Uncertain Programming Model for Solid Transportation Problem

Transcription

1 INFORMATION Volume 15, Number 12, pp ISSN c 212 International Information Institute Uncertain Programming Model for Solid Transportation Problem Qing Cui 1, Yuhong Sheng 2 1. School of Information Science and Engineering, Xinjiang University, Urumchi 8346, China 2. College of Mathematical and System Sciences, Xinjiang University, Urumchi 8346, China cuiqing@xju.edu.cn, shengyuhong1@sina.com Abstract The solid transportation problem is an important extension of the traditional transportation problem. Solid transportation problem with uncertain variables as its parameters is called uncertain solid transportation problem. In this paper, the expected-constrained programming for an uncertain solid transportation problem is given based on uncertainty theory. According to inverse uncertainty distribution, this model can be transformed to its deterministic form. Finally, in order to solve the uncertain solid transportation problem, a numerical example was given to show the application of the model. Keywords: transportation problem, uncertain programming, uncertain variable 1 Introduction The traditional transportation problem (TP) is a well-known optimization problem in operational research, in which two kinds of constraints are taken into consideration, i.e., source constraint and destination constraint. But in the real system, we always deal with other constraints besides of source constraint and destination constraint, such as product type constraint or transportation mode constraint. For such case, the traditional TP turns into the solid transportation problem (STP). As a generalization of the traditional TP, the STP was introduced by Haley [1] in Recently, the STP obtained much attention and many models and algorithms under both crisp environment and uncertain environment have been investigated. For examples, Bit et al.[2] presented the fuzzy programming model for a multi-objective STP, Jimėnez and Verdegay [3] studied two kinds of uncertain STP, that is, the supplies, demands and conveyance capacities are interval numbers and fuzzy numbers, respectively. In the paper [4], Jimėnez and Verdegay designed an evolutionary algorithm based on parametric approach to solve fuzzy STP. In addition, Li et al.[5] designed a neural network approach for multi-criteria STP, and they also presented an improved genetic algorithm to solve multi-objective STP with fuzzy numbers in [6]. And Gen et al.[7] gave a genetic algorithm for solving bicriteria FSTP. It is easy to see from the literature that the research of STP under fuzzy environment is very popular in recent years. One of the reasons is due to the development of the fuzzy set theory so that the ability of dealing with fuzziness is improved. As we know, fuzzy set theory was introduced by Zadeh [8] to deal with fuzziness. Up to now, fuzzy set theory has been applied to a broad fields. For the development of fuzzy set theory, we may refer to the papers of Kaufmann [9], Dubois and Prade [1] and so on. As an important extension of the traditional transportation problem is the fixed charge transportation problem (FCTP) [11]. The FCTP has also been studied by many researchers such as Steinberg [12], Sun et al.[13] and so on. In this paper, the cost solid transportation problem(cstp) is modeled based on uncertainty theory. Uncertainty theory was founded by Liu [14] in 27 and refined by Liu [26] in 211, which is a branch of mathematics based on normality, duality, subadditivity, and product axioms. Since then significant work has been done by researchers based on the uncertainty theory both in theoretical and practical aspects. If the cost parameters of the transportation problem are uncertain variables, we call the problem uncertain cost transportation problem(uctp)[15][16]. This paper mainly deals with uncertain cost solid transportation problem(ucstp). Along with the global economics development, production and demand have more and more importance. The importance of goods transportation is also increasingly reflected. Transportation model in logistics and supply management to reduce costs and improve service quality plays an important role. In reality, due to changes in market supply and demand, weather conditions, road conditions and other uncertainty factors, such that uncertainty transportation problem is particularly important. Therefore studying uncertainty transportation problem has theoretical and practical significance. In order to construct model for STP in uncertain environment, we shall first introduce some knowledge of uncertainty theory. In theoretical aspect, Liu proposed uncertain process which is a sequence of uncertain variables indexed by 342

