- Email: [email protected]
Fuzzy multi-product constraint newsboy problem
Applied Mathematics and Computation 180 (2006) 7–15 www.elsevier.com/locate/amc
Fuzzy multi-product constraint newsboy problem Zhen Shao a, Xiaoyu Ji a
b,*
School of Management, Graduate School of the Chinese Academy of Sciences, Beijing 100080, China b Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China
Abstract In this paper, we consider the multi-product newsboy problem with fuzzy demands under budget constraint. Since the demands of products are often fuzzy in real life, the profit of the newsboy is fuzzy too. We develop three types of models under different criteria: EVM model, DCP model and CCP model. In these models, the objective functions are to maximize the expected profit of newsboy, the chance of achieving a target profit and the profit which satisfies some chance constraints with at least some given confidence level, respectively. Furthermore, the hybrid intelligent algorithm based on genetic algorithm and fuzzy simulation is designed for these models. And some illustrating examples are given in order to show the application of these proposed models and algorithm. Ó 2006 Published by Elsevier Inc. Keywords: Newsboy problem; Single-period inventory; Genetic algorithm; Fuzzy programming
1. Introduction The newsboy problem, also known as a single-period inventory problem, is to find the newspaper order quantity so that maximizes the expected profit of newsboy with random demand. In real life, many products have a limited selling period, so newsboy problem provides very useful framework to manage capacity or make decision of advanced booking of ordering in a number of practical contexts, such as in fashion, sporting and service industries. When a newsboy orders multiple products under a budget or storage constraint, we call this kind of newsboy problem as constraint multi-product newsboy problem. Hadley and Whitin [5] are credited for their initial work on it. They developed a numerical approach based on dynamic programming to solve such a problem. From then on, this problem has been extensively studied in many literatures. Lau and Lau [8,9] studied the multi-product newsboy problem with different constraint. Erlebacher [2] developed optimal and heuristic solutions for the capacitated newsboy problem. In [20], Vairaktarakis addressed a robust newsboy model with interval and discrete demand. Most of the extensions of newsboy problem have been made in the probabilistic framework, in which the uncertainty of demand is characterized by the random demands. In these literatures, the objective function is *
Corresponding author. E-mail address: [email protected] (X. Ji).
0096-3003/$ - see front matter Ó 2006 Published by Elsevier Inc. doi:10.1016/j.amc.2005.11.123
8
Z. Shao, X. Ji / Applied Mathematics and Computation 180 (2006) 7–15
used to maximizing the expected profit or probability measure of achieving a target profit. However, randomness is not the unique uncertainty in real world, sometimes the probability distributions of the demands for products are difficult to acquire due to lack of information and historical data. In this case, the demands are approximately specified based on the experience and managerial subjective judgement, and fuzzy theory offers a powerful tool to deal with this case. Fuzzy theory was proposed by Zadeh [21], and developed by many researchers such as Dubois and Prade [1]. Some papers have dealt newsboy problem using fuzzy theory. Ishii and Konno [7] studied the newsboy problem with the fuzzy shortage cost, in which an optimal order quantity was obtained by fuzzy maximal order. Petrovic et al. [19] gave a newsboy model in which the demand was represented by a group of fuzzy sets and inventory cost was given by a fuzzy number, and in order to obtain an optimal order quantity, the defuzzification method was used. In order to minimize the fuzzy cost, Kao and Hsu [4] compared the area of fuzzy numbers to obtain the optimal order quantity. Li et al. [10] studied the fuzzy newsboy problem in different cases to maximize the profit through ordering fuzzy numbers with respect to their total integral values. Almost all of the papers which studied the fuzzy newsboy problem have been used the possibility measure or necessary measure and the objective function is to maximize the profit or minimize the total cost. As we all know, a fuzzy event may fail even though its possibility achieves 1, and hold even though its necessity is 0. In this paper, we adopt the concept of credibility measure in credibility theory proposed by Liu [11,12], in which the fuzzy event must hold if its credibility is 1 and fail if its credibility is 0. Based on credibility measure, we present some new models according to different decision criteria: the expected value model, chance-constrained programming and dependent-chance programming. Furthermore, a genetic algorithm integrating fuzzy simulation is designed for solving the proposed fuzzy programming models. The remainder of the paper is organized as follows. In the next section, we present some basic knowledge of credibility theory. Section 3 describes constraint newsboy problem with fuzzy demand. Section 4 proposes three types of fuzzy programming models according to different decision criteria. Nextly, a hybrid intelligent algorithm integrating simulation technology and genetic algorithm is designed for solving these fuzzy programming models. Lastly, some numerical examples are provided for illustrating the effectiveness of the algorithm. 2. Preliminaries In this paper, we adopt the concepts of credibility theory [11,12] including possibility, necessity, credibility of fuzzy event and the expected value of a fuzzy variable. Definition 1. Let n be a fuzzy variable with the membership function l(x). Then the possibility, necessity, credibility measure of the fuzzy event {n P r} can be represented, respectively, by Posfn P rg ¼ sup lðuÞ; uPr
Necfn P rg ¼ 1 sup lðuÞ; u
1 Crfn P rg ¼ ½Posfn P rg þ Necfn P rg. 2 Example 1. A trapezoidal fuzzy variable n determined by quadruplet (a, b, c, d) of crisp numbers with a < b < c < d, whose membership function is given by Fig. 1. According to Definition 1, {n P x}, respectively. 8 1; > > > < d x Posfn P xg ¼ ; > d c > > : 0;
it is easy to obtain the possibility, necessity and credibility of fuzzy event if x 6 c if c < x 6 d if x > d;
Z. Shao, X. Ji / Applied Mathematics and Computation 180 (2006) 7–15
9
Fig. 1. Membership function of trapezoidal fuzzy variable.
