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A simple parametric method to generate all optimal solutions of fuzzy solid transportation problem
Accepted Manuscript
A simple parametric method to generate all optimal solutions of Fuzzy Solid Transportation Problem Hale Gonce Kocken , Mustafa Sivri PII: DOI: Reference:
S0307-904X(15)00745-3 10.1016/j.apm.2015.10.053 APM 10895
To appear in:
Applied Mathematical Modelling
Received date: Revised date: Accepted date:
4 May 2012 7 April 2015 30 October 2015
Please cite this article as: Hale Gonce Kocken , Mustafa Sivri , A simple parametric method to generate all optimal solutions of Fuzzy Solid Transportation Problem, Applied Mathematical Modelling (2015), doi: 10.1016/j.apm.2015.10.053
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ACCEPTED MANUSCRIPT Highlights
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A new method is proposed to generate all optimal solutions of the fully fuzzy version of the solid transportation problem. Auxilary programs construct the parametric form of the solutions. A decision maker can attain the level of fuzzy supply-demand-conveyance quantities and also the fuzzy costs. With our method, it is easy to reflect qualitative factors, market analysis, economic conditions, etc. for making real life decisions
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A simple parametric method to generate all optimal solutions of Fuzzy Solid Transportation Problem Hale Gonce Kocken, Mustafa Sivri (1)
Abstract
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Department of Mathematical Engineering, Faculty of Chemistry-Metallurgy, Yildiz Technical University, Davutpasa, Istanbul, Turkey
This paper deals with the Fuzzy Solid Transportation Problem (FSTP) that has fuzzy cost coefficients, fuzzy supplies, fuzzy demands and fuzzy conveyances. All these fuzzy quantities of FSTP are assumed to be triangular fuzzy numbers. For this problem, we propose
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an approach to generate all optimal solutions parametrically. The first stage of our approach is to determine the feasibility range based on fuzzy supply-demand-conveyance quantities. In the second stage, the breaking points of fuzzy costs are found by intersecting the membership functions of the fuzzy costs. The last stage constructs the optimal solutions of FSTP by means of some proposed auxiliary programs. Also a numerical example has been provided to
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illustrate our solution procedure.
Programming.
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1. Introduction
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Keywords: Solid transportation problem, Fuzzy mathematical programming, Parametric
Transportation models play an important role in logistics and supply chain management
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for reducing cost and improving service. The classical Transportation Problem (TP) is a special type of linear programming problem. The purpose of TP is to transport the goods
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from sources to destinations. TP is also used in inventory control, manpower planning, personnel allocation, etc. The Solid Transportation Problem (STP) is a generalization of the classical TP. The
necessity of considering this special type of TP arises when there exist different type of product and also when heterogeneous transportation modes called conveyances are available for the shipments of goods. Thus, three item properties (called parameters) are taken into account in STP instead of two (source and destination). The source quantities ( ai ) may be production facilities, warehouses or supply points whereas the destination quantities ( b j )
ACCEPTED MANUSCRIPT may be consumption facilities, warehouses, sales outlets or demand points. And the conveyances may be trucks, air freights, freight trains or ships. In practice, the parameters of TP or STP are not always exactly known and stable. This imprecision may follow from the lack of exact information, changeable economic conditions, uncontrollable factors, the nature of the parameters, etc.. A frequently used way of expressing the imprecision is to use the fuzzy numbers. It enables us to consider tolerances for the model parameters in a more natural and direct way. Therefore, TP or STP with fuzzy parameters seems to be more
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realistic and reliable.
For TP, Chanas and Kuchta (1996) proposed a concept of the optimal solution of the transportation problem with fuzzy cost coefficients and an algorithm determining this solution. Das et al. (1999) focused on the solution procedure of the multi objective version of TP where all the parameters have been expressed as interval values by the decision maker
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(DM). Ahlatçıoğlu et al. (2002) proposed a model for solving the transportation problem that supply and demand quantities are given as triangular fuzzy numbers bounded from below and above, respectively. Basing on extension principle, Liu and Kao (2004) developed a procedure to derive the fuzzy objective value of the fuzzy transportation problem where the
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cost coefficients, supply and demand quantities are fuzzy numbers. Using signed distance ranking, defuzzification by signed distance, interval-valued fuzzy sets and statistical data, Chiang (2005) get the transportation problem in the fuzzy sense. Ammar and Youness (2005)
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examined the solution of multi objective TP which has fuzzy cost, source and destination parameters. They introduced the concepts of fuzzy efficient and parametric efficient
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solutions. And Barough (2011) presented a two stage procedure for fuzzy transportation problem in which the cost coefficients and supply and demand quantities are fuzzy numbers.
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Ojha et al. (2011) formulated single and multi-objective transportation models with fuzzy relations under the fuzzy logic. In that paper, the parameters of models are stated by verbal
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words such as ‘very high’, ‘high’, ‘medium’, ‘low’ and ‘very low’. And both models are solved with Real coded Genetic Algorithms. For STP, Jimenez and Verdegay (1997) propose a Genetic Algorithm based solution
method to FSTP in the case in which the fuzziness affects only in the constraint set and a fuzzy solution required. And they improve this paper in Jimenez and Verdegay (1999). Also in (1998), they analyzed two uncertain models for the STP in the names of Interval and Fuzzy STP. Both models are extensions of Interval and Fuzzy TP respectively in which three item properties are assumed. Liu (2006) develops a method that is able to derive the fuzzy
ACCEPTED MANUSCRIPT objective value of the FSTP. Based on the extension principle, the FSTP is transformed into a pair of mathematical programs that is employed to calculate the lower and upper bounds of the fuzzy total transportation cost at possibility level . From different values of , the membership function of the objective value is approximated. In Ojha et al. (2009), a multiobjective solid transportation problem is considered with generalised fuzzy transportation costs. In the proposed problem the objective functions are expressed in fuzzy equality sense through a possibility measure and the entropy function was considered as an additional
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objective function. Ojha et al. (2010)1 considered a STP for an item with fixed charge, vehicle cost and price discounted varying charge. To solve the problem, genetic algorithm which is based on roulette wheel selection, arithmetic crossover and uniform mutation was suitably developed and applied. Ojha et al. (2010)2 investigates the best optimal policy for a multicriteria solid transportation problem with nested discounts in transportation costs. Using multi
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objective genetic algorithm, first a set of pareto optimal solutions is obtained and then the best one solution is chosen using AHP.