2 UNCERTAIN PROGRAMMING MODEL FOR SOLID TRANSPORTATION PROBLEM time or space and uncertain differential equation which is a type of differential equation driven by canonical process in 28 [17]. Liu established uncertain calculus to deal with dynamic uncertain phenomena in 29 [18]. In the mean time, the concept of uncertain logic was proposed by Li and Liu [2] to describe uncertain knowledge, uncertain inference was introduced by Liu [21] via conditional uncertain measure. Meanwhile, some other theoretical properties of uncertain measure are studied [22] [23]. In practical aspect, Liu [18] founded uncertain programming that is a type of mathematical programming involving uncertain variables [18], which has been used to model system reliability design, project scheduling problem, vehicle routing problem and facility location problem [24] [25]. In financial mathematics, Liu [21] gave an uncertain stock model and European option price formula. Now, uncertainty theory has become a mathematical tool to model the indeterminate phenomenon in our real world. It has been developed to a fairly complete mathematical system [26]. In this paper, the STP is modeled based on uncertainty theory. This paper consists of 6 sections, and its frame is organized as follows: In Section 2, some basic concepts and properties in uncertainty theory used throughout this paper are introduced. In Section 3, a model is constructed for the UCSTP. In Section 4, according to the uncertainty theory, several crisp equivalences the model can be transformed to its deterministic form, and then we can find their solution by simplex method. A numerical experiment is given in Section 5. At the end of the paper, a brief summary is presented in Section 6. 2 Preliminaries In this paper, the USCTP is modeled based on uncertainty theory. In order to construct model for USCTP in uncertain environment, we shall first introduce some basic concepts of uncertainty theory. Definition 1 (Liu[14]) Let ξ is an uncertain variable. Then the expected value of ξ is defined by E[ξ] = + M{ξ r}dr M{ξ r}dr provided that at least one of the two integrals is finite. Let ξ is uncertain variable with uncertainty distribution Φ. If the expected value exists, then E[ξ] = 1 Φ 1 (α)dα. In fact, the expected value operator is linear. Let ξ and η are independent uncertain variables with expected values. Then for any real numbers a, b, we have E[aξ + bη] = ae[ξ] + be[η]. Theorem 1 (Liu [14]) Let ξ 1, ξ 2,, ξ n are independent uncertain variables with uncertainty distributions Φ 1, Φ 2,, Φ n, respectively. If f is a strictly increasing function, then ξ = f(ξ 1, ξ 2,, ξ n ) is an uncertain variable with inverse uncertainty distribution Ψ 1 (α) = f(φ 1 1 (α), Φ 1 2 (α),, Φ 1 n (α)). Theorem 2 (Liu [14]) Let ξ 1, ξ 2,, ξ n are independent uncertain variables with uncertainty distributions Φ 1, Φ 2,, Φ n, respectively. If f is a strictly decreasing function, then ξ = f(ξ 1, ξ 2,, ξ n )) is an uncertain variable with inverse uncertainty distribution Ψ 1 (α) = f(φ 1 1 (1 α), Φ 1 2 (1 α),, Φ 1 n (1 α)). 343

3 QING CUI, YUHONG SHENG 3 Uncertain solid transportation model In this section, we shall introduce some knowledge of non-balance solid transportation problem and uncertainty theory. As we know, the STP involves how to transport homogeneous products from i sources to j destinations by k conveyances so that the total transportation cost is minimized. Actually, STP is the generalization of the well-known traditional transportation problem, in which three item properties (source, destination and conveyance) are considered in the constraints instead of two item properties (source and destination). In the balance STP, the sum of supplies, the sum of demands and the sum of conveyance capacities are supposed to be equal to each other. But in the real systems, the balance condition does not always hold. It suffices to suppose that there are enough products in the sources to satisfy the demand of each destination, and the conveyances have ability to transport products to satisfy the demand of each destination. Let m be the number of the sources of the STP, let n be the number of destinations of the STP, and let l be the number of conveyances of the STP. The amount of products in source i which can be transported to destination j is denoted by a i, the minimal demand of products in destination j is denoted by b j, the transportation capacities of conveyance k is denoted by c k, where i = 1, 2,, m, j = 1, 2,, n, k = 1, 2,, l. The cost of unit product transported from source i to destination j by conveyance k of is denoted by ξ ijk, the quantity transported from source i to destination j by conveyance k of is denoted by x ijk, where i = 1, 2,, m, j = 1, 2,, n, k = 1, 2,, l. In order to describe the problems conveniently, we denote the cost function of model by n f(x, ξ) = ξ ijk x ijk where x, ξ denote the vectors consisting of x ijk, ξ ijk, i = 1, 2,, m, j = 1, 2,, n, k = 1, 2,, l respectively. Therefore, the CSTP can be formulated as follows: n min ξ ijk x ijk subject to : n x ijk a i, x ijk b j, n x ijk c k, i = 1, 2,, m j = 1, 2,, n k = 1, 2,, l x ijk, i = 1, 2,, m, j = 1, 2,, n, k = 1, 2,, l. In this model, the first constraint implies that the total amount transported from source i is no more than a i ; the second constraint implies that the total amount transported from source i should satisfy the demand of destination j; the third constraint states that the total amount transported by conveyance k is no more than its transportation capacity. The above model is constructed under certain conditions, that is, the parameters in the model are all fixed quantities. But due to the complexity of the real world, we may always meet uncertain phenomena in constructing mathematical model. For such condition, we generally add the uncertain variables to the model. Hence, in this paper, we assume that the unit cost, the capacity of each source and that of each destination are all uncertain variables and denoted by ξ ijk, ã i, b j, c k respectively. The model is called excepted-constrained programming and is constructed by Liu in 211. The main idea of uncertain programming model is to optimize the expected value of objective function under the chance constraints. Definition 2 (Liu [14]) Assume that f(x, ξ) is an objective function, and g j (x, ξ) are constraints functions, j = 1, 2,, p. A solution x is feasible if and only if M{g j (x, ξ) } α j for j = 1, 2,, p. A solution x is an optimal solution to the uncertain programming model if E[f(x, ξ)] E[f(x, ξ)] for any feasible solution x. (1) 344

