paper is an inter-disciplinary research of inventory models and intelligent optimization algorithms. The main contributions Table 2(I) Results comparison of two methods.

Model-I Model-II

(Q, r)

Cr{C(Q, r) < TC}

MðQ  ; r  Þ

(333.44, 96.499) (352.97, 96.527)

0.8400 0.8550

1724.67 1727.38

Table 2(II) Comparison of total cost gained by two methods. (D, dL)

(2400, 70)

(3600, 85)

(3900, 100)

Model-I Model-II

1541.91 1559.02

1754.82 1755.99

2494.65 2452.06

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L. Wang et al. / Expert Systems with Applications 39 (2012) 4181–4189

Table 3 The results of sensitivity analysis. ~ d L

Model

e ¼(3300, 3450, 3600) D ⁄

⁄

(Q , r ) (50, 65, 100)

e ¼(2900, 3400, 4200) D

MðQ  ; r  Þ

⁄

⁄

Cr{}

(Q , r )

e ¼(2400, 3600, 3900) D Cr{}

(Q⁄, r⁄)

MðQ  ; r  Þ

Cr{  }

2042.52 2045.60

0.811 0.818

TC = 1880 (329, 91) (352, 92)

2042.52 2044.54

0.858 0.868

MðQ  ; r Þ

I II

TC = 1815 (338, 92) (339, 92)

2042.52 2042.57

0.812 0.814

TC = 1900 (329, 91) (356, 92)

I II

TC = 1725 (333, 73) (338, 73)

1703.77 1703.98

0.794 0.800

TC = 1810 (333, 73) (355, 73)

1703.77 1707.27

0.807 0.816

TC = 1780 (333, 73) (347, 73)

1703.77 1705.24

0.777 0.793

I II

TC = 1750 (333, 88) (338, 88)

1728.55 1728.74

0.782 0.796

TC = 1835 (333, 88) (352, 88)

1728.55 1731.41

0.807 0.815

TC = 1810 (333, 88) (348, 88)

1728.55 1730.45

0.815 0.828

(55, 65, 75)

(40, 80, 90)

are as follows. (1) We develop two new and practical continuous review inventory models with a mixture of backorders and lost sales under a fuzzy average annual demand and fuzzy demand during lead-time under different decision situations. Two models designed to give suggestions under different situations are compared and corresponding advices about cash flow of inventory management are given. To our best knowledge, this paper presents a DCP model about continuous review inventory for the first time, and this will expand the scope of inventory management research. (2) The proposed hybrid intelligent algorithms using fuzzy simulation and DE/SDE/MDE algorithm can evaluate the credibility in a highly robust and reliable way without considering the shapes of fuzzy membership functions and extend the application field of DE algorithm. (3) At last, we give some advices about under what circumstance the first decision method should be used, and under what circumstance decision makers should adopt the second one. It is very useful to help inventory managers make the right decision under uncertain environment. Future research may investigate the following aspects: (1) The use of DE algorithm to solve more difficult inventory problem such as managing the continuous review inventory in mixed fuzzy and stochastic environment. DE algorithms may provide better solutions to these problems that result in a cost savings when compared to other approaches. (2) We have indicated that DCP model may be improved. A multi-objective DCP model, the first object is maximize the credibility of an event such that the total cost in the planning periods does not exceed a certain budget level and the second one is minimizing the total cost, can be studied.

h i

~  rÞþ ðs þ pð1  bÞÞ DþDþ4D ; ¼ Q23 A þ Mðd 6 h

i rdL @ 2 MðQ ;rÞ sþpð1bÞ DþDþ4D ¼ hð1  bÞ þ ; 2 2 6 Q @r ðdL dL Þ





2 4rd þ3d2 2 2 r DþDþ4D @ MðQ ;rÞ  ¼  1 þ 2ðd dL Þ2 L ½sþpð1bÞ ; 2 6 @Q @r Q L L 

2

2   ~L  rÞþ ¼ dL þdL þ4dL þ dL r rdL  r 1  1 rdL : Mðd 2 dL dL 6 dL dL 3 @ 2 MðQ ;rÞ @Q 2

Hence,

!2 @ 2 MðQ; rÞ @ 2 MðQ ; rÞ @ 2 MðQ ; rÞ  @r 2 @Q@r @Q 2 ( " "  2  dL þ dL þ 4dL dL  r r  dL 2 ¼ A þ þ 3 6 dL  dL Q3 !) #  2 !# 1 r  dL D þ D þ 4D ðs þ pð1  bÞÞ r 1 2 dL  dL 6 ( " !#) r  dL s þ pð1  bÞ D þ D þ 4D hð1  bÞ þ  Q 6 ðdL  dL Þ2 ( ! ! 2 2 2 r  4rdL þ 3dL ½s þ pð1  bÞ   1þ Q2 2ðdL  dL Þ2 !)2 D þ D þ 4D  6 (2) Situation 2 The sufficient condition is where



2

@ 2 MðQ ;rÞ @Q @r

>0

h i

~  rÞþ ðs þ pð1  bÞÞ DþDþ4D ; ¼ Q23 A þ Mðd 6 h

i  rÞ @ 2 MðQ ;rÞ sþpð1bÞ DþDþ4D L ¼ ðdðdd ; 2 hð1  bÞ þ 6 Q @r 2 Þ L L



2 2  DþDþ4D @ 2 MðQ ;rÞ ½sþpð1bÞ ; ¼  2ððddL rÞ 2 6 @Q @r Q2 L dL Þ

2 ~L  rÞþ ¼ dL r dL r : Mðd  d 6 d @ 2 MðQ ;rÞ @Q 2

Acknowledgement The authors are very grateful to anonymous referees and editors for their constructive comments and suggestions. This research is partially supported by National Natural Science Foundation of China (No. 70801030), Key Program of National Natural Science Foundation of China (No. 71131004), Humanities and Social Sciences Foundation of Chinese Ministry of Education (No. 11YJC630275), Educational Commission of Hubei Province of China (No. D20112201) and Fundamental Research Funds for the Central Universities (No. 2010MS133).

Appendix A ~  rÞþ can The sufficient condition to attain the minimum of Mðd be deduced as follows (1) Situation 1 The sufficient condition is where

@ 2 MðQ ;rÞ @ 2 MðQ ;rÞ  @r 2 @Q 2

@ 2 MðQ ;rÞ @ 2 MðQ ;rÞ @r 2 @Q 2





2

@ 2 MðQ ;rÞ @Q @r

>0

L

L

Hence,

!2 @ 2 MðQ ; rÞ @ 2 MðQ; rÞ @ 2 MðQ; rÞ  @r2 @Q @r @Q 2 2 3 ! ! ! L  r L  r 2 d d 2 4 D þ D þ 4D 5 ¼ 3 Aþ L  dL ðs þ pð1  bÞÞ 6 6 d Q " !# L  rÞ ðd s þ pð1  bÞ D þ D þ 4D   hð1  bÞ þ   d Þ2 Q 6 ðd L L ( !)2 ! L  rÞ2 ½s þ pð1  bÞ2 ðd D þ D þ 4D   L  dL Þ2 6 Q2 2ðd

L. Wang et al. / Expert Systems with Applications 39 (2012) 4181–4189

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