8 0; if x P b > > < bx Necfn P xg ¼ ; if a 6 x < b > > :b a 1; otherwise, 8 1; if x < a > > > > 2b a x > > ; if a 6 x < b > > > 2ðb aÞ > > < 1 Crfn P xg ¼ ; if b 6 x 6 c > 2 > > > > d x > > ; if c < x 6 d > > 2ðd cÞ > > : 0; otherwise. Based on the credibility measure, we adopt the following definition of expected value operator. Definition 2 (Liu and Liu [16]). Let n be a fuzzy variable. The expected value of n is defined as Z 1 Z 0 E½n ¼ Crfn P rgdr Crfn 6 rgdr; 0
ð1Þ
ð2Þ
1
provided that at least one of the two integrals is finite. Example 2. The expected value of a trapezoidal fuzzy variable n = (a, b, c, d) is 1 E½n ¼ ða þ b þ c þ dÞ. 4 Definition 3 (Liu and Liu [12]). Let n be a fuzzy variable. Then the optimistic function of n is defined as nsup ðaÞ ¼ supfrjCrfn P rg P ag;
a 2 ð0; 1.
ð3Þ
3. The fuzzy constraint multi-product newsboy problem In order to introduce the constraint multi-product newsboy problem clearly, we give the following list of symbols that will be used throughout this paper: n ci ri si xi ni w f(xi, ni) F(x, n)
number of products, cost per unit for the ith product, retail price per unit (ri P ci) for the ith product, salvage value per unit (si 6 ci) for the ith product, order quantity for the ith product, the decision variables, demand for the ith product, fuzzy variables, maximum available budget for all product, the profit function for the order quantity xi and demand ni, the total profit function for all products, in which x = (x1, x2, . . . , xn), n = (n1, n2, . . . , nn).
10
Z. Shao, X. Ji / Applied Mathematics and Computation 180 (2006) 7–15
The profit function for ith product is given by if xi 6 ni ðri ci Þxi ; f ðxi ; ni Þ ¼ ðri si Þni ðci si Þxi ; if xi P ni . Then the total profit of the newsboy is as follows: n X f ðxi ; ni Þ; F ðx; nÞ ¼ i
subject to Xn i
ci xi 6 w.
In real life, the decision-maker need to make decision in different situation. Based on the credibility measure, we give the following definitions. Definition 4. If E[F(x*, n)] P E[F(x, n)] for all feasible order quantity x, then we call E[F(x*, n)] as the maximum expected profit (MEP) of the newsboy. In some cases, the decision-maker sets a confidence level a as an appropriate safety margin, and hopes to maximize a critical value F with CrfF ðx; nÞ P F g P a. For this case, we propose the concept of a-maximum profit of the newsboy (a-MP) as follows. Definition 5. If the following condition is satisfied, maxfF jCrfF ðx ; nÞ P F g P ag P maxfF jCrfF ðx; nÞ P F g P ag; then we call F as a-maximum profit (a-MP) of the newsboy, in which a is a predetermined confidence level. Sometimes, the decision-maker considers the profit with greatest credibility to be greater than or equal to the given profit. Suppose the decision maker provides a profit infremum F0 and hopes that the credibility not less than F0 will be maximized as possible. For this case, we propose the concept of the most maximum profit as follows. Definition 6. For a given predetermined profit F0, if CrfF ðx ; nÞ P F 0 g P CrfF ðx; nÞ P F 0 g; then we call F(x*, n) as the most maximum profit (MMP) of the newsboy. 4. New models According to decision criteria of decision maker, three types of models are formulated as follows. Firstly, expected value model is employed to find the maximum expected profit of newsboy. The model can be formulated as follows: 8 max E½F ðx; nÞ > > > > > > < subject to: n P > w ci xi P 0 > > > i > > : xi P 0; in which E denotes the expected value operator of fuzzy event. The second model is called chance-constrained programming (CCP). Chance-constrained programming offers us a powerful means for modelling fuzzy decision systems [13–15]. When the decision-maker is interested in the profit which satisfies some chance constraints with at least some given confidence level, the following model is suitable.
Z. Shao, X. Ji / Applied Mathematics and Computation 180 (2006) 7–15
11
8 max F > > > > > subject to: > > < CrfF ðx; nÞ P F g P a n > > P > ci xi 6 w > > > i > : xi P 0; where a is predetermined confidence levels provided as appropriate margin by the decision-maker. The third model is called dependent chance programming. Very often a practical situation is required not to less than a predetermined profit which decision-maker can accept in practice. And the decision-maker wants to maximize the credibility not less than the profit. This is just the essential idea of dependent chance programming [17,18], we give the model as follows: 8 max CrfF ðx; nÞ P F 0 g > > > > < subject to: n P > ci xi 6 w > > i > : xi P 0; where F0 is the predetermined profit which decision-maker can accept. 5. Hybrid intelligent algorithm In order to solve the newsboy problem under different criteria, we develop a hybrid intelligent algorithm integrating fuzzy simulation and genetic algorithm. 5.1. Fuzzy simulation In this subsection, we give the detailed procedure to estimate the uncertain functions in the fuzzy models for newsboy problem by simulation technique. Now, we note n as n = (n1, n2, . . . , nn), where n is the number of products. And we denote that l is the membership function of n and li are the membership functions of ni, i = 1, 2, . . . , n, respectively. Randomly generate uik from the e-level sets of fuzzy variables ni, i = 1, 2, . . . , n, k = 1, 2, . . . , N respectively, where e is a sufficiently small positive number, N is a sufficiently large positive number. Set uk = (u1k, u2k, . . . , unk) and l(uk) = l1(u1k) ^ l2(u2k) ^ ^ lm(unk). At first, we simulate the uncertain function U 1 : CrfF ðx; nÞ P F 0 g. According to the concept of credibility measure (1), the credibility Cr{F(x, n) P F0} can be obtained approximately by the following formula 1 max lðuk Þ F ðx; uk Þ P F 0 þ min 1 lðuk Þ F ðx; uk Þ < F 0 . L¼ 16k6N 2 16k6N Therefore, we have the following procedure to simulate this type of uncertain function can be run as follows: Step 1. Let k = 1. Step 2. Randomly generate uik from the e-level sets of fuzzy variables ni, and set uk = (u1k, u2k, . . . , unk) and l(uk) = l1(u1k) ^ l2(u2k) ^ ^ lm(unk). Step 3. k k + 1. If k 6 N, go to Step 2, else, go to Step 4. Step 4. Return L.