In this paper, we focus on the solution procedure of the Solid Transportation Problem with fuzzy parameters, i.e. fuzzy cost coefficients, fuzzy supply, fuzzy demand and fuzzy
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conveyance quantities. Because of its fuzzy parameters, this transportation problem is very complicated and also due to the fuzziness in the costs it has a non-linear structure. To
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overcome these difficulties, we give an approach that generates all possible solutions of FSTP with regard to cuts of fuzzy parameters. Also, we note that our method works only for single
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objective function case and the fuzzy parameters in given triangular forms. In the first stage of our method, the feasibility range based on fuzzy supply-demand quantities is obtained. Then,
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breaking points of cost coefficients (i.e. the possible values of cost-satisfaction level that can change the optimal solution set) are found by intersecting the membership functions of cost coefficient. Finally, considering all the breaking points and the feasibility range, the optimal
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solutions of FSTP is constructed by some proposed auxiliary programs which are based on parametric programming techniques. A numerical example has been provided to illustrate our solution procedure.
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Ojha, A., Das, B., Mondal, S.K., Maiti, M., (2010), “A solid transportation problem for an item with fixed charge, vechicle cost and price discounted varying charge using genetic algorithm”, Applied Soft Computing, 10: 100–110. 2 Ojha, A., Das, B., Mondal, S.K., Maiti, M., (2010) ,“A stochastic discounted multi-objective solid transportation problem for breakable items using analytical hierarchy process”, Applied Mathematical Modeling, 34: 2256–2271.
ACCEPTED MANUSCRIPT This paper is organized as follows. After having presented brief information about fuzzy mathematics in the next section, the mathematical model of FSTP is given in Section 3. Section 4 introduces the proposed procedure with three stages. Section 5 gives an illustrative numerical example. Finally, Section 6 includes some results. 2. Preliminaries In this paper, we assumed that the parameters of FSTP are expressed as triangular fuzzy
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numbers. In this section, brief information about the fuzzy numbers especially triangular fuzzy numbers are presented. For more detailed information, the reader should check Zimmermann (1993).
Definition 2.1: A fuzzy number a is an upper semi-continuous normal and convex fuzzy
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subset of the real line R .
Definition 2.2: A fuzzy number a a1 , a2 , a3 is said to be a Triangular Fuzzy Number (TFN) if its membership function is given by
, x a1
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, a1 x a2
(1)
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, x a2
, a2 x a3
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0 x a1 a2 a1 a ( x) 1 a x 3 a3 a2 0
, x a3
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where a1 , a2 , a3 R and a1 a2 a3 . The figure of the fuzzy number a is given in Figure 1.
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In this paper, we called these ordered elements as characteristic points of a .
Figure 1- The membership function of a .
ACCEPTED MANUSCRIPT Some algebraic operations on TFNs that will use in this paper are defined as follows: Let a a1 , a2 , a3 and b b1 , b2 , b3 be TFNs. (i) Addition:
a b a1 b1 , a2 b2 , a3 b3
(2)
(ii) Multiplication with a positive crisp number k 0 :
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k a ka1 , ka2 , ka3
(3)
Definition 2.3: The set of elements that belong to the fuzzy number a at least to the degree
is called the cut of a .
L R a x X a x a , a .
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According to this definition, cut of a fuzzy number is a closed interval.
(4)
Definition 2.4: Let a fuzzy number a be positive if its membership function is such that
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a x 0, x 0 .
3. Fuzzy Solid Transportation Problem
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Assume that there are m sources (supply points), n destinations and K conveyances. At each source, let ai be the amount of homogenous products which are transported to n
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destinations to satisfy the demand for b j units of the product there. And let ek be the units of this product which can be carried by K conveyances. A cost (penalty) cijk is associated
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with transportation of a unit of the product from source i to destination j by means of the kth conveyance for the objective function F ( x) . Then, the mathematical model of STP can
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be given as follows: m
min F (x) n
K
j 1
k 1
m
K
i 1
k 1
K
j 1
k 1
c
i 1
s.t.
n
x
ijk
x
ijk
x ,
(5)
ijk ijk
ai ,
i 1, 2,
,m
bj ,
j 1, 2,
,n
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n
i 1
j 1
x
ijk
ek ,
xijk 0 , i 1, 2,
k 1, 2,
,K
, m ; j 1, 2,
, n ; k 1, 2,
,K
where xijk is the decision variable which refers to product quantity that transported from source i to destination j by means of the k-th conveyance. Here we pointed out that the constraints of STP do not always have to be in the form of equality. Each constraint
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(supply-demand-conveyance) may be in the form of inequality " " or " " . Through this diversity of constraint types, the variants of STP arise. These variants have been discussed by several authors [Patel and Tripathy(1989), Bit et al.(1993)]. This paper only aims to present a procedure to generate all optimal solutions of STP with fuzzy parameters, so we will not deal with these variants.
