Title: Uncertain Goal Programming Models for Bicriteria Solid Transportation Problem

Transcription

1 Title: Uncertain Goal Programming Models for Bicriteria Solid Transportation Problem Author: Lin Chen Jin Peng Bo Zhang PII: S (16) DOI: Reference: ASOC 3917 To appear in: Applied Soft Computing Received date: Revised date: Accepted date: Please cite this article as: Lin Chen, Jin Peng, Bo Zhang, Uncertain Goal Programming Models for Bicriteria Solid Transportation Problem, <![CDATA[Applied Soft Computing Journal]]> (216), This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

2 *Graphical abstract (for review) Graphical Abstract Uncertainty Theory BSTP Uncertain Goal Programming Expected Value Goal Programming Model Uncertain BSTP Chance-constrained Goal Programming Model Page 1 of 25

3 *Highlights (for review) Highlights 1) Expected value goal programming model and chance-constrained goal programming model for the bicriteria solid transportation problem are constructed within the framework of uncertain goal programming. 2) Taking advantage of some properties of uncertainty theory, the models are respectively transformed into the corresponding deterministic equivalents which are solved by optimization packages conveniently. 3) Some numerical examples are given to show the applications of the models, and some large-scale computational experiments are performed to further show the efficiency and feasibility of the models. Page 2 of 25

4 Uncertain Goal Programming Models for Bicriteria Solid Transportation Problem Lin Chen a,b, Jin Peng b,, Bo Zhang c a College of Mathematics and Sciences, Shanghai Normal University, Shanghai 2234, China b Institute of Uncertain Systems, Huanggang Normal University, Hubei 43899, China c School of Statistics and Mathematics, Zhongnan University of Economics and Law, Hubei 4373, China Abstract: This paper investigates an uncertain bicriteria solid transportation problem. The supplies, demands, conveyance capacities, transportation cost and transportation time are regarded as uncertain variables. According to two types of methods to rank the uncertain variables, expected value goal programming model and chance-constrained goal programming model for the bicriteria solid transportation problem are constructed. It is proved that the expected value goal programming model and chanceconstrained goal programming model can be respectively transformed into the corresponding deterministic equivalents by taking advantage of some properties of uncertainty theory. Based on these equivalence relations, the optimal transportation plans of the uncertain goal programming models can be obtained. Several numerical experiments are presented to illustrate the applications of the models. Keywords: Bicriteria solid transportation problem; Combinatorial optimization; Uncertainty theory; Uncertain goal programming 1 Introduction As one of the useful combinatorial optimization problems, the solid transportation problem (STP) was first stated by Schell [33] in 1955 and developed by Haley [11] in The aim of STP is to transport homogenous products from sources to destinations by different types of conveyances such that the total transportation cost is minimized. However, in the real-world applications, the decision makers always take the transportation cost and transportation time into account at the same time (Gen et al. [9], Kundu et al. [18] and Tao and Xu [37]). (BSTP) has a real significance. So the study of the bicriteria solid transportation problem Many researchers have investigated the transportation problem in the past decades, such as Hitchcock [12], Juman and Hoque [15], Lotfi and Tavakkoli-Moghaddam [29], Ojha et al. [31] and Yang and Gen [41], etc. The above researchers have studied the transportation problem in deterministic environment, and have regarded the relevant parameters in the problem as constant numbers. However, due to the complexity of the social and economic environment in the real-world situations, the supplies, demands, conveyance capacities, transportation cost and transportation time often meet with uncertainties. Therefore, it is unsuitable to characterize these relevant parameters in the problem as constant numbers. Corresponding author pengjin1@tsinghua.org.cn. Lin Chen s current affiliation is the College of Management and Economics, Tianjin University, Tianjin 372, China. 1 Page 3 of 25

5 In the relevant literature, random variables have been used to characterize the stochastic factors in the transportation problem. For example, Williams [38] assumed that the demands were random variables and proposed a stochastic transportation model. After that, under considering the penalties that paid for each oversupplied and undersupplied unit of product, Szwarc [36] investigated a stochastic transportation model in which the total transportation cost plus total expected penalty costs were minimized. What s more, Yang and Feng [39] studied a BSTP with fixed charge under stochastic environment and proposed three models for the problem. On the basis of fuzzy set theory which was proposed by Zadeh [44], several scholars have employed fuzzy set theory to deal with the fuzzy factors in the transportation problems. For instance, Jiménez and Verdegay [13] considered the fuzzy STP in which the supplies, demands and conveyance capacities were trapezoidal fuzzy numbers. In the paper [14], Jiménez and Verdegay dealt with the fuzzy STP in which the fuzziness affected the constraint set. In 1995, Gen et al. [9] investigated an implementation of genetic algorithm to solve the fuzzy BSTP, where the supplies, demands and conveyance capacities were presented as fuzzy numbers. Following that, Yang and Liu [4] presented the fixed charge solid transportation problem under fuzzy environment, in which the direct costs, fixed charges, supplies, demands and conveyance capacities were supposed to be fuzzy variables. In 213, Kundu et al. [17] modeled the multi-objective multi-item solid transportation problem with fuzzy coefficients for the objectives and constraints, and provided two different methods to solve the problem. In 214, Kundu et al. [18] investigated the multi-objective solid transportation problem under fuzzy environment, where the transportation penalties and cost were regarded as fuzzy variables. In 215, Kundu et al. [19] formulated the multi-item solid transportation problem under the assumption that the supplies, demands and transportation cost were regraded as type-2 triangular fuzzy variables. For more studies of transportation problem under fuzzy environment, we may consult Bit et al. [1], Giri et al. [1], Kaur and Kumar [16] and Liu et al. [26]. In addition, several researchers have also investigated the BSTP problem based on the rough theory in which the relevant parameters were supposed to be rough variables, see Kundu et al. [2], Tao and Xu [37] and Yu et al. [43] for instance. Our research paper is different from these papers in a fundamental way. We establish our models based on the goal programming which is a commonly used tool to solve transportation problems with multiple conflicting objectives. We are particularly interested in how to make the transportation plan to meet with the three kinds of capacity constraints when the supplies, demands, conveyance capacities, unit transportation cost and transportation time are uncertain variables with specific uncertainty distributions. If uncertain factors come from the decision-makers empirical estimations, then it is unsuitable to use random variables, fuzzy variables or rough variables to characterize this kind of uncertain factors (Liu [24]). This statement would be especially true for new products that have not been in the market for long. In order to deal with human data such as empirical estimations and expert data, uncertainty theory was proposed by Liu [21] in 27 and refined by Liu [24] in 21. When no samples are available in a transportation problem, we have to invite some domain experts to evaluate the belief degrees about the supplies, demands, conveyance capacities, transportation cost and transportation time, etc. Uncertainty theory is a branch of axiomatic mathematics for dealing with belief degrees (Liu [24]). So the meaning of studying the bicriteria solid transportation problem within the framework of uncertainty theory is significant. The significance of uncertainty has motivated several scholars contribute to investigate the trans- 2 Page 4 of 25