12
Z. Shao, X. Ji / Applied Mathematics and Computation 180 (2006) 7–15
The second type of uncertain function in our optimization problem is U 2 : x ! maxfF CrfF ðx; nÞ P F g P ag. Set 1 LðrÞ ¼ 2
max lðuk Þ F ðx; uk Þ P r þ max 1 lðuk Þ F ðx; uk Þ < r .
16k6N
16k6N
Then the process of fuzzy simulation can be performed as follows: Step 1. Let k = 1. Step 2. Randomly generate uik from the e-level sets of fuzzy variables ni, and set uk = (u1k, u2k, . . . , unk) and l(uk) = l1(u1k) ^ l2(u2k) ^ ^ lm(unk). Step 3. k k + 1. If k 6 N, go to Step 2, else, go to Step 4. Step 4. Return the maximal r satisfying L(r) P a. At last, we estimate the following function: U 3 : x ! E½F ðx; nÞ. According the definition of expected value of fuzzy variable, we have Z 1 Z 0 E½F ðx; nÞ ¼ CrfF ðx; nÞ P rgdr CrfF ðx; nÞ 6 rgdr. 0
1
For any number r P 0, Cr{F(x, uk) P r} can be estimated by 1 max fl jF ðx; uk Þ P rg þ 1 max flk jF ðx; uk Þ < rg CrfF ðx; uk Þ P rg ¼ k¼1;2;...;N 2 k¼1;2;...;N k and for any number r < 0, the credibility Cr{F(x, uk) 6 r} can be estimated by 1 max fl jF ðx; uk Þ 6 rg þ 1 max flk jF ðx; uk Þ > rg CrfF ðx; uk Þ 6 rg ¼ k¼1;2;...;N 2 k¼1;2;...;N k provided that N is sufficiently large. We can write the procedure for estimating function U3 as follows: Step Step Step Step Step Step Step Step
1. 2. 3. 4. 5. 6. 7. 8.
Set E = 0. Randomly generate uik from the e-level sets of fuzzy variables ni, and set uk = (u1k, u2k, . . . , unk). Set a = F(x, u1) ^ F(x, u2) ^ ^ F(x, uN), b = F(x, u1) _ F(x, u2) _ _ F(x, uN). Randomly generate r from [a, b]. If r P 0, then E E + Cr{F(x, n) P r}. If r < 0, then E E Cr{F(x, n) 6 r}. Repeat the fourth to sixth steps for N times. E[F(x, n)] = a _ 0 + b ^ 0 + E Æ (b a)/N.
5.2. Genetic algorithm Ever since Holland [6] proposed the genetic algorithm to tackle complex problems, genetic algorithm has been looked as one of the most efficient stochastic solution search methods for solving various network problem [3].
Z. Shao, X. Ji / Applied Mathematics and Computation 180 (2006) 7–15
13
In order to solve our newsboy problem models, genetic algorithm is employed to find the best solutions under different criteria. The representation structure, initialization, crossover and mutation operations are as follows. 5.2.1. Representation and initialization Let x = (x1, x2, . . . , xn) be the chromosomes in a population, in which xi represents the order quantities for ith product and is generated randomly in its feasible domain. In a population, there are pop-size chromosomes, where pop-size is the number of total members in the population. 5.2.2. Fitness value for each chromosome For each chromosome x, the values of objective functions are calculated by fuzzy simulation. The rankbased evaluation function is defined as Evalðxj Þ ¼ að1 aÞ
j1
; j ¼ 1; 2; . . . ; pop size;
where the chromosomes are assumed to have been rearranged from good to bad according to their objective values and a 2 (0, 1) is a parameter in the genetic system. 5.2.3. Selection process for mutation Before selecting the chromosomes for the new population, we calculate the cumulative probability pi for each chromosome xi as follows: p0 ¼ 0;
pi ¼
i X
Evalðxj Þ;
i ¼ 1; 2; . . . ; pop size.
j¼1
Then spin the roulette wheel pop_size times to select chromosomes for the new population. The select process is as the following way: (a) generate a random number r in (0,ppop-size). (b) If pi1 < r 6 pi, then select the ith chromosome xi, i = 1, 2, . . . , pop-size. 5.2.4. Mutation operation Set the mutation probability of genetic system be Pm. We determine the parents for mutation operation firstly. Generating randomly a real number r from the interval [0, 1], if r < Pm, then the chromosome xi is selected as a parent, repeat this for pop-size times. Let M be an appropriate large positive number, we choose a mutation direction d randomly, then the child of the parent x is xchild = x + M * d. 5.2.5. Crossover operation Set the crossover probability of genetic system be Pc. In a similar procedure as mutation to determine the parents for crossover operation, we generate randomly a real number r from the interval [0, 1], if r < Pc, then the chromosome xi is selected as a parent, repeat this for popsize times. Then, we divided these parents into pairs such as ðx01 ; x02 Þ; ðx03 ; x04 Þ; . . ., and crossover on each pair. We illustrate the crossover operator on the pair ðx01 ; x02 Þ as follows. Generate a random number c from the interval (0, 1), we can produce two children xchild1, xchild2, xchild1 ¼ cx01 þ ð1 cÞx02 ; xchild2 ¼ ð1 cÞx01 þ cx02 . 5.2.6. Genetic algorithm procedure Step 1. Initialize chromosomes randomly and check their feasibility. Step 2. Update the chromosomes by crossover and mutation operations and check their feasibility. Step 3. Calculate the objective functions values for all chromosomes according to above fuzzy simulation. Step 4. Compute the fitness of each chromosome according to the objective values. Step 5. Select the chromosomes by spinning the roulette wheel. Step 6. Repeat the steps 2–5 until a terminal condition is satisfied. Step 7. Return the best chromosome.