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In practice, the parameters of STP are not always exactly known and stable. Thus, assuming the parameters as fuzzy numbers enables to consider tolerances for them. Therefore, STP with fuzzy parameters seems to be more realistic and reliable. The solution procedure of fuzzy transportation problem / fuzzy solid transportation problem (which has fuzzy supply, demand and/or conveyance quantities) can be classified into two groups: In the first group,
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additively to the parameters, decision variables are assumed as fuzzy numbers (triangular, trapezoidal, LR type, generalized fuzzy numbers or fuzzy intervals) and accordingly a fuzzy
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solution is obtained (Ammar and Youness, (2005); Kaur and Kumar (2013)). In the second group, decision variables are assumed as crisp numbers and so a crisp solution is found for
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fuzzy supply, demand and/or conveyance quantities (Chanas, and Kuchta (1996); Liu, and Kao (2004)). The aim this group can be interpreted as finding crisp solution of problem in a
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fuzzy environment. In real life applications, being fuzzy of supply, demand and conveyance quantities are more applicable, however a DM wants to obtain a crisp solution to behave. So
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in our paper, we begin with assumptions of the second group and give a simple method to generate all optimal solutions of the problem with regard to cuts of fuzzy parameters which are in the following triangular forms: ai (ai1 , ai2 , ai3 ) ,
b j (b1j , b2j , b3j ) ,
ek (e1k , ek2 , ek3 ) ,
2 3 cijk (, cijk , cijk ).
The membership functions of ai , b j , ek can be constructed similar to (1). The membership function of the fuzzy cost cijk and its figure are given in (6) and Figure 2, respectively.
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parameters in given triangular forms. 4. Generating all optimal solutions of FSTP
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Figure 2- The membership function of the fuzzy cost cijk . Also, we note that our method works only for single objective function case and the fuzzy
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In this section, we will describe a procedure with three stages to generate all optimal solutions of FSTP. The first stage aims to determine maximal feasibility level based on fuzzy supply-demand-conveyance quantities. The second stage is obtaining the breaking points (i.e. the possible values of cost-satisfaction level that can change the optimal solution set) by intersecting the membership functions of cost coefficients. The last stage constructs
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the optimal solutions of FSTP by means of parametric programming techniques.
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4.1. Feasibility range based on fuzzy supply-demand-conveyance quantities The first stage of our procedure is to determine the feasibility range based on fuzzy supplydemand-conveyance quantities by means of these fuzzy quantities’ cuts. Using (4), the
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cut of the fuzzy supply-demand and conveyance quantities are:
ai ai , ai ai1 ai2 ai1
b
, b
bj
L
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j
, ai3 ai3 ai2 ,
(6.a)
b1 b2 b1 , b3 b3 b 2 , j j j j j j
(6.b)
R
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L
j
R
ek ek , ek e1j e2j e1j L
R
, e3j e3j e2j .
(6.c)
We note that value represents the possibility level of fuzzy supply-demand-conveyance quantities. Using (6.a), supply constraints can be rearranged as follows:
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K
j 1 k 1
ai xijk ai1 ai2 ai1 , ai3 ai3 ai2 , i 1, 2, n
x
ijk
K
j 1 k 1
ai1 ai2 ai1
n
K
x j 1 k 1
ijk
ai3 ai3 ai2 , i 1, 2,
, m 0,1 , m 0,1
Similarly, demand-conveyance constraints can be constituted using (6.b) and (6.c). Thus,
K
j 1
k 1
c
i 1
n
n
K
x
s.t.
j 1
k 1
m
K
ijk
L R ai , ai ,
i 1, 2,
, m , 0,1
, b
j 1, 2,
, n , 0,1
b ijk j
x i 1
k 1
m
n
x i 1
j 1
ijk
x ,
ijk ijk
L
j
R
,
L R ek , ek , k 1, 2,
xijk 0 , i 1, 2,
, K , 0,1
, n ; k 1, 2,
,K .
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, m ; j 1, 2,
(7)
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m
min F (x)
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the mathematical formulation of FSTP can be written as:
Considering the rearranged form of equations (6.a), (6.b) and (6.c), the maximal feasibility
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level based on fuzzy supply-demand-conveyance quantities is determined with the following problem:
(8)
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max
ai1 ai2 ai1
n
ijk
ai3 ai3 ai2 ,
i 1, 2,
,m
ijk
b3j b3j b2j ,
j 1, 2,
,n
e3j e3j e2j ,
k 1, 2,
,K
K
x j 1 k 1
x
e1j e2j e1j
m
x
xijk 0 , i 1, 2,
, m ; j 1, 2,
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b1j b2j b1j
m
K
i 1 k 1
n
i 1 j 1
ijk
, n ; k 1, 2,
,K
0,1 .
The constraints of (8) represent the feasibility range of the transported quantities
x
ijk
j
x
ijk
i
k
and
x
ijk
i
,
k
, i.e. supply-demand-conveyance quantities, respectively. Thus (8)
j
determines the maximal feasibility level. Let the optimal solution set of (8) is , xijk . Here,
ACCEPTED MANUSCRIPT is the maximal feasibility level based on fuzzy supply-demand-conveyance quantities. If is chosen in the interval 0, , then a feasible solution can be obtained. If is chosen in the interval ,1 , then it is impossible to find a feasible solution due to some negative values of supply-demand-conveyance quantities. If costs coefficients of (7)’s objective function are crisp, for all 0, , the feasible
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solutions of (8) correspond to the cut of the fuzzy set of feasible solutions of (7). Using the optimal xijk values, the maximal value of fuzzy supply-demand-conveyance quantities are calculated as ai xijk , b j xijk , ek xijk . j
i
k
k
j
k
(9)
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As a result of lying in the interval 0, , values of ai , b j , ek will vary depending on . Therefore, they should be written in terms of as follows:
x
ai2 ai ai1 ai2 ai1 , i 1, 2,
,m
(10.a)
x
ai2 ai ai3 ai3 ai2 , i 1, 2,
,m
(10.b)
x
ai2 ai ai2 , i 1, 2,
ijk
k
ijk
j
k
ijk
k
,m
(10.c)
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j
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j
Similarly, b j and ek can be written in terms of . Let the parametric forms of ai , b j , ek
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denoted by ai , b j and ek . Before solving FSTP, it is needed providing a balance between total supply, demand and conveyance capacities. By means of adding dummy supply,
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demand or conveyance point, the balance equality
a b e i
j
j
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i
k
(11)
k
holds for FSTP. 4.2. Breaking points of fuzzy cost coefficients
The membership function of cijk is monotone decreasing in the interval 0,1 . So, there exists at least one parameter 0,1 satisfying
c (cijk ) ijk
cijk cijk1 .