6 portation problem under an uncertain environment. According to the idea of expected value of the objective function under the chance constraints, the expected-constrained programming model for an uncertain solid transportation problem was studied by Cui and Sheng [6]. In their paper [6], Cui and Sheng regarded the cost of unit product transported from source to destination as an uncertain variable. After that, Sheng and Yao [34] proposed a model without path choice or capacity constraint, in which the supply capacity, demand amount and unit transportation cost were regarded as uncertain variables. Later, Sheng and Yao [35] investigated the fixed charge transportation problem within the framework of uncertainty theory. In their paper [35], the authors considered direct cost and fixed charge, where the direct cost was the fee with respect to per unit transportation amount and the fixed charge occurred when the transportation activity took place in the corresponding source-destination pair. Motivated by the emergency scheduling in a transportation network, Mou et al. [3] considered the transportation problem in which the truck time and transportation cost were assumed to be uncertain variables. Recently, Gao et al. [7] investigated the frequency service network design problem in a railway freight transportation system in which the fixed charge and transportation cost were both depicted by uncertain variables. Gao and Qin [8] studied the p-hub center location problem under uncertain setting by characterizing the travel times as uncertain variables. The above studies only take the total transportation cost into consideration. However, in the real-life decision making problems, the decision makers should take several objectives into account simultaneously. In the actual transportation problem, it is inevitable to deliver the products which are not easy to store, such as aquatic products, perishable vegetables, and so on. Thus, we should consider the total transportation cost and total transportation time at the same time. This paper investigates an uncertain bicriteria solid transportation problem, in which the supplies, demands, conveyance capacities, unit transportation cost and transportation time are supposed to be uncertain variables. Compared with the prior studies in the area of uncertain transportation problem, the innovations of this research are as follows. Different from the prior studies in the literature [6, 7, 8, 3, 34, 35], we take the total transportation cost and total transportation time into account simultaneously. We employe the uncertain goal programming to model the bicriteria solid transportation problem under an uncertain environment. Our paper is also related to the literature pool of goal programming. The concept of goal programming was first put forward in the seminar work of Charnes and Cooper [3] and subsequently studied by many researchers. After that, goal programming under an indeterminacy environment was introduced and widely investigated. Contini [5] was the first one to propose the stochastic goal programming with random parameters. As is known to all of us, the probability theory is based on the frequency of events and on the collected empirical data along the way. Thus we have to rely on a great quantity of historical data to get the probability distribution. In many cases, however, gathered data are not credible enough, or there are no available samples to estimate the probability distribution. Then some domain experts are invited to evaluate the degrees of belief that each event may happen. In order to handle the goal programming with belief degrees which are given by domain experts, Liu and Chen [25] proposed an uncertain goal programming. Motivated by this new kind of goal programming, we intend to consider an uncertain bicriteria solid transportation problem by means of the uncertainty theory in this paper. Since uncertainty theory is a powerful tool to deal with the incomplete information of the relevant parameters in the transportation problem, the uncertain bicriteria solid transportation problem discussed in this paper is typically a novel problem in the existing literature within the framework of uncertain goal programming. 3 Page 5 of 25

7 Generally speaking, there exists two kinds of uncertain goal programming models. One is the expected value goal programming model and another is the chance-constrained goal programming model. These two uncertain programming models have their corresponding application scopes. The former one is based on the expected value criterion which is to optimize the expected value of the decision maker s profit, while the latter one is based on the critical value criterion which is to optimize the critical value of the decision maker s profit with a predetermined confidence level. On one hand, based on the priority structure, the main idea of the expected value goal programming model is to balance multiple conflicting objectives under some expected constraints. On the other hand, according to the priority structure, the essence of the chance-constrained goal programming model is to balance the multiple conflicting objectives with a given confidence level subject to some chance constraints. Motivated by the considerations above, we establish two types of uncertain goal programming models for the uncertain bicriteria solid transportation problem. According to the operational law of independent uncertain variables, the proposed models can be respectively transformed into their corresponding deterministic equivalents which can be solved by the optimization software package LINGO. The rest of the paper is organized as follows. Section 2 briefly reviews the preliminaries of uncertainty theory. In Section 3, we present the problem considered in this paper. In Section 4, the expected value goal programming model and the chance-constrained goal programming model are constructed for the bicriteria solid transportation problem. Section 5 investigates several deterministic equivalents for the different models. In Section 6, several numerical examples are presented. The main conclusions are pointed out in Section 7. 2 Preliminary Up to now, researchers have actively studied uncertainty theory and its applications. In order to calculate the expected value of monotone function of independent uncertain variables, Liu and Ha [27] gave the important formula for expected value. Peng [32] proposed two types of risk metrics of loss function for uncertain system. Recently, Chen et al. [4] studied some properties of uncertain sequences and derived several convergence theorems. What s more, Yao [42] investigated sine entropy for uncertain set, and derived a formula to calculate the sine entropy via inverse membership function. From a practical point of view, Liu [23] presented uncertain programming which is a type of mathematical programming involving uncertain variables. On the basis of uncertain programming, Liu and Chen [25] developed uncertain multiobjective programming and uncertain goal programming to model multiobjective decision-making problems with uncertain parameters. For the recent developments about the uncertainty theory, the interested readers may consult the book of Liu [24]. In this section, some necessary preliminaries of uncertainty theory are briefly introduced. Let Γ be a nonempty set and L be a σ-algebra over Γ. Liu [21] presented an axiomatic uncertain measure M{Λ} to indicate the belief degree that uncertain event Λ occurs. The uncertain measure M{Λ} meets with the following three axioms: Axiom 1. M{Γ} = 1 for the universal set Γ; Axiom 2. M{Λ} + M{Λ c } = 1 for any Λ L, where Λ c is a complement of Λ; 4 Page 6 of 25