14
Z. Shao, X. Ji / Applied Mathematics and Computation 180 (2006) 7–15
Table 1 The numerical data and results for example Item
ri ($/unit)
si ($/unit)
ci ($/unit)
Fuzzy demand
xi (unit)
x0:8 (unit) i
xM i (unit)
1 2 3 4 5 6
7 12 30 30 40 45
1 2 4 4 2 5
4 8 20 10 13 15
(180, 190, 210, 220) (210, 220, 230, 240) (100, 110, 115, 125) (80, 90, 110, 120) (60, 70, 80, 90) (20, 25, 35, 40)
78.44 58.20 30.08 82.75 70.92 26.29
80.22 60.00 31.71 80.45 72.80 20.78
79.00 62.43 29.98 83.45 69.78 22.47
6. Numerical examples We consider a constraint newsboy problem with six products and the budget constraint is $ 3500. The demands of multi-product are trapezoidal fuzzy variables and shown in the following Table 1. We run the hybrid intelligent algorithm for solving these models, and the order quantities of these six products are shown in Table 1. We can find the EMP is 5475. When F0 = 5800, the MMP is 5583. When a = 0.9, the a-MP is 5560. 7. Conclusion In this paper, the constraint newsboy problem with fuzzy demands has been presented. Three types of models and hybrid intelligent algorithm are developed, this work is very helpful and useful for decision-makers in real life. The contributions of this paper are as the following aspects. Firstly, three types of fuzzy programming model were presented for the first time: expected profit model, the most profit model and a-profit model. Then, to solve these fuzzy models, a hybrid intelligent algorithm based on genetic algorithm and fuzzy simulation was also developed. Finally, some numerical examples were given to show the performance of the hybrid intelligent algorithm. Acknowledgments This work was supported by National Natural Science Foundation of China (No. 60425309), and Specialized Research Fund for the Doctoral Program of Higher Education (No. 20020003009). References [1] D. Dubois, H. Prade, Possibility Theory, Plenum, New York, 1988. [2] S.J. Erlebacher, Optimal and heuristic solutions for the multi-item newsvendor problem with a single capacity constraint, Production and Operations Management 9 (3) (2000) 303–318. [3] M. Gen, R. Cheng, Genetic Algorithms and Engineer Optimization, John Wiley and Sons, Inc., New York, 2000. [4] C. Kao, W. Hsu, A single-period inventory model with fuzzy demand, Computers and Mathematics with Applications 43 (2002) 841– 848. [5] G. Hadley, T.M. Whitin, Analysis of Inventory Systems, Prentice-Hall, Englewood Cliffs, NJ, 1963. [6] J. Holland, Adaptation in Natural and Artificial System, University of Michigan Press, Ann Arbor, MI, 1975. [7] H. Ishii, T. Konno, A stochastic inventory problem with fuzzy shortage cost, European Journal of Operational Research 106 (1998) 90–94. [8] H.S. Lau, A.H.L. Lau, The multi-product multi-constraint newsboy problem: applications formulation and solution, Journal of Operations Management 13 (1995) 153–162. [9] H.S. Lau, A.H.L. Lau, The newsstand problem: a capacitated multiple-product single-period inventory problem, European Journal of Operational Research 94 (1996) 29–42. [10] L. Li, S.N. Kabadi, K.P.K. Nair, Fuzzy models for single-period inventory problem, Fuzzy Sets and Systems 132 (3) (2002) 273–289. [11] B. Liu, Theory and Practice of Uncertain Programming, Physica-Verlag, Heidelberg, 2002. [12] B. Liu, Uncertainty Theory: An Introduction to its Axiomatic Foundations, Springer-Verlag, Berlin, 2004. [13] B. Liu, K. Iwamura, Chance constrained programming with fuzzy parameters, Fuzzy Sets and Systems 94 (1998) 227–237. [14] B. Liu, K. Iwamura, A note on chance constrained programming with fuzzy coefficients, Fuzzy Sets and Systems 100 (1998) 229–233. [15] B. Liu, Minimax chance constrained programming models for fuzzy decision systems, Information Sciences 112 (1998) 25–38.
Z. Shao, X. Ji / Applied Mathematics and Computation 180 (2006) 7–15
15
[16] B. Liu, Y.K. Liu, Expected value of fuzzy variable and fuzzy expected value model, IEEE Transactions on Fuzzy Systems 10 (2002) 445–450. [17] B. Liu, Dependent-chance programming with fuzzy decisions, IEEE Transactions on Fuzzy Systems 7 (3) (1999) 354–360. [18] B. Liu, Dependent-chance programming in fuzzy environments, Fuzzy Sets and Systems 109 (1) (2000) 97–106. [19] D. Petrovic, R. Petrovic, M. Vujosevic, Fuzzy models for the newsboy problem, International Journal of Production Economics 45 (1996) 435–441. [20] G.L. Vairaktarakis, Robust multi-item newsboy models with a budget constraint, International Journal of Production Economics 66 (2000) 213–226. [21] L.A. Zadeh, Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems 1 (1978) 3–28.