4 UNCERTAIN PROGRAMMING MODEL FOR SOLID TRANSPORTATION PROBLEM So the cost of the solid transportation problem model is converted to the following equivalence model: n min E ξ ijk x ijk subject to : n M x ijk ã i β i, i = 1, 2,, m { } M bj x ijk γ j, j = 1, 2,, n n M x ijk c k η k, k = 1, 2,, l x ijk, i = 1, 2,, m, j = 1, 2,, n, k = 1, 2,, l. where β i, γ j, η k are specified confidence levels for i = 1, 2,, m, j = 1, 2,, n, k = 1, 2,, l. The first constraint implies that total amount transported from source i should be no more than its supply capacity at the confidence level β i ; the second constraint implies that the total amount transported from source i should satisfy the requirement of destination j at the credibility level γ j ; the third constraint states that the total amount transported by conveyance k should be no more than its transportation capacity at the confidence level η k. (2) 4 Crisp equivalences of models In this section, we shall induce the crisp equivalences for the model under some special conditions. Theorem 3 Assume that ξ ij, ã i, b j, c k are independent uncertain variables with uncertainty distributions Φ ξij, Φãi,,. Then the model (2) is converted to equivalence model: Φ bj Φ ck min n subject to : n Φ 1 b j M 1 x ijk Φ 1 ξ ijk (α)dα x ijk Φ 1 ã i (1 β i ), i = 1, 2,, m (γ j ) x ijk, j = 1, 2,, n n x ijk c k η k, k = 1, 2,, l x ijk, i = 1, 2,, m, j = 1, 2,, n, k = 1, 2,, l. Proof: Since ξ ijk, ã i, b j, c k are independent uncertain variables with uncertainty distributions Φ ξijk, Φãi,, respectively. According to the linearity of expected value operator, we have Φ bj Φ ck n E n ξ ijk x ijk = x ijk E [ξ ijk ] (3) where E [ξ ijk ] = 1 Φ 1 ξ ijk (α)dα, i = 1, 2,, m, j = 1, 2,, n, k = 1, 2,, l. 345