8 Axiom 3. For every countable sequence of events Λ 1, Λ 2,..., we have { } M Λ i M{Λ i }. i=1 i=1 To provide an operational law on the uncertainty space (Γ, L, M), the product uncertain measure was defined by Liu [22] as the following product axiom: Axiom 4. Let (Γ k, L k, M k ) be uncertainty spaces for k = 1, 2,... Then the product uncertain measure M is an uncertain measure satisfying { } M Λ k = M k {Λ k }, k=1 where Λ k are arbitrarily chosen events from L k for k = 1, 2,..., respectively. An uncertain variable (Liu [21]) is a measurable function ξ from an uncertainty space (Γ, L, M) to the set of real numbers. On the basis of the concept of uncertain variable, Liu [21] defined the uncertainty distribution of the uncertain variable ξ as Φ(x) = M {ξ x}. Definition 2.1 (Liu [24]) An uncertainty distribution Φ(x) is said to be regular if it is a continuous and strictly increasing function with respect to x at which < Φ(x) < 1, and lim x Φ(x) =, lim k=1 x + Φ(x) = 1. Based on Definition 2.1, the inverse uncertainty distribution was defined by Liu [24] as follows. Definition 2.2 (Liu [24]) Let ξ be an uncertain variable with regular uncertainty distribution Φ(x). Then the inverse function (α) is called the inverse uncertainty distribution of ξ. According to inverse uncertainty distribution, Liu [24] gave the operational law of the strictly monotone function of uncertain variables as follows. Theorem 2.3 (Liu [24]) Let ξ 1, ξ 2,..., ξ n be independent uncertain variables with regular uncertainty distributions Φ 1, Φ 2,..., Φ n, respectively. If the function f(x 1, x 2,..., x n ) is strictly increasing with respect to x 1, x 2,..., x m and strictly decreasing with respect to x m+1, x m+2,..., x n, then the uncertain variable ξ = f(ξ 1, ξ 2,..., ξ n ) has an inverse uncertainty distribution Ψ 1 (α) = f ( (α),..., Φ 1 m (α), 1 m+1 (1 α),..., Φ 1 n (1 α) ). Liu [22] introduced the concept of independent uncertain variables in the following way. The uncertain variables ξ 1, ξ 2,..., ξ n are said to be independent if { n } n M (ξ i B i ) = M{ξ i B i } i=1 i=1 for any Borel sets B 1, B 2,..., B n. The expected value of uncertain variable ξ was defined by Liu [21] as E[ξ] = + M{ξ x}dx M{ξ x}dx 5 Page 7 of 25

9 provided that at least one of the two integrals is finite. As a useful representation of expected value, it has been proved by Liu [21] that E[ξ] = (α)dα, where is the inverse uncertainty distribution of uncertain variable ξ. Theorem 2.4 (Liu [24]) Let ξ and η be independent uncertain variables with finite expected values. Then for any real numbers a and b, we have E[aξ + bη] = ae[ξ] + be[η]. Example 2.5 (Liu [24]) Suppose that ξ Z(a, b, c) is a zigzag uncertain variable, where a, b and c are real numbers with a < b < c. Its uncertainty distribution and inverse uncertainty distribution are given by, if x a (x a)/2(b a), if a < x b Φ(x) = (x + c 2b)/2(c b), if b < x c 1, if x > c, and (1 2α)a + 2αb, if α <.5 (α) = (2 2α)b + (2α 1)c, if α.5, respectively. Furthermore, the expected value of ξ is E[ξ] = (a + 2b + c)/4. 3 Problem Description In this section, we describe the BSTP particularly. It is well-known that the aim of classical STP is to transport homogenous products from m sources to n destinations by k conveyances such that the total transportation cost is minimized. But in the real-world decision making problems, the decision makers always take several objectives into account simultaneously. In the actual transportation problem, it is inevitable to deliver the products which are not easy to store, such as aquatic products, perishable vegetables, and so on. Thus, the total transportation cost and total transportation time are considered at the same time in this paper. The required notations are summarized in Table 1. Notation Description Table 1 List of notation i The index for sources which belongs to {1, 2,..., m} j The index for destinations which belongs to {1, 2,..., n} k The index for conveyances which belongs to {1, 2,..., l} a i b j c k c ijk t ijk The amount of products in source i which can be transported to n destinations The minimal demand of products at destination j The transportation ability of conveyance k The cost of unit transportation amount from source i to destination j by conveyance k The transportation time of transportation activity from source i to destination j by conveyance k 6 Page 8 of 25