Fuzzy multi-product constraint newsboy problem Zhen Shao a, Xiaoyu Ji a
b,*
School of Management, Graduate School of the Chinese Academy of Sciences, Beijing 100080, China b Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China
Abstract In this paper, we consider the multi-product newsboy problem with fuzzy demands under budget constraint. Since the demands of products are often fuzzy in real life, the profit of the newsboy is fuzzy too. We develop three types of models under different criteria: EVM model, DCP model and CCP model. In these models, the objective functions are to maximize the expected profit of newsboy, the chance of achieving a target profit and the profit which satisfies some chance constraints with at least some given confidence level, respectively. Furthermore, the hybrid intelligent algorithm based on genetic algorithm and fuzzy simulation is designed for these models. And some illustrating examples are given in order to show the application of these proposed models and algorithm. Ó 2006 Published by Elsevier Inc. Keywords: Newsboy problem; Single-period inventory; Genetic algorithm; Fuzzy programming
1. Introduction The newsboy problem, also known as a single-period inventory problem, is to find the newspaper order quantity so that maximizes the expected profit of newsboy with random demand. In real life, many products have a limited selling period, so newsboy problem provides very useful framework to manage capacity or make decision of advanced booking of ordering in a number of practical contexts, such as in fashion, sporting and service industries. When a newsboy orders multiple products under a budget or storage constraint, we call this kind of newsboy problem as constraint multi-product newsboy problem. Hadley and Whitin [5] are credited for their initial work on it. They developed a numerical approach based on dynamic programming to solve such a problem. From then on, this problem has been extensively studied in many literatures. Lau and Lau [8,9] studied the multi-product newsboy problem with different constraint. Erlebacher [2] developed optimal and heuristic solutions for the capacitated newsboy problem. In [20], Vairaktarakis addressed a robust newsboy model with interval and discrete demand. Most of the extensions of newsboy problem have been made in the probabilistic framework, in which the uncertainty of demand is characterized by the random demands. In these literatures, the objective function is *
Corresponding author. E-mail address: [email protected] (X. Ji).
0096-3003/$ - see front matter Ó 2006 Published by Elsevier Inc. doi:10.1016/j.amc.2005.11.123
8
Z. Shao, X. Ji / Applied Mathematics and Computation 180 (2006) 7–15
used to maximizing the expected profit or probability measure of achieving a target profit. However, randomness is not the unique uncertainty in real world, sometimes the probability distributions of the demands for products are difficult to acquire due to lack of information and historical data. In this case, the demands are approximately specified based on the experience and managerial subjective judgement, and fuzzy theory offers a powerful tool to deal with this case. Fuzzy theory was proposed by Zadeh [21], and developed by many researchers such as Dubois and Prade [1]. Some papers have dealt newsboy problem using fuzzy theory. Ishii and Konno [7] studied the newsboy problem with the fuzzy shortage cost, in which an optimal order quantity was obtained by fuzzy maximal order. Petrovic et al. [19] gave a newsboy model in which the demand was represented by a group of fuzzy sets and inventory cost was given by a fuzzy number, and in order to obtain an optimal order quantity, the defuzzification method was used. In order to minimize the fuzzy cost, Kao and Hsu [4] compared the area of fuzzy numbers to obtain the optimal order quantity. Li et al. [10] studied the fuzzy newsboy problem in different cases to maximize the profit through ordering fuzzy numbers with respect to their total integral values. Almost all of the papers which studied the fuzzy newsboy problem have been used the possibility measure or necessary measure and the objective function is to maximize the profit or minimize the total cost. As we all know, a fuzzy event may fail even though its possibility achieves 1, and hold even though its necessity is 0. In this paper, we adopt the concept of credibility measure in credibility theory proposed by Liu [11,12], in which the fuzzy event must hold if its credibility is 1 and fail if its credibility is 0. Based on credibility measure, we present some new models according to different decision criteria: the expected value model, chance-constrained programming and dependent-chance programming. Furthermore, a genetic algorithm integrating fuzzy simulation is designed for solving the proposed fuzzy programming models. The remainder of the paper is organized as follows. In the next section, we present some basic knowledge of credibility theory. Section 3 describes constraint newsboy problem with fuzzy demand. Section 4 proposes three types of fuzzy programming models according to different decision criteria. Nextly, a hybrid intelligent algorithm integrating simulation technology and genetic algorithm is designed for solving these fuzzy programming models. Lastly, some numerical examples are provided for illustrating the effectiveness of the algorithm. 2. Preliminaries In this paper, we adopt the concepts of credibility theory [11,12] including possibility, necessity, credibility of fuzzy event and the expected value of a fuzzy variable. Definition 1. Let n be a fuzzy variable with the membership function l(x). Then the possibility, necessity, credibility measure of the fuzzy event {n P r} can be represented, respectively, by Posfn P rg ¼ sup lðuÞ; uPr
Necfn P rg ¼ 1 sup lðuÞ; u
1 Crfn P rg ¼ ½Posfn P rg þ Necfn P rg. 2 Example 1. A trapezoidal fuzzy variable n determined by quadruplet (a, b, c, d) of crisp numbers with a < b < c < d, whose membership function is given by Fig. 1. According to Definition 1, {n P x}, respectively. 8 1; > > > < d x Posfn P xg ¼ ; > d c > > : 0;
it is easy to obtain the possibility, necessity and credibility of fuzzy event if x 6 c if c < x 6 d if x > d;
Z. Shao, X. Ji / Applied Mathematics and Computation 180 (2006) 7–15
9
Fig. 1. Membership function of trapezoidal fuzzy variable.