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By introducing the membership function of cijk , we get the following parametric form of
cijk i 1, 2, , m; j 1, 2, , n; k 1, 2, , K :
3 cijk cijk
3 3 2 cijk cijk cijk cijk .
c c 3 ijk
2 ijk
(12)
For 0,1 , the order of costs ( cijk ) and accordingly the optimal solution set of FSTP would
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possibly change. To that end, breaking points of the costs will be determined through intersection of cijk ’s membership functions. Here breaking points are defined as the values of
0,1 that change the order of costs. We note that each breaking point will not change certainly the optimal solution set.
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Let BPC denotes the set of costs’ breaking points. Being empty of the set BPC means that there is a unique optimal solution set regardless of the costs or the value of . According to the definition of breaking points, for the interval form of BPC ’s consecutive
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elements q , q 1 , the optimal solution of FSTP would remain the same. ( q is index of the set BPC ).
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4.3 Proposed Auxilary programs
In the first two stage (Subsection 4.1 and 4.2), the maximal feasibility level based on fuzzy
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supply-demand-conveyance quantities
and the set of breaking points
BPC
are
determined. To construct the optimal solutions of FSTP, in each interval form of BPC ’s
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consecutive elements
q
, q 1 , some auxiliary programs based on parametric
programming are solved for the interval 0, . Obtained results will depend on the parameter
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of supply-demand-conveyance quantities’ satisfactory parameter owing to the original constraints of (7). While variations of change the optimal solution, non-negativity constraints may be violated. The 0, values that violated the feasibility of optimal solution are called as breaking points of supply-demand-conveyance quantities. If a breaking point of are found, relevant variable is removed from the basis and a new variable enters into the basis by means of auxiliary programs. Thus, a new optimal solution is determined.
ACCEPTED MANUSCRIPT These operations will be continued until the maximal feasibility level based on fuzzy supplydemand-conveyance quantities. The procedure of obtaining all optimal solutions can be given with following algorithm: Step 0: (Initialization) Set q 0 . Step 1: If q 1, STOP. Else choose a representative point from the interval q , q 1 .
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Set q 0 . Solve the crisp STP for , q . Let denote the optimal solution of this crisp q* problem as xijk . Determine variable index sets BV and NBV . The set BV is the index set
of non-negative variables in the optimal solution and NBV V BV where V is the set of all indexes of the variables xijk .
We note that, since the optimal solution set doesn’t change between consecutive
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breaking points of costs, a representative point can be chosen arbitrarily for each interval. Substituting this value in costs, the fuzzy costs of FSTP are converted to crisp form for the relevant interval.
Step 2: (AP1) Solve the following auxiliary program: m
n
K
n
K
j 1
k 1
m
K
x
s.t.
ijk
x i 1
k 1
m
n
x
i 1, 2,
,m
bj ,
j 1, 2,
,n
j 1
ijk
ek ,
k 1, 2,
,K
CE
i 1
(AP1)
ai ,
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ijk
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i 1 j 1 k 1
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min F (x) cijk xijk
is free , i, j, k BV xijk
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0 , i, j, k NBV xijk
where ai
db dai de , b j j , ek k . d d d
We note that AP1 aims to find the parametric optimal solution. Let denote the optimal q * . Determine the sets BV and NBV according to xijk q * . Update the solution of AP1 as xijk * overall optimal solution as xijk xijkq* xijk q* .
ACCEPTED MANUSCRIPT * Step 3: (Feasibility Control) Check the optimal solution xijk , i, j, k whether it
satisfies the feasibility condition for q , . * If xijk satisfies the feasibility condition, q q 1 and go to Step 1, else determine the
breaking point of and update q with the breaking point of . Let the index of variable
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which determines the breaking point of be i , j , k . Step 4: (AP2) If q , than q q 1 and go to Step 1. Else, solve the following auxiliary program for the relevant variable of q : m
n
K
min F (x) cijk xijk
(AP2)
n
K
j 1
k 1
m
K
i 1
k 1
m
n
i 1
j 1
x
ijk
x
ijk
x
ijk
ai ,
i 1, 2,
,m
b j ,
j 1, 2,
,n
ek ,
k 1, 2,
,K
M
s.t.
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i 1 j 1 k 1
is free , i, j, k BV xijk
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0 , i, j, k NBV xijk
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xi j k 0 (Leaving variable)
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0 kk 0 ,i i 0 ,j j where ai , bj , ek . 1 , i i 1 , j j 1 k k We note that AP2 aims to find the entering variable not to deteriorate the optimality
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* condition. Determine the new optimal solution xijk within the optimal solution xijk q* of
AP2. Go to Step 3. The flow diagram of Stage 3 can be summarized in Figure 3.
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q=0
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α=0
AP1
q =q +1
YES
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Feasiblity Control ?
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NO
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AP2
Figure 3- The flow chart of Stage 3.