10 In order to construct the bicriteria solid transportation problem mathematically, the following decision variables are needed. We define x ijk to represent the quantity transported from source i to destination j by conveyance k, and denote the binary variables y ijk to stand for the transportation activity from source i to destination j by conveyance k for i = 1, 2,..., m, j = 1, 2,..., n and k = 1, 2,..., l. If x ijk >, then y ijk = 1, while if x ijk, then y ijk =. The all decision variables are denoted by vectors x and y, respectively. Let f 1 (x) and f 2 (y) denote the total transportation cost and total transportation time after the transportation activity occurs, respectively. Then the following mathematical relations can be easily obtained: f 1 (x) = c ijk x ijk, f 2 (y) = t ijk y ijk. In the previous studies, many methods were presented to find the optimal solution for the optimization problem with bi-objective function, such as multiobjective programming and goal programming. In this paper, we employe the goal programming to deal with the BSTP. That is, we need to seek the optimal transportation plan meeting with the given goal values. For the total transportation cost and total transportation time, the optimal values cannot be obtained at the same solution since the two objectives always conflict with each other. In order to balance this two conflicting objectives, the decision-makers may establish a hierarchy of importance among the two incompatible goals so as to satisfy as many goals as possible. In the real-world situations, the decision makers may firstly take the total transportation cost into consideration in the process of making decision. Thus, we shall minimize the total transportation cost at the first priority level and minimize the total transportation time at the second priority level. Specifically, at the first priority level, the total transportation cost f 1 (x) should not be larger than the predetermined goal value C. Thus we have the following goal constraint: f 1 (x) + d 1 d+ 1 = C, in which C is a predetermined total transportation cost, and d + 1 and d 1 are the positive deviation and negative deviation from the target C, respectively. At the second priority level, the total transportation time f 2 (y) should not be larger than the predetermined goal value T. Then we have the following goal constraint: f 2 (y) + d 2 d+ 2 = T, where T is a predetermined total transportation time, and d + 2 and d 2 are the positive deviation and negative deviation from the target T, respectively. 7 Page 9 of 25

11 Therefore, we formulate the following goal programming model for the BSTP: { lexmin d + 1, d + } 2 subject to: j=1 k=1 i=1 k=1 i=1 j=1 c ijk x ijk + d 1 d+ 1 = C t ijk y ijk + d 2 d+ 2 = T x ijk a i, x ijk b j, x ijk c k, i = 1, 2,..., m j = 1, 2,..., n k = 1, 2,..., l x ijk, y ijk {, 1}, i = 1, 2,..., m; j = 1, 2,..., n; k = 1, 2,..., l d 1, d+ 1, d 2, d+ 2 where the lexmin represents lexicographically minimizing the objective vector, y ijk is defined as a function by y ijk = 1 if x ijk >, or y ijk = otherwise. The third constraint indicates that the total volume transported from source i is not larger than its supply capacity. The fourth constraint states that the total amount transported from sources should meet with the required demand of destination j. The fifth constraint implies that the total amount transported by each conveyance is not larger than its conveyance ability. It is necessary to point out that in the above model, the parameters are all constant numbers. But in the real-world applications, the transportation plan always be made ahead of time, no samples are available for the decision makers. So the BSTP arises in an indeterminacy environment. As we mentioned in Introduction part, the uncertainty theory is a useful tool to deal with such kind of problems in an indeterminacy environment. In order to discuss the BSTP under an uncertain environment, we assume that the supplies, demands, conveyance capacities, unit transportation cost and transportation time are all uncertain variables, and denote them as ã i, b j, c k, ξ ijk and η ijk, respectively. We suppose that all the uncertain variables are positive and independent with each other. Such independent assumption is widely used in the area of transportation problem with uncertain parameters ([6, 7, 3, 34, 35]). Then the BSTP becomes uncertain bicriteria solid transportation problem, which can be denoted by UBSTP in short., 4 Uncertain Bicriteria Solid Transportation Models (1) In recent years, studying the transportation problem by using goal programming is a hot research topic. The goal programming was first put forward by Charnes and Cooper [3] and subsequently studied by many researchers. In the beginning, the parameters in the goal programming are all deterministic. However, the decision environment is usually full of indeterminacy. In the past decades, several researchers have investigated the goal programming under indeterminacy environments (Charnes and Cooper [2], Gen et al. [9], Kundu et al. [2] and Yang and Feng [39]). On the other hand, several researchers have studied the bicriteria solid transportation problem by using the chance-constrained goal programming 8 Page 1 of 25

12 within the framework of different theories, that is, probability theory, fuzzy set theory and rough theory (Kundu et al. [17], Yang and Feng [39] and Yu et al. [43]). However, these three theories have their applicable scopes. When we encounter the case of lacking sample data, the existing methods are unapplicable ([6, 7, 3, 34, 35]). In this case, we have to invite some domain experts to evaluate the degrees of belief that each event may occur. In order to handle the goal programming with belief degrees, Liu and Chen [25] proposed an uncertain goal programming to deal with multiobjective optimization problems with conflicting objectives. The uncertain goal programming has a distinctive advantage, that is, by using some properties of uncertainty theory, the uncertain goal programming can be transformed into the corresponding deterministic equivalent model which can be solved by the classical methods conveniently. From this point of view, the uncertain goal programming is a convenient and simple tool to model the bicriteria solid transportation problem with uncertain parameters which are given by the domain experts. Motivated by these considerations, we study the same problem, i.e., bicriteria solid transportation problem, by using the goal programming within the framework of uncertainty theory. We have assumed that the supplies, demands, conveyance capacities, unit transportation cost and transportation time are all uncertain variables in Section 3. So the total transportation cost and total transportation time are also uncertain variables. In the process of finding the optimal transportation plan, it is meaningless to minimize the total transportation cost and total transportation time since we cannot rank uncertain variables directly. In order to optimize the objective function, it is inevitable to rank uncertain variables according to some decision criteria. Generally speaking, there exists two kinds of decision criteria to rank the uncertain variables: one is the expected value criterion and another is the critical value criterion. Based on this two decision criteria, we construct two new models for the UBSTP within the framework of uncertainty theory. One is the expected value goal programming model and another is the chance-constrained goal programming model. 4.1 Expected Value Goal Programming Model Optimizing expected objective value is a preferably choice for decision makers in an indeterminacy environment. The essence of the expected value goal programming model is to balance multiple conflicting objectives under some expected constraints. According to the priority structure and target levels set by the decision makers, we may formulate the following expected value goal programming model for the 9 Page 11 of 25