8 0; if x P b > > < bx Necfn P xg ¼ ; if a 6 x < b > > :b a 1; otherwise, 8 1; if x < a > > > > 2b a x > > ; if a 6 x < b > > > 2ðb aÞ > > < 1 Crfn P xg ¼ ; if b 6 x 6 c > 2 > > > > d x > > ; if c < x 6 d > > 2ðd cÞ > > : 0; otherwise. Based on the credibility measure, we adopt the following definition of expected value operator. Definition 2 (Liu and Liu [16]). Let n be a fuzzy variable. The expected value of n is defined as Z 1 Z 0 E½n ¼ Crfn P rgdr Crfn 6 rgdr; 0
ð1Þ
ð2Þ
1
provided that at least one of the two integrals is finite. Example 2. The expected value of a trapezoidal fuzzy variable n = (a, b, c, d) is 1 E½n ¼ ða þ b þ c þ dÞ. 4 Definition 3 (Liu and Liu [12]). Let n be a fuzzy variable. Then the optimistic function of n is defined as nsup ðaÞ ¼ supfrjCrfn P rg P ag;
a 2 ð0; 1.
ð3Þ
3. The fuzzy constraint multi-product newsboy problem In order to introduce the constraint multi-product newsboy problem clearly, we give the following list of symbols that will be used throughout this paper: n ci ri si xi ni w f(xi, ni) F(x, n)
number of products, cost per unit for the ith product, retail price per unit (ri P ci) for the ith product, salvage value per unit (si 6 ci) for the ith product, order quantity for the ith product, the decision variables, demand for the ith product, fuzzy variables, maximum available budget for all product, the profit function for the order quantity xi and demand ni, the total profit function for all products, in which x = (x1, x2, . . . , xn), n = (n1, n2, . . . , nn).
10
Z. Shao, X. Ji / Applied Mathematics and Computation 180 (2006) 7–15
The profit function for ith product is given by if xi 6 ni ðri ci Þxi ; f ðxi ; ni Þ ¼ ðri si Þni ðci si Þxi ; if xi P ni . Then the total profit of the newsboy is as follows: n X f ðxi ; ni Þ; F ðx; nÞ ¼ i
subject to Xn i
ci xi 6 w.
In real life, the decision-maker need to make decision in different situation. Based on the credibility measure, we give the following definitions. Definition 4. If E[F(x*, n)] P E[F(x, n)] for all feasible order quantity x, then we call E[F(x*, n)] as the maximum expected profit (MEP) of the newsboy. In some cases, the decision-maker sets a confidence level a as an appropriate safety margin, and hopes to maximize a critical value F with CrfF ðx; nÞ P F g P a. For this case, we propose the concept of a-maximum profit of the newsboy (a-MP) as follows. Definition 5. If the following condition is satisfied, maxfF jCrfF ðx ; nÞ P F g P ag P maxfF jCrfF ðx; nÞ P F g P ag; then we call F as a-maximum profit (a-MP) of the newsboy, in which a is a predetermined confidence level. Sometimes, the decision-maker considers the profit with greatest credibility to be greater than or equal to the given profit. Suppose the decision maker provides a profit infremum F0 and hopes that the credibility not less than F0 will be maximized as possible. For this case, we propose the concept of the most maximum profit as follows. Definition 6. For a given predetermined profit F0, if CrfF ðx ; nÞ P F 0 g P CrfF ðx; nÞ P F 0 g; then we call F(x*, n) as the most maximum profit (MMP) of the newsboy. 4. New models According to decision criteria of decision maker, three types of models are formulated as follows. Firstly, expected value model is employed to find the maximum expected profit of newsboy. The model can be formulated as follows: 8 max E½F ðx; nÞ > > > > > > < subject to: n P > w ci xi P 0 > > > i > > : xi P 0; in which E denotes the expected value operator of fuzzy event. The second model is called chance-constrained programming (CCP). Chance-constrained programming offers us a powerful means for modelling fuzzy decision systems [13–15]. When the decision-maker is interested in the profit which satisfies some chance constraints with at least some given confidence level, the following model is suitable.
Z. Shao, X. Ji / Applied Mathematics and Computation 180 (2006) 7–15
11
8 max F > > > > > subject to: > > < CrfF ðx; nÞ P F g P a n > > P > ci xi 6 w > > > i > : xi P 0; where a is predetermined confidence levels provided as appropriate margin by the decision-maker. The third model is called dependent chance programming. Very often a practical situation is required not to less than a predetermined profit which decision-maker can accept in practice. And the decision-maker wants to maximize the credibility not less than the profit. This is just the essential idea of dependent chance programming [17,18], we give the model as follows: 8 max CrfF ðx; nÞ P F 0 g > > > > < subject to: n P > ci xi 6 w > > i > : xi P 0; where F0 is the predetermined profit which decision-maker can accept. 5. Hybrid intelligent algorithm In order to solve the newsboy problem under different criteria, we develop a hybrid intelligent algorithm integrating fuzzy simulation and genetic algorithm. 5.1. Fuzzy simulation In this subsection, we give the detailed procedure to estimate the uncertain functions in the fuzzy models for newsboy problem by simulation technique. Now, we note n as n = (n1, n2, . . . , nn), where n is the number of products. And we denote that l is the membership function of n and li are the membership functions of ni, i = 1, 2, . . . , n, respectively. Randomly generate uik from the e-level sets of fuzzy variables ni, i = 1, 2, . . . , n, k = 1, 2, . . . , N respectively, where e is a sufficiently small positive number, N is a sufficiently large positive number. Set uk = (u1k, u2k, . . . , unk) and l(uk) = l1(u1k) ^ l2(u2k) ^ ^ lm(unk). At first, we simulate the uncertain function U 1 : CrfF ðx; nÞ P F 0 g. According to the concept of credibility measure (1), the credibility Cr{F(x, n) P F0} can be obtained approximately by the following formula 1 max lðuk Þ F ðx; uk Þ P F 0 þ min 1 lðuk Þ F ðx; uk Þ < F 0 . L¼ 16k6N 2 16k6N Therefore, we have the following procedure to simulate this type of uncertain function can be run as follows: Step 1. Let k = 1. Step 2. Randomly generate uik from the e-level sets of fuzzy variables ni, and set uk = (u1k, u2k, . . . , unk) and l(uk) = l1(u1k) ^ l2(u2k) ^ ^ lm(unk). Step 3. k k + 1. If k 6 N, go to Step 2, else, go to Step 4. Step 4. Return L.