5 QING CUI, YUHONG SHENG According to Theorems 1, 2, we have n M x ijk ã i β i { M bj } x ijk n γ j Φ 1 (γ j ) b j n M x ijk c k η k x ijk Φ 1 ã i (1 β i ), i = 1, 2,, m, x ijk, j = 1, 2,, n, x ijk Φ 1 c k (1 η k ), k = 1, 2,, l. The model (4) is linear programming. Hence, we may find easily its solution by simplex method. 5 Numerical experiments In order to show the applications of the models, we shall present an example of coal transportation problem in this section. Coal is a kind of crucial energy source in the development of economy and society. Accordingly, how to transport the coal from mines to the different areas economically is also an important issue in the coal transportation. For the convenience of description, we summarize the problem as follows. Suppose that there are four coal mines to supply the coal for six cities. During the process of transportation, two kinds of conveyances are available to be selected, i.e., train and cargo ship. Now, the task for the decision-maker is to make the transportation plan for the next month. At the beginning of this task, the decision maker needs to obtain the basic data, such as supply capacity, demand, transportation cost of unit product, and so on. In fact, since the transportation plan is made in advance, we generally cannot get these data exactly. For this condition, the usual way is to obtain the uncertain data by means of experience evaluation or expert advice. Assume that all uncertain variables are normal uncertain variables, ξ ijk N(e ijk, σ ijk ), i = 1, 2, 3, 4, j = 1, 2, 3, 4, 5, 6, k = 1, 2, ã i N(e i, σ i ), i = 1, 2, 3, 4, b j N(e j, σ j), j = 1, 2, 3, 4, 5, 6, c k N(e k, σ k), k = 1, 2. And the corresponding uncertain data are listed as follows: Table 4.1: Parameters of normal distribution N(e ij1, σ ij1 ) of costs by train (e ij1,σ ij1 ) (2,2) (2,2) (2,2) (19,1.5) (2,2) (1,1.5) 2 (1,1) (11,1.5) (7,1.5) (2,2) (2,1.5) (2,1.5) 3 (1,1.5) (18,1.5) (8,1.5) (12,1.5) (2,1.5) (22,1.5) 4 (21,1.5) (14,1.5) (18,1.5) (12,1.5) (13,1.5) (22,2) Table 4.2: Parameters of normal distribution N(e ij2, σ ij2 ) of costs by cargo ship (e ij2,σ ij2 ) (4,2) (4,2) (4,2) (39,1.5) (4,2) (3,1.5) 2 (2,1) (31,1.5) (27,1.5) (4,2) (4,1.5) (4,1.5) 3 (2,1.5) (38,1.5) (28,1.5) (32,1.5) (4,1.5) (42,1.5) 4 (41,1.5) (34,1.5) (38,1.5) (32,1.5) (33,1.5) (42,2) Table 4.3: Parameters of normal distribution N(e i, σ i ) of supplies (e i,σ i ) (25,1.5) (3,1.5) (32,2) (28,2) 346

6 UNCERTAIN PROGRAMMING MODEL FOR SOLID TRANSPORTATION PROBLEM Table 4.4: Parameters of normal distribution N(e j, σ j ) of demands (e j, σ j ) (1,1.5) (14,1) (22,1) (18,1) (16,1) (12,1) Table 4.5: Parameters of normal distribution N(e j, σ j ) of transportation capacities (e k, σ k ) 1 2 (4,1.5) (6,1) Then the model (3) is equivalent to the following model: min x ijk e ijk subject to : 6 2 x ijk [e i + σ i 3 π ln 1 β i ], i = 1, 2, 3, 4 β j=1 i k=1 [e j + σ j 3 γ j 4 2 ln ] x ijk, j = 1, 2, 3, 4, 5, 6 π 1 γ j 4 6 x ijk [e k + k σ 3 ln 1 η k ], k = 1, 2 π η k x ijk, i = 1, 2, 3, 4, j = 1, 2, 3, 4, 5, 6, k = 1, 2. (4) The transportation cost is , and the corresponding transportation plan is 6 Conclusions x 161 = , x 231 = , x 341 = , x 451 = , x 222 = , x 232 = , x 312 = , x 332 = , x 442 = This paper mainly investigated uncertain cost solid transportation problem based on uncertainty theory. As a result, a decision model under criteria was presented. The construction of expected-constrained programming model was according to the idea of expected value of the objective under the chance constraints. Since there are many uncertain variables in the model, computing objective value and checking feasibility become complex in general. In order to solve the model conveniently, we discussed the crisp equivalences for the model under the condition that the parameters are special uncertain variables. Under uncertainty theory, this model can be transformed to its deterministic form, and then we can find its solution by simplex method. Finally, as an application of the model, we presented a coal transportation problem as example, excepted-constrained programming model was employed as the experimental model. References [1] Haley, K.B., The Solid Transportation Problem. Operational Research, 11 (1962) [2] Bit, A.K., Biswal, M.P., and Alam, S.S., Fuzzy Programming Approach to Multiobjective Solid Transportation Problem. Fuzzy Sets Systems, 57 (1993) [3] Jimėnez, F., Verdegay, J.L., Solid Transportation Problems. Fuzzy Sets Systems, 1 (1998) [4] Jimėnez, F., Verdegay, J.L., Solving Fuzzy Solid Transportation Problems by an Evolutionary Algorithm Based Parametric Approach. European Journal of Operational Research, 117 (1999) [5] Li, Y., Ida, K., Gen, M., Kobuchi, R., Neural Network Approach for Multicriteria Solid Transportation Problem. Computer and Industrial Engeering, 33 (1997) [6] Li, Y., Ida, K., Gen, M., Improved Genetic Algorithm for Solving Multiobjective Solid Transportation Problem with Fuzzy Numbers. Computer and Industrial Engeering, 33 (1997)

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