13 UBSTP: lexmin { d + 1, } d+ 2 subject to: E ξ ijk x ijk + d 1 d+ 1 = C E E E j=1 k=1 [ m i=1 k=1 i=1 j=1 η ijk y ijk + d 2 d+ 2 = T x ijk ã i, ] x ijk b j, j = 1, 2,..., n E x ijk c k, i = 1, 2,..., m (2) k = 1, 2,..., l x ijk, y ijk {, 1}, i = 1, 2,..., m; j = 1, 2,..., n; k = 1, 2,..., l d 1, d+ 1, d 2, d+ 2 where y ijk is defined as a function by y ijk = 1 if x ijk >, or y ijk = otherwise. The third constraint implies that the total product transported from the source i should not be larger than the expected value of its supply ability. The forth constraint states that the total product transported from sources should meet with the expected value of the demand of destination j. The fifth constraint indicates that the total amount transported by conveyance k should not exceed the expected value of its conveyance ability. 4.2 Chance-constrained Goal Programming Model In order to construct the chance-constrained goal programming model for the UBSTP, we may have the following priority structure and target levels. For the first priority level, the total transportation cost should not be more than the given value C with the predetermined confidence level α. Then we have the first chance constraint: M, ξ ijk x ijk d + 1 C α. For the second priority level, the total transportation time should not exceed the given value T with the predetermined confidence level β. Then we have the second chance constraint: M η ijk y ijk d + 2 T β. In order to find a suitable transportation plan, we formulate the following chance-constrained goal 1 Page 12 of 25

14 programming model for the UBSTP: lexmin { d + 1, } d+ 2 subject to: M M M M j=1 k=1 { m i=1 k=1 i=1 j=1 ξ ijk x ijk d + 1 C α η ijk y ijk d + 2 T β x ijk ã i α i, } x ijk b j β j, M x ijk c k γ k, i = 1, 2,..., m (3) j = 1, 2,..., n k = 1, 2,..., l x ijk, y ijk {, 1}, i = 1, 2,..., m; j = 1, 2,..., n; k = 1, 2,..., l d + 1, d+ 2, where α, β, α i, β j and γ k are the predetermined confidence levels for i = 1, 2,..., m, j = 1, 2,..., n and k = 1, 2,..., l. 5 Deterministic Transformations If we want to find the optimal solutions of the above models, then we need to compute the expected values or uncertain measures of the uncertain variables. Generally, computing these values are actually time-consuming. Therefore, we take advantage of some properties of uncertainty theory to transfer the models into their corresponding deterministic equivalents under some special conditions. In this section, we induce the deterministic equivalents for the different models. Theorem 5.1 If ã i, b j, c k, ξ ijk and η ijk are independent uncertain variables with regular uncertainty distributions Φãi,, Φ bj Φ ck, Φ ξijk and Φ ηijk, i = 1, 2,..., m, j = 1, 2,..., n, k = 1, 2,..., l, respectively, 11 Page 13 of 25

15 then the model (2) is equivalent to the following one { lexmin d + 1, d + } 2 subject to : j=1 k=1 i=1 k=1 i=1 j=1 x ijk y ijk x ijk x ijk x ijk ξ ijk (α)dα + d 1 d+ 1 η ijk (α)dα + d 2 d+ 2 ã i (α)dα, = C = T i = 1, 2,..., m bj (α)dα, j = 1, 2,..., n c k (α)dα, k = 1, 2,..., l x ijk, y ijk {, 1}, i = 1, 2,..., m; j = 1, 2,..., n; k = 1, 2,..., l d 1, d+ 1, d 2, d+ 2, where y ijk is a function defined by y ijk = 1 if x ijk >, or y ijk = otherwise. Proof. It follows from Theorem 2.4 that E ξ ijk x ijk = where the expected value can be expressed in more explicit form Thus we have and E[ξ ijk ] = E x ijk E[ξ ijk ], ξ ijk (α)dα, i = 1, 2,..., m; j = 1, 2,..., n; k = 1, 2,..., l. E Now, it can be easily seen that E ξ ijk x ijk = η ijk y ijk = j=1 k=1 x ijk ã i, x ijk y ijk ξ ijk (α)dα, i = 1, 2,..., m η ijk (α)dα. (4) is equivalent to Similarly, j=1 k=1 E x ijk [ m i=1 k=1 ã i (α)dα, i = 1, 2,..., m. ] x ijk b j, j = 1, 2,..., n 12 Page 14 of 25