12
Z. Shao, X. Ji / Applied Mathematics and Computation 180 (2006) 7–15
The second type of uncertain function in our optimization problem is U 2 : x ! maxfF CrfF ðx; nÞ P F g P ag. Set 1 LðrÞ ¼ 2
max lðuk Þ F ðx; uk Þ P r þ max 1 lðuk Þ F ðx; uk Þ < r .
16k6N
16k6N
Then the process of fuzzy simulation can be performed as follows: Step 1. Let k = 1. Step 2. Randomly generate uik from the e-level sets of fuzzy variables ni, and set uk = (u1k, u2k, . . . , unk) and l(uk) = l1(u1k) ^ l2(u2k) ^ ^ lm(unk). Step 3. k k + 1. If k 6 N, go to Step 2, else, go to Step 4. Step 4. Return the maximal r satisfying L(r) P a. At last, we estimate the following function: U 3 : x ! E½F ðx; nÞ. According the definition of expected value of fuzzy variable, we have Z 1 Z 0 E½F ðx; nÞ ¼ CrfF ðx; nÞ P rgdr CrfF ðx; nÞ 6 rgdr. 0
1
For any number r P 0, Cr{F(x, uk) P r} can be estimated by 1 max fl jF ðx; uk Þ P rg þ 1 max flk jF ðx; uk Þ < rg CrfF ðx; uk Þ P rg ¼ k¼1;2;...;N 2 k¼1;2;...;N k and for any number r < 0, the credibility Cr{F(x, uk) 6 r} can be estimated by 1 max fl jF ðx; uk Þ 6 rg þ 1 max flk jF ðx; uk Þ > rg CrfF ðx; uk Þ 6 rg ¼ k¼1;2;...;N 2 k¼1;2;...;N k provided that N is sufficiently large. We can write the procedure for estimating function U3 as follows: Step Step Step Step Step Step Step Step
1. 2. 3. 4. 5. 6. 7. 8.
Set E = 0. Randomly generate uik from the e-level sets of fuzzy variables ni, and set uk = (u1k, u2k, . . . , unk). Set a = F(x, u1) ^ F(x, u2) ^ ^ F(x, uN), b = F(x, u1) _ F(x, u2) _ _ F(x, uN). Randomly generate r from [a, b]. If r P 0, then E E + Cr{F(x, n) P r}. If r < 0, then E E Cr{F(x, n) 6 r}. Repeat the fourth to sixth steps for N times. E[F(x, n)] = a _ 0 + b ^ 0 + E Æ (b a)/N.
5.2. Genetic algorithm Ever since Holland [6] proposed the genetic algorithm to tackle complex problems, genetic algorithm has been looked as one of the most efficient stochastic solution search methods for solving various network problem [3].
Z. Shao, X. Ji / Applied Mathematics and Computation 180 (2006) 7–15
13
In order to solve our newsboy problem models, genetic algorithm is employed to find the best solutions under different criteria. The representation structure, initialization, crossover and mutation operations are as follows. 5.2.1. Representation and initialization Let x = (x1, x2, . . . , xn) be the chromosomes in a population, in which xi represents the order quantities for ith product and is generated randomly in its feasible domain. In a population, there are pop-size chromosomes, where pop-size is the number of total members in the population. 5.2.2. Fitness value for each chromosome For each chromosome x, the values of objective functions are calculated by fuzzy simulation. The rankbased evaluation function is defined as Evalðxj Þ ¼ að1 aÞ
j1
; j ¼ 1; 2; . . . ; pop size;
where the chromosomes are assumed to have been rearranged from good to bad according to their objective values and a 2 (0, 1) is a parameter in the genetic system. 5.2.3. Selection process for mutation Before selecting the chromosomes for the new population, we calculate the cumulative probability pi for each chromosome xi as follows: p0 ¼ 0;
pi ¼
i X
Evalðxj Þ;
i ¼ 1; 2; . . . ; pop size.
j¼1
Then spin the roulette wheel pop_size times to select chromosomes for the new population. The select process is as the following way: (a) generate a random number r in (0,ppop-size). (b) If pi1 < r 6 pi, then select the ith chromosome xi, i = 1, 2, . . . , pop-size. 5.2.4. Mutation operation Set the mutation probability of genetic system be Pm. We determine the parents for mutation operation firstly. Generating randomly a real number r from the interval [0, 1], if r < Pm, then the chromosome xi is selected as a parent, repeat this for pop-size times. Let M be an appropriate large positive number, we choose a mutation direction d randomly, then the child of the parent x is xchild = x + M * d. 5.2.5. Crossover operation Set the crossover probability of genetic system be Pc. In a similar procedure as mutation to determine the parents for crossover operation, we generate randomly a real number r from the interval [0, 1], if r < Pc, then the chromosome xi is selected as a parent, repeat this for popsize times. Then, we divided these parents into pairs such as ðx01 ; x02 Þ; ðx03 ; x04 Þ; . . ., and crossover on each pair. We illustrate the crossover operator on the pair ðx01 ; x02 Þ as follows. Generate a random number c from the interval (0, 1), we can produce two children xchild1, xchild2, xchild1 ¼ cx01 þ ð1 cÞx02 ; xchild2 ¼ ð1 cÞx01 þ cx02 . 5.2.6. Genetic algorithm procedure Step 1. Initialize chromosomes randomly and check their feasibility. Step 2. Update the chromosomes by crossover and mutation operations and check their feasibility. Step 3. Calculate the objective functions values for all chromosomes according to above fuzzy simulation. Step 4. Compute the fitness of each chromosome according to the objective values. Step 5. Select the chromosomes by spinning the roulette wheel. Step 6. Repeat the steps 2–5 until a terminal condition is satisfied. Step 7. Return the best chromosome.