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5. Illustrative Example:
Let us consider a FSTP with the following characteristics:
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Supplies: a1 20, 24,30 ; a2 6,8,12 ; a3 23, 27,30
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Demands: b1 35, 40, 45 ; b2 14,17, 22 ; Conveyances capacities: e1 15,16, 20 ; e2 20, 26,32 ; e3 10,15,17
b1
a1 a2 a3
,3,5 ,7,9 ,9,13
Table 1: Penalties of the objective cijk
b2
,6,7 ,9,15 ,9,16 Conveyance 1 e1
b1
,13,18 ,10,18 ,9,10
b2
,5, 7 ,8,10 , 6,8 Conveyance 2 e2
b1
,17,18 ,1,3 ,5,10
b2
,3, 4 ,9,11 ,5, 6 Conveyance 3 e3
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Stage 1: Using (6.a), (6.b) and (6.c), cuts of the fuzzy supply-demand and conveyance (conv.) quantities are constructed and thus the problem corresponding to (8) are obtained as follows: max
(13) 3
1 jk
j 1 k 1 2
3
x
23 4
3 jk
j 1 k 1 3
30 6 ,
6 2
30 3 ,
3
14 3
i 1 k 1
3
2
x i 1 j 1
10 5
3
ij1
x i 1 j 1
ij 3
3
x i 1 k 1
20 4 ,
2
3
20 6
3
2
xijk 0 , i 1, 2,3 ; j 1, 2 ; k 1, 2,3 .
0,1 .
i 2k
x i 1 j 1
17 2
2 jk
12 4 ,
ij 2
22 5 ,
32 6 ,
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15
3
j 1 k 1
xi1k 45 5 ,
35 5
2
x
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2
x
20 4
By solving (13), the maximal feasibility level based on fuzzy supply-demand-conveyance
15 0 0 0 (Conv. 3) 0 0
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8.6 0 0.5 7.3 (Conv. 2), 0 10.2
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0 0 0 0 (Conv. 1), 16.4 0
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quantities are obtained as 0.9 , and the optimal xijk values:
Using the optimal xijk values and (9), the maximal value of fuzzy supply-demand-conveyance
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quantities are calculated as ai 23.6 7.8 26.6 , b j 40.5 17.5 , ek 16.4 26.6 15 . With (9.a)-(9.c), a1 can be constructed as follows:
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20 a1 23.6 24 a1 20 4 .
After the other supply, demand and conveyance capacities are written in terms of in a similar way, it is needed to constitute total quantities:
a 49 10 , b 67 10 , e 67 10 i
i
k
j
j
k
To satisfy the balance equality (11), these summations must be made equal for 0,0.9 . To this end, a dummy supply point with the following quantity is added to problem:
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e a 67 10 49 10 18 20 a k
Thus
4
i
k
18 20
i
a 49 10 18 20 67 10 i
is obtained and (11) holds.
i
Stage 2: Parametric forms of costs that are constructed using (12) are given in Table 2. Table 2: The parametric form of the fuzzy costs. b1 b2 b2
b1
b1
b2
5 2
7
18 5
7 2
18
4
a2
9 2
15 6
18 8
10 2
3 2
11 2
a3
13 4
16 7
10
8 2
10 5
6
0
0
a4
0
0
e1
0
0
e3
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e2
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a1
The breaking point set of costs are constructed by the twice intersection of cijk ’s membership functions as follows:
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1 2 3 BPC 0, , , ,1 , q 0,1, 2,3, 4 . 3 3 4
Stage 3: We now have the maximal feasibility level and the breaking point set of costs. So
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our algorithm can be applied to this balanced problem.
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q 0 , 0 0 . Let us choose the representative point as the arithmetic mean of the boundaries 1 1 1 of interval 0 , 1 0, : 0 2 . 3 6 3
0 0 (Conv. 1), 0 0
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20 0 0 0
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1 0 0 . The optimal solution xijk0* of the crisp STP for , 0 0 is: 6
0 0 0 0 (Conv. 2), 1 13 18 0
0 6 0 0
0 0 (Conv. 3). 9 0
1 Also the objective function value is F , 0 271.33 . Variable index sets are 6
BV 1,1,1 , 3,1, 2 , 3, 2, 2 , 4,1, 2 , 2,1,3 , 3, 2,3 and NBV V BV . The auxiliary program AP1 are constructed as follows:
ACCEPTED MANUSCRIPT 4 2 3 1 min F (x) cijk xijk 6 i 1 j 1 k 1 3
j 1
k 1
4
3
i 1
k 1
4
2
i 1
j 1
ijk
x
ijk
x
ijk
ai ,
i 1, 2,3, 4
bj ,
j 1, 2
ek ,
k 1, 2,3
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2
x
s.t.
(14)
is free , i, j, k BV xijk 0 , i, j, k NBV xijk
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4 , i 1 4 , k 1 2 , i 2 5 , j 1 where ai , b j , ek 6 , k 2 . 5 , j 2 0 ,k 3 4 , i 3 20 , i 4
0* of (14) is: The optimal solution xijk
0 0 0 0 (Conveyance 2), 17 3 20 0
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0 0 (Conveyance 1), 0 0
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4 0 0 0
0 8 2 0 (Conveyance 3). 0 10 0 0
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* Using the equation xijk xijkq* xijk 0* , the parametric form of the corresponding solution
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is given in Table 3.
a1
20 4 a2 6 2 a3
23 4 a4 18 20
Table 3: The parametric solution for 1 6 . b1 b2 b1 b1 b2 b2 45 5 22 5 20 - 4α 8α 7 18 5 7 2 18 4 5 2 6+2α 9 2 15 6 18 8 10 2 3 2 11 2 1+17α 13-3α 9-10α 13 4 16 7 10 10 5 6 8 2 18-20α 0 0 0 0 0 0 e2 32 6 e1 20 4 e3 15
ACCEPTED MANUSCRIPT 1 and the corresponding objective value is F , 271.33 103.17 . The optimal 6 solution given in Table 3 satisfies the feasibility condition for 0,0.9 . Thus this solution 1 is valid for the interval 0 , 1 0, and 0,0.9 . 3 1 2 For the next interval 1 , 2 , , the representative point is obtained as 3 3
1* of the crisp STP for , 1 0 2 . 1 0 . The optimal solution xijk 3 3 2 2 1
2
is: 0 0 (Conv. 1), 0 0
0 0 0 0 (Conv. 2), 0 14 18 0
0 6 1 0
0 0 (Conv. 3) 8 0
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20 0 0 0
1
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1
1 Also the objective function value is F , 0 241.50 . Variable index sets are 2 BV 1,1,1 , 3, 2, 2 , 4,1, 2 , 2,1,3 , 3,1,3 , 3, 2,3 and NBV V BV . The auxiliary
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program AP1 are constructed as follows:
3
j 1
k 1
x 4
ijk
3
x k 1
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i 1
4
2
i 1
j 1
ijk
x
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ai ,
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s.t.