16 is equivalent to And the constraint i=1 k=1 x ijk i=1 j=1 E x ijk c k, bj (α)dα, j = 1, 2,..., n. k = 1, 2,..., l is equivalent to The theorem is proved. i=1 j=1 x ijk c k (α)dα, k = 1, 2,..., l. Theorem 5.2 If ã i, b j, c k, ξ ijk and η ijk are independent uncertain variables with regular uncertainty distributions Φãi, Φ bj, Φ ck, Φ ξijk then the model (3) is equivalent to the following model { lexmin d + 1, d + } 2 subject to : j=1 k=1 bj and Φ ηijk, i = 1, 2,..., m, j = 1, 2,..., n, k = 1, 2,..., l, respectively, x ijk ξ ijk (α) d + 1 C y ijk η ijk (β) d + 2 T x ijk ã i (1 α i ), (β j ) i=1 j=1 i=1 k=1 x ijk, x ijk c k (1 γ k ), i = 1, 2,..., m j = 1, 2,..., n k = 1, 2,..., l x ijk, y ijk {, 1}, i = 1, 2,..., m; j = 1, 2,..., n; k = 1, 2,..., l d + 1, d+ 2, where y ijk is a function defined by y ijk = 1 if x ijk >, or y ijk = otherwise. Proof. It follows from Theorem 2.3 that we can transform the constraint M ξ ijk x ijk d + 1 C α into a deterministic constraint Similarly, the constraint M x ijk ξ ijk (α) d + 1 C. η ijk y ijk d + 2 T β can be transformed into a deterministic constraint y ijk η ijk (β) d + 2 T. (5) 13 Page 15 of 25

17 Moreover, the constraint M j=1 k=1 x ijk ã i α i, i = 1, 2,..., m is equivalent to Similarly, the constraint is equivalent to And the constraint is equivalent to The theorem is verified. j=1 k=1 M bj { m x ijk ã i (1 α i ), i = 1, 2,..., m. i=1 k=1 (β j ) i=1 j=1 } x ijk b j β j, j = 1, 2,..., n i=1 k=1 M x ijk c k γ k, i=1 j=1 x ijk, j = 1, 2,..., n. k = 1, 2,..., l x ijk c k (1 γ k ), k = 1, 2,..., l. It is necessary to point out that the models (4) and (5) are both -1 programming models. Generally, solving such models are actually time consuming. However, with the aid of some well developed optimization software packages, for example, LINGO, CPLEX and MATLAB, we can solve the optimization models with moderate size or even large size easily and effectively. Similar to the mainstream literature in the area of transportation problem (Cui and Sheng [6], Giri et al. [1], Kundu et al. [18, 19] and Sheng and Yao [34, 35]), we solve the proposed models by using LINGO. 6 Numerical Examples In this section, we give several numerical examples to illustrate the applications and the sensitivity analysis of the models as mentioned above. The developed optimization software package LINGO 11. will be employed in the examples to produce the optimal transportation schedule. The computational experiments are carried out on a personal computer (Dell with Intel(R) Core(TM) i5-245m CPU 2.5GHZ and RAM 2.5GB), using the Microsoft Windows 7 operating system. 6.1 Case Study Consider a transportation plan in which four coal mines supply the coal for six cities by means of two types of conveyances, i.e., cargo ship and goods train. Now, the project managers should make a transportation schedule for the next quarter such that the total transportation cost and total transportation time are minimized at the same time. Due to the complexity of the social and economic reasons, and the 14 Page 16 of 25

18 unexpected circumstances, it is common for the decision makers to be uncertain about the basic information, such as the supply, demand, conveyance capacity, unit transportation cost and transportation time, etc. This would be especially true for new products that have not been transported for long. In this case, requesting domain experts estimations on the subject in question could be an alternative approach. Assume that the supplies ã i, demands b j and transportation capacities c k (i = 1, 2, 3, 4, j = 1, 2, 3, 4, 5, 6 and k = 1, 2) are all zigzag uncertain variables, their uncertainty distributions are listed in Tables 2-4, respectively. We take the following goal targets: C = 1972 and T = 142. Table 2 The supplies ã i of four coal mines i ã i Z(25, 33, 39) Z(28, 33, 38) Z(28, 32, 35) Z(3, 33, 36) Table 3 The demands b j of six cities j b j Z(1, 13, 15) Z(12, 15, 18) Z(8, 12, 14) Z(11, 13, 17) Z(12, 15, 18) Z(9, 13, 16) Table 4 The transportation capacities c k of two conveyances k 1 2 c k Z(52, 56, 6) Z(58, 65, 7) Similarly, the unit transportation costs ξ ijk and transportation time η ijk (i = 1, 2, 3, 4, j = 1, 2, 3, 4, 5, 6 and k = 1, 2) are also assumed to be zigzag uncertain variables, their uncertainty distributions are listed in Tables 5-8, respectively. Table 5 The unit transportation costs by goods train (ξ ij1) Mines\Cities Z(25, 33, 39) Z(16, 18, 21) Z(27, 32, 35) Z(25, 29, 33) Z(15, 18, 2) Z(15, 17, 2) 2 Z(25, 29, 33) Z(2, 26, 3) Z(25, 28, 32) Z(3, 32, 35) Z(27, 28, 33) Z(21, 23, 26) 3 Z(28, 32, 38) Z(18, 23, 27) Z(17, 22, 3) Z(26, 29, 33) Z(3, 33, 37) Z(21, 24, 29) 4 Z(25, 29, 33) Z(22, 24, 26) Z(2, 23, 27) Z(27, 32, 35) Z(18, 2, 25) Z(21, 24, 28) Table 6 The unit transportation costs by cargo ship (ξ ij2 ) Mines\Cities Z(36, 39, 43) Z(37, 42, 45) Z(3, 33, 37) Z(25, 27, 3) Z(25, 28, 3) Z(28, 32, 37) 2 Z(4, 42, 47) Z(28, 3, 35) Z(3, 31, 34) Z(37, 38, 43) Z(37, 38, 43) Z(32, 33, 36) 3 Z(4, 43, 48) Z(25, 3, 33) Z(31, 34, 39) Z(31, 35, 39) Z(32, 33, 36) Z(35, 39, 43) 4 Z(4, 42, 46) Z(25, 3, 34) Z(37, 38, 43) Z(27, 3, 32) Z(3, 35, 4) Z(3, 36, 4) Table 7 The transportation time by goods train (η ij1 ) Mines\Cities Z(16, 2, 25) Z(1, 12, 14) Z(8, 12, 13) Z(2, 23, 25) Z(18, 2, 23) Z(1, 13, 17) 2 Z(18, 22, 26) Z(1, 13, 15) Z(11, 13, 15) Z(2, 23, 26) Z(15, 2, 24) Z(16, 19, 24) 3 Z(2, 25, 28) Z(9, 12, 15) Z(8, 13, 15) Z(21, 24, 26) Z(12, 15, 18) Z(23, 25, 28) 4 Z(16, 19, 23) Z(9, 1, 12) Z(12, 15, 16) Z(17, 2, 24) Z(1, 15, 18) Z(2, 24, 26) 15 Page 17 of 25