14
Z. Shao, X. Ji / Applied Mathematics and Computation 180 (2006) 7–15
Table 1 The numerical data and results for example Item
ri ($/unit)
si ($/unit)
ci ($/unit)
Fuzzy demand
xi (unit)
x0:8 (unit) i
xM i (unit)
1 2 3 4 5 6
7 12 30 30 40 45
1 2 4 4 2 5
4 8 20 10 13 15
(180, 190, 210, 220) (210, 220, 230, 240) (100, 110, 115, 125) (80, 90, 110, 120) (60, 70, 80, 90) (20, 25, 35, 40)
78.44 58.20 30.08 82.75 70.92 26.29
80.22 60.00 31.71 80.45 72.80 20.78
79.00 62.43 29.98 83.45 69.78 22.47
6. Numerical examples We consider a constraint newsboy problem with six products and the budget constraint is $ 3500. The demands of multi-product are trapezoidal fuzzy variables and shown in the following Table 1. We run the hybrid intelligent algorithm for solving these models, and the order quantities of these six products are shown in Table 1. We can find the EMP is 5475. When F0 = 5800, the MMP is 5583. When a = 0.9, the a-MP is 5560. 7. Conclusion In this paper, the constraint newsboy problem with fuzzy demands has been presented. Three types of models and hybrid intelligent algorithm are developed, this work is very helpful and useful for decision-makers in real life. The contributions of this paper are as the following aspects. Firstly, three types of fuzzy programming model were presented for the first time: expected profit model, the most profit model and a-profit model. Then, to solve these fuzzy models, a hybrid intelligent algorithm based on genetic algorithm and fuzzy simulation was also developed. Finally, some numerical examples were given to show the performance of the hybrid intelligent algorithm. Acknowledgments This work was supported by National Natural Science Foundation of China (No. 60425309), and Specialized Research Fund for the Doctoral Program of Higher Education (No. 20020003009). References [1] D. Dubois, H. Prade, Possibility Theory, Plenum, New York, 1988. [2] S.J. Erlebacher, Optimal and heuristic solutions for the multi-item newsvendor problem with a single capacity constraint, Production and Operations Management 9 (3) (2000) 303–318. [3] M. Gen, R. Cheng, Genetic Algorithms and Engineer Optimization, John Wiley and Sons, Inc., New York, 2000. [4] C. Kao, W. Hsu, A single-period inventory model with fuzzy demand, Computers and Mathematics with Applications 43 (2002) 841– 848. [5] G. Hadley, T.M. Whitin, Analysis of Inventory Systems, Prentice-Hall, Englewood Cliffs, NJ, 1963. [6] J. Holland, Adaptation in Natural and Artificial System, University of Michigan Press, Ann Arbor, MI, 1975. [7] H. Ishii, T. Konno, A stochastic inventory problem with fuzzy shortage cost, European Journal of Operational Research 106 (1998) 90–94. [8] H.S. Lau, A.H.L. Lau, The multi-product multi-constraint newsboy problem: applications formulation and solution, Journal of Operations Management 13 (1995) 153–162. [9] H.S. Lau, A.H.L. Lau, The newsstand problem: a capacitated multiple-product single-period inventory problem, European Journal of Operational Research 94 (1996) 29–42. [10] L. Li, S.N. Kabadi, K.P.K. Nair, Fuzzy models for single-period inventory problem, Fuzzy Sets and Systems 132 (3) (2002) 273–289. [11] B. Liu, Theory and Practice of Uncertain Programming, Physica-Verlag, Heidelberg, 2002. [12] B. Liu, Uncertainty Theory: An Introduction to its Axiomatic Foundations, Springer-Verlag, Berlin, 2004. [13] B. Liu, K. Iwamura, Chance constrained programming with fuzzy parameters, Fuzzy Sets and Systems 94 (1998) 227–237. [14] B. Liu, K. Iwamura, A note on chance constrained programming with fuzzy coefficients, Fuzzy Sets and Systems 100 (1998) 229–233. [15] B. Liu, Minimax chance constrained programming models for fuzzy decision systems, Information Sciences 112 (1998) 25–38.
Z. Shao, X. Ji / Applied Mathematics and Computation 180 (2006) 7–15
15
[16] B. Liu, Y.K. Liu, Expected value of fuzzy variable and fuzzy expected value model, IEEE Transactions on Fuzzy Systems 10 (2002) 445–450. [17] B. Liu, Dependent-chance programming with fuzzy decisions, IEEE Transactions on Fuzzy Systems 7 (3) (1999) 354–360. [18] B. Liu, Dependent-chance programming in fuzzy environments, Fuzzy Sets and Systems 109 (1) (2000) 97–106. [19] D. Petrovic, R. Petrovic, M. Vujosevic, Fuzzy models for the newsboy problem, International Journal of Production Economics 45 (1996) 435–441. [20] G.L. Vairaktarakis, Robust multi-item newsboy models with a budget constraint, International Journal of Production Economics 66 (2000) 213–226. [21] L.A. Zadeh, Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems 1 (1978) 3–28.