2
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4 2 3 1 min F (x) cijk xijk 2 i 1 j 1 k 1
ijk
i 1, 2,3, 4
bj ,
j 1, 2
ek ,
k 1, 2,3
is free , i, j, k BV xijk
0 , i, j, k NBV xijk
4 , i 1 4 , k 1 2 , i 2 5 , j 1 where ai , b j , ek 6 , k 2 . 5 , j 2 0 ,k 3 4 , i 3 20 , i 4
1* of (15) is: The optimal solution xijk
(15)
ACCEPTED MANUSCRIPT 4 0 0 0
0 0 (Conveyance 1), 0 0
0 0 0 0 (Conveyance 2), 0 14 20 0
8 0 2 0 (Conveyance 3) 17 27 0 0
And the parametric form of the corresponding solution is given in Table 4. Variable index sets are BV 1,1,1 , 3, 2, 2 , 4,1, 2 , 1, 2,3 , 2,1,3 , 3,1,3 , 3, 2, 3 and NBV V BV .
20 4 a2
15 6
9 2
6 2 a3 23 4 a4
13 4
16 7
18 20
0
0
18 8
b2
8α 4
10 2
11 2
14 + 14α 8 2
1 + 17α 10 5
8 -27α 6
0
0
10 18 - 20α 0 0 e2 32 6
e1 20 4
b1
18 6 + 2α 3 2
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a1
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Table 4: The parametric solution for 1 2 . b1 b2 b1 b2 45 5 22 5 20 - 4α 7 18 5 7 2 5 2
e3 15
27
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8 27 0 8
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1 1* and the corresponding objective value is F , 241.50 93 . Since x323 8 27 2 does not satisfy the feasibility condition for 0,0.9 , the breaking point of is determined as:
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The optimal solution given in Table 4 is valid for 0,0.2963 . But for 0.2963 , a new optimal solution should be obtained. To this end, AP2 is constructed for the leaving variable
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with the index i , j , k 3, 2,3 . In other words, let give an t increment in the leaving variable’s cell and create the optimal cycle with AP2: 4 2 3 1 min F (x) cijk xijk 2 i 1 j 1 k 1
s.t.
2
3
j 1
k 1
4
3
x
ijk
x i 1
k 1
ijk
ai ,
i 1, 2,3, 4
b j ,
j 1, 2
(16)
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2
i 1
j 1
x
ijk
ek ,
k 1, 2,3
is free , i, j, k BV xijk
0 , i, j, k NBV xijk
0 (Leaving variable) x323
1* of (16) is: The optimal solution xijk 0 0 (Conveyance 1), 0 0
0 0 0 0 (Conveyance 2), 1 1 0 0
0 0 1 0
0 0 (Conveyance 3) 0 0
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0 0 0 0
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0 ,i 3 0 ,j2 0 k 3 where ai , bj , ek . 1 , j 2 1 , i 3 1 k 3
1* is given in Table 5. The optimal cycle that constructed within xijk
6 2 a3 23 4 a4
13 4
15 6
16 7
0 0 e1 20 4
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18 20
9 2
7
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20 4 a2
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a1
45 5 20 - 4α 5 2
18 5
7 2
18 8 10 2 +t 14 + 14α - t 10 8 2 18 - 20α 0 0 e2 32 6
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b1
Table 5: The optimal cycle for 1 2 . b2 b1 b2 22 5
b1
b2 8α 4
18 6 + 2α 3 2 1 + 17α – t 10 5
11 2 8 -27α + t 6
0
0
e3 15
The value of increment t is determined as follows:
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8 27 t 0 t 27 8 .
Thus, new optimal solution is obtained and given in Table 6. And the corresponding objective
1 value for 0.2963,0.9 is F , 241.50 93 . 2
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20 4 a2 6 2 a3
23 4 a4 18 20
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a1
Table 6: New optimal solution for 1 2 , 0.2963 . b1 b2 b1 b1 b2 b2 45 5 22 5 20 - 4α 8α 7 18 5 7 2 18 4 5 2 6 + 2α 9 2 15 6 18 8 10 2 11 2 3 2 27α - 8 22 -13α 9 - 10α . 6 13 4 16 7 10 10 5 8 2 18 - 20α 0 0 0 0 0 0 e2 32 6 e1 20 4 e3 15
The optimal solution given in Table 6 satisfies the feasibility condition for 0.2963,0.9 .
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1 2 Thus this solution is valid for the interval 1 , 2 , and 0.2963,0.9 . 3 3 Our proposed procedure can be continued similarly. The results of the procedure are given in
Table 7.
5.1. Analysing the features of the obtained solutions
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Almost all existing methods to FSTP generate a single or a few crisp or fuzzy compromise solution for both single and multi-objective case. Some of them are Liu and Kao (2004),
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Ammar and Youness (2005), Manjot and Kumar (2013). However, the optimal distributions (solutions) of the proposed algorithm are in terms of which corresponds to the possibility
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level of fuzzy supply-demand-conveyance quantities. And also the level of costs (objective function coefficients) are expressed the parameter . Thus, our method generates a detailed
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solution table based on the level of and . This enables the DM to make a relationship between the level of quantities and the optimal distribution easily. Owing to the table of the
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optimal distributions of the proposed algorithm, the DM can guess without difficulty the possibility level of fuzzy quantities considering such as qualitative factors, market analysis, economic conditions, etc. Finally, it can be stated that our method supports considerably the application of the obtained solution for making real life decisions. 6. Conclusions This paper deals with the Fuzzy Solid Transportation Problem (FSTP) that has fuzzy cost coefficients, fuzzy supplies, fuzzy demands and fuzzy conveyances. FSTP is a wellknown problem in the literature. Existing procedures for analyzing the FSTP fall into two
ACCEPTED MANUSCRIPT general categories. These methods either generate all optimal solutions or they construct a single (or a part) compromise solution. In this paper, we mainly aim to find all optimal solutions of FSTP based on the satisfaction levels. Our procedure with three stages generates all optimal solutions corresponding to all possible values of fuzzy parameters and thus allows a more general perspective to DM. Besides that, our paper has some limitations in terms of including only one objective function and considering the fuzzy parameters in given triangular forms. For future work, we
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try our method to solve multi objective form of STP and to modify for other type fuzzy numbers. References
Ahlatcıoğlu, M., Sivri, M. ve Güzel, N., (2002), “Transportation of the fuzzy amounts using the fuzzy cost”, Journal of Marmara for Pure and Applied Sciences, 18.
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Ammar, E.E. and Youness, E. A., (2005), “Study on multiobjective transportation problem with fuzzy numbers”, Applied Mathematics and Computation, 166 : 241-253. Bit, A.K., Biswal, M.P. and Alam, S.S., (1993), “Fuzzy programming approach to multiobjective solid transportation problem”, Fuzzy Sets and Systems, 57(2): 183-194. Chanas, S. and Kuchta, D., (1996), “A concept of the optimal solution of the transportation problem with fuzzy cost coefficients”, Fuzzy Sets and Systems, 82(3): 299-305.
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Chiang, J., (2005), “The Optimal Solution of the Transportation Problem with Fuzzy Demand and Fuzzy Product”, Journal of Information Science And Engineering, 21: 439-451.
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Das, S.K., Goswami, A. ve Alam, S.S., (1999), “Multiobjective transportation problem with interval cost, source and destination parameters”, European Journal of Operational Research, 117: 100-112.
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Liu, S.T. and Kao, C., (2004) “Solving fuzzy transportation problems based on extension principle”, European Journal of Operational Research 153 (3): 661–674.
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Barough, A. H., (2011), “Fuzzy Cost Analysis in a Fuzzy Transportation System a Study of the Supply Chain Management in a General Contractor Company”, The Journal of Mathematics and Computer Science, 2(1): 186196.
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Jimenez, F. and Verdegay, J.L., (1997), “Obtaining fuzzy solutions to the fuzzy solid transportation problem with genetic algorithms”, Fuzzy Systems, Proceedings of the Sixth IEEE International Conference on , vol 3, pp.1657-1663, 1-5 Jul 1997 (doi: 10.1109/FUZZY.1997.619789) Jimenez, F. and Verdegay, J.L., (1998), “Uncertain solid transportation problems”, Fuzzy Sets and Systems, 100(1-3): 45-57. Jimenez, F. and Verdegay, J.L., (1999), “Solving fuzzy solid transportation problems by an evolutionary algorithm based parametric approach”, European Journal of Operational Research, 117(3): 485-510. Kaur, Manjot, and Amit Kumar. "Method for solving unbalanced fully fuzzy multi-objective solid minimal cost flow problems." Applied Intelligence 38.2 (2013): 239-254. Liu, S.T., (2006), “Fuzzy total transportation cost measures for fuzzy solid transportation problem”, Applied Mathematics and Computation, 174(2): 927-941.
ACCEPTED MANUSCRIPT Patel, G. and Tripathy, J., (1989), “The solid transportation problem and its variants”, International Journal Management and Systems, 5: 17-36. Ojha, A., Das, B., Mondal, S.K., Maiti, M., (2009), “An Entropy based Solid Transportation Problem for General Fuzzy Costs and Time with Fuzzy Equality”, Mathematical and Computer Modelling, 50: 166–178. Ojha, A., Das, B., Mondal, S.K., Maiti, M., (2010), “A solid transportation problem for an item with fixed charge, vechicle cost and price discounted varying charge using genetic algorithm”, Applied Soft Computing, 10: 100–110.
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Ojha, A., Das, B., Mondal, S.K., Maiti, M., (2010) ,“A stochastic discounted multi-objective solid transportation problem for breakable items using analytical hierarchy process”, Applied Mathematical Modeling, 34: 2256– 2271. Ojha, A., Mondal, S.K., Maiti, M., (2011), “Transportation policies for single and multi-objective transportation problem using fuzzy logic”, Mathematical and Computer Modelling, 53: 1637-1646.
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Zimmermann, H.-J.,(1993). Fuzzy Set Theory and its Applications, Second, Revised Edition, Sixth Printing , Kluwer Academic Publishers, Boston/ Dordrecht/ London.
ACCEPTED MANUSCRIPT Table 7: The results of FSTP in terms of ve . α
8 0, 27
x111 20 4 x312 1 17 x322 13 3
x111 20 4 x312 1 17 x322 13 3
x412 18 20 x123 8 x213 6 2 x323 9 10
x412 18 20 x123 8 x213 6 2 x323 9 10
x412 18 20 x123 8 x213 6 2 x323 9 10
x111 20 4 x322 14 14
x111 20 4 x312 27 8 x322 22 13
x111 20 4 x312 27 8 x322 22 13
x412 18 20 x123 8 x213 6 2 x313 9 10
x412 18 20 x123 8 x213 6 2 x313 9 10
x111 20 4 x122 27 8 x322 22 13
x111 20 4
x412 18 20 x123 8 x213 6 2 x313 1 17 x323 8 27
1 2 3 , 3
x111 20 4 x322 14 14
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x412 18 20 x123 8 x213 6 2 x313 1 17 x323 8 27
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2 3 3 , 4
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CE
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x111 20 4 x322 14 14
3 4 ,1
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1 0, 3
8 9 19 , 10
x111 20 4 x312 1 17 x322 13 3
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γ
8 8 27 , 19
x412 18 20 x123 8 x213 6 2 x313 1 17 x323 8 27
x412 18 20 x123 8 19 x213 6 2 x313 1 17 x111 20 4 x122 27 8 x322 22 13
x412 18 20 x123 8 19 x213 6 2 x313 1 17
x122 8 , x312 19 8 x322 22 13 x412 18 20 x213 6 2 x313 9 2 x111 20 4
x122 8 , x312 19 8 x322 22 13 x412 18 20 x213 6 2 x313 9 2
A simple parametric method to generate all optimal solutions of Fuzzy Solid Transportation Problem Hale Gonce Kocken , Mustafa Sivri PII: DOI: Reference:
S0307-904X(15)00745-3 10.1016/j.apm.2015.10.053 APM 10895
To appear in:
Applied Mathematical Modelling
Received date: Revised date: Accepted date:
4 May 2012 7 April 2015 30 October 2015
Please cite this article as: Hale Gonce Kocken , Mustafa Sivri , A simple parametric method to generate all optimal solutions of Fuzzy Solid Transportation Problem, Applied Mathematical Modelling (2015), doi: 10.1016/j.apm.2015.10.053
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ACCEPTED MANUSCRIPT Highlights
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A new method is proposed to generate all optimal solutions of the fully fuzzy version of the solid transportation problem. Auxilary programs construct the parametric form of the solutions. A decision maker can attain the level of fuzzy supply-demand-conveyance quantities and also the fuzzy costs. With our method, it is easy to reflect qualitative factors, market analysis, economic conditions, etc. for making real life decisions
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A simple parametric method to generate all optimal solutions of Fuzzy Solid Transportation Problem Hale Gonce Kocken, Mustafa Sivri (1)
Abstract
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Department of Mathematical Engineering, Faculty of Chemistry-Metallurgy, Yildiz Technical University, Davutpasa, Istanbul, Turkey
This paper deals with the Fuzzy Solid Transportation Problem (FSTP) that has fuzzy cost coefficients, fuzzy supplies, fuzzy demands and fuzzy conveyances. All these fuzzy quantities of FSTP are assumed to be triangular fuzzy numbers. For this problem, we propose
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an approach to generate all optimal solutions parametrically. The first stage of our approach is to determine the feasibility range based on fuzzy supply-demand-conveyance quantities. In the second stage, the breaking points of fuzzy costs are found by intersecting the membership functions of the fuzzy costs. The last stage constructs the optimal solutions of FSTP by means of some proposed auxiliary programs. Also a numerical example has been provided to
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illustrate our solution procedure.
Programming.
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1. Introduction
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Keywords: Solid transportation problem, Fuzzy mathematical programming, Parametric
Transportation models play an important role in logistics and supply chain management
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for reducing cost and improving service. The classical Transportation Problem (TP) is a special type of linear programming problem. The purpose of TP is to transport the goods
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from sources to destinations. TP is also used in inventory control, manpower planning, personnel allocation, etc. The Solid Transportation Problem (STP) is a generalization of the classical TP. The
necessity of considering this special type of TP arises when there exist different type of product and also when heterogeneous transportation modes called conveyances are available for the shipments of goods. Thus, three item properties (called parameters) are taken into account in STP instead of two (source and destination). The source quantities ( ai ) may be production facilities, warehouses or supply points whereas the destination quantities ( b j )
ACCEPTED MANUSCRIPT may be consumption facilities, warehouses, sales outlets or demand points. And the conveyances may be trucks, air freights, freight trains or ships. In practice, the parameters of TP or STP are not always exactly known and stable. This imprecision may follow from the lack of exact information, changeable economic conditions, uncontrollable factors, the nature of the parameters, etc.. A frequently used way of expressing the imprecision is to use the fuzzy numbers. It enables us to consider tolerances for the model parameters in a more natural and direct way. Therefore, TP or STP with fuzzy parameters seems to be more
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realistic and reliable.
For TP, Chanas and Kuchta (1996) proposed a concept of the optimal solution of the transportation problem with fuzzy cost coefficients and an algorithm determining this solution. Das et al. (1999) focused on the solution procedure of the multi objective version of TP where all the parameters have been expressed as interval values by the decision maker
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(DM). Ahlatçıoğlu et al. (2002) proposed a model for solving the transportation problem that supply and demand quantities are given as triangular fuzzy numbers bounded from below and above, respectively. Basing on extension principle, Liu and Kao (2004) developed a procedure to derive the fuzzy objective value of the fuzzy transportation problem where the
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cost coefficients, supply and demand quantities are fuzzy numbers. Using signed distance ranking, defuzzification by signed distance, interval-valued fuzzy sets and statistical data, Chiang (2005) get the transportation problem in the fuzzy sense. Ammar and Youness (2005)
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examined the solution of multi objective TP which has fuzzy cost, source and destination parameters. They introduced the concepts of fuzzy efficient and parametric efficient
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solutions. And Barough (2011) presented a two stage procedure for fuzzy transportation problem in which the cost coefficients and supply and demand quantities are fuzzy numbers.
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Ojha et al. (2011) formulated single and multi-objective transportation models with fuzzy relations under the fuzzy logic. In that paper, the parameters of models are stated by verbal
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words such as ‘very high’, ‘high’, ‘medium’, ‘low’ and ‘very low’. And both models are solved with Real coded Genetic Algorithms. For STP, Jimenez and Verdegay (1997) propose a Genetic Algorithm based solution
method to FSTP in the