19 Table 8 The transportation time by cargo ship (η ij2 ) Mines\Cities Z(18, 21, 26) Z(12, 15, 18) Z(14, 17, 2) Z(1, 14, 16) Z(19, 21, 25) Z(14, 17, 19) 2 Z(25, 27, 3) Z(22, 25, 27) Z(14, 16, 18) Z(18, 22, 25) Z(11, 14, 16) Z(12, 15, 19) 3 Z(22, 26, 28) Z(22, 25, 3) Z(13, 16, 2) Z(16, 18, 2) Z(18, 21, 24) Z(1, 13, 16) 4 Z(12, 15, 19) Z(22, 24, 29) Z(1, 14, 17) Z(1, 15, 17) Z(18, 21, 22) Z(12, 15, 18) In order to solve the proposed models in which the parameters are given above, we employ the developed optimization software package LINGO to obtain the optimal solutions. In the following, we will test the solutions generated by expected value goal programming and chance-constrained goal programming, respectively Solutions Generated by Expected Value Goal Programming According to Theorem 5.1, the expected value goal programming model for the UBSTP is equivalent to the following one lexmin subject to: 4 { d + 1, d + 2 } j=1 k=1 4 i=1 k=1 4 i=1 j=1 2 x ijk 2 y ijk 2 x ijk 2 x ijk 6 x ijk ξ ijk (α)dα + d 1 d+ 1 = 1972 η ijk (α)dα + d 2 d+ 2 = 142 ã i (α)dα, i = 1, 2, 3, 4 bj (α)dα, j = 1, 2, 3, 4, 5, 6 c k (α)dα, k = 1, 2 x ijk, y ijk {, 1}, i = 1, 2, 3, 4; j = 1, 2, 3, 4, 5, 6; k = 1, 2 d 1, d+ 1, d 2, d+ 2, where y ijk is a function defined by y ijk = 1 if x ijk >, or y ijk = otherwise. In order to find an optimal transportation plan, the objective function of Model (6) is rewritten as min x,y P 1 d P 2d + 2, where P i, i = 1, 2, are the preemptive priority factors which express the relative importance of various targets, respectively, with P 1 P 2. By using mathematical software LINGO, we get the optimal value of { d + 1, d + } 2 is {,.9375}, the corresponding transportation plan is x121 = 3.25, x 151 = 15.25, x 161 = 12.75, (6) x 211 = 12.75, x 331 = 1.75, x 431 =.75, x 142 = 1.25, x 322 = 11.5, x 442 = 12, and the values of other decision variables are equal to. That is to say, according to the optimal transportation plan, the first objective can be achieved, but the second objective cannot be achieved. For better understanding the optimal transportation plan, we summarize the detailed optimal solution for expected value goal programming model in Table 9. We describe the optimal transportation plan from the coal mines to the cities by different conveyances by a network diagram which is shown in Figure Page 18 of 25

20 Table 9 The optimal solution for the expected value goal programming model (6) Mines\Cities k = k = Coal Mines Cities Goods Train Coal Mines Cities Cargo Ship Figure 1: Network diagram for the expected value goal programming model (6) Solutions Generated by Chance-constrained Goal Programming Assume that the decision makers give the priority structure as follows. (1) The total transportation cost is not more than 1972 at a confidence level.9. (2) The total transportation time does not exceed 142 at a confidence level.9. The other confidence levels are set in the following way: α i =.9, β j =.9, γ k =.9, for i = 1, 2,..., m, j = 1, 2,..., n and k = 1, 2,..., l. According to Theorem 5.2, the chance-constrained goal programming model for the UBSTP is equivalent to the following one lexmin { d + 1, } d+ 2 subject to: x ijk ξ ijk (.9) 1972 d j=1 k=1 (.9) bj 4 6 i=1 j=1 2 y ijk η ijk (.9) 142 d + 2 x ijk ã i (.1), i = 1, 2, 3, 4 4 i=1 k=1 2 x ijk, j = 1, 2, 3, 4, 5, 6 x ijk c k (.1), k = 1, 2 x ijk, y ijk {, 1}, i = 1, 2, 3, 4; j = 1, 2, 3, 4, 5, 6; k = 1, 2 d + 1, d+ 2, (7) where y ijk is a function defined by y ijk = 1 if x ijk >, or y ijk = otherwise. 17 Page 19 of 25

21 Similarly, the objective function of the model (7) can also be expressed by min x,y P 1 d P 2d + 2, where P i, i = 1, 2, are the preemptive priorities assigned to the two objectives, respectively, with P 1 P 2, which express the relative importance of various targets. Taking by mathematical software LINGO, we get the optimal value of { d + 1, } d+ 2 is {, }, the corresponding transportation plan is x121 = 3.2, x 151 = 5.4, x 161 = 15.4, x 211 = 8.1, x 411 = 4.4, x 451 = 5.49, x 232 = 1.6, x 322 = 8.8, x 442 = 1.4, and the values of other decision variables are equal to. That is to say, according to the optimal transportation plan, the first objective can be achieved, but the second objective cannot be achieved. In fact, for different confidence levels, the optimal solution may be different. For better understanding the optimal transportation plan, we summarize the detailed optimal solution for chance-constrained goal programming model in Table 1. We also depict the optimal transportation plan from the coal mines to the cities by different conveyances by a network diagram which is shown in Figure 2. Table 1 The optimal solution for the chance-constrained goal programming model (7) Mines\Cities k = k = Coal Mines Coal Mines Cities Cities Goods Train Cargo Ship Figure 2: Network diagram for the chance-constrained goal programming model (7) In the above examples, we can see that the expected value goal programming model and chanceconstrained goal programming model are employed to deal with coal transportation problem under an uncertain environment, respectively. It is noteworthy that the main focus of this paper is to present the different decision-making methods for bicriteria solid transportation problem under an uncertain environment. From the results shown in Figures 1 and 2, we may find that there exists a relatively large difference between the two solutions corresponding to two kinds of uncertain goal programming models. This is essentially due to the fact that different ranking criteria transform their corresponding models with particular capacity constraints, leading to different best solutions. Generally, it is not easy 18 Page 2 of 25

22 to conclude which model is better in the process of decision-making. Actually, the usage of different models are completely based on the decision-makers preferences. As mentioned in Section 5, the optimal objective value of the chance-constrained goal programming model is dependent on the values of confidence levels α, β, α i, β j and γ k for i = 1, 2,..., m, j = 1, 2,..., n and k = 1, 2,..., l. Thus it is meaningful to conduct a sensitivity analysis on the optimal objective value with respect to these parameters. In the following, we investigate the parameters influence on the solution quality of Model (7). We shall discuss this issue by using two parameter groups, that is, Group A and Group B. In Group A, we set the parameters α and β as variables, and the other parameters are fixed. In Group B, we set the parameters α i, β j and γ k as variables, and the other parameters are fixed. Table 11 gives the two parameter groups and the optimal objective values obtained by Model (7). Table 11 Sensitivity analysis of the model (7) with respect to confidence levels Group No. α β α i β j γ k Optimal objective value A (,.8765) (, ) (, ) (, ) (, ) B (, ) (, ) (, ) (, 6.159) (, ) It can be seen from Table 11 that in the chance-constrained goal programming model, for different confidence levels, the optimal solution may be different. Thus, the sensitivity analysis can help the decision makers to make decisions when they are lack of basic information, such as the supply, demand, conveyance capacity, unit transportation cost, transportation time, etc. 6.2 Large-scale Experiments In this part, we further validate the applications of the models using numerical experiments by considering the UBSTP with large-scale. The independent zigzag uncertain variables ã i, b j, c k, ξ ijk and η ijk are generated according to Table 12. Table 12 The range of uncertain parameters Parameters ã i bj c k ξ ijk η ijk Range [25, 4] [5, 2] [5, 7] [15, 4] [5, 3] Similar to the mainstream literature in the area of transportation problem (Giri et al. [1], Kaur and Kumar [16], Liu et al. [26] and Yu et al. [43]), we randomly generated parameters to further show the performances of the proposed models. Take the supply capacities b j as an example. The uncertain demand data of uncertain variables b j are selected from the interval [5, 2]. Since we use the well developed software package MATLAB to solve the proposed models, the particular steps to produce the uncertain demand data are designed as follows. Step 1. Randomly generate three numbers a, b and c in the interval [5, 2]. 19 Page 21 of 25

23 Step 2. If a, b and c are not meet with the inequality a < b < c, then exchange the values of a, b and c such that the inequality a < b < c holds. Step 3. Treating the zigzag uncertain variables Z(a, b, c) as the uncertain demand data of uncertain variables b j. The procedures of generating the other relevant uncertain variables are similar to the above steps. In this example, we employe the model (4) as an experimental model. In order to show the applications of the proposed models with large size, 9 groups numerical experiments with different numbers of coal mines and cities as well as conveyances are designed. For each experiment, we list the goal targets C and T in the fifth column of Table 13. The corresponding objective values are also listed in the last column of Table 13. Table 13 The goal targets and the objective values of different cases No. Source Destination Conveyance (C, T ) (d + 1, d+ 2 ) (315, 223) (,.8975) (479, 32) (,.8965) (64, 425) (,.9355) (9, 632) (,.897) (12,3, 72) (,.9365) (15,, 85) (,.937) (32,, 24,2) (,.9375) (96,, 34,5) (,.941) (15,, 72,) (,.9372) It is easy to see from Table 13 that when the numbers of coal mines and cities are increased, the objective values are close to the result which is obtained in the experiment of expected value goal programming model in Subsection 6.1. Similarly, when we do the experiments in the chance-constrained goal programming model (5), we can also obtain the similar results. This fact shows that the proposed models can effectively solve the complex bicriteria solid transportation problem under an uncertain environment with large-scale. 7 Conclusions In this paper, we have investigated the bicriteria solid transportation problem under an uncertain environment, where the supplies, demands, conveyance capacities, transportation cost and transportation time were supposed to be uncertain variables. Comparison of the present paper with the previous studies, the main contributions of this paper can be summarized as follows. (1) In traditional STP, the researchers always only consider a single objective in a model, but in the real-world decision making problems, the decision makers should take several objectives into account simultaneously. In this paper, the total transportation cost and total transportation time were considered at the same time. Because this two objectives were incommensurate and conflicting with each other, so we employed the uncertain goal programming model to construct the mathematical model. Two new models were constructed for uncertain bicriteria solid transportation problem with different modeling ideas, that is, expected value goal programming and chance-constrained goal programming. (2) It has proved that the expected value goal programming model and chance-constrained goal programming model could be respectively transformed into the corresponding deterministic equivalents 2 Page 22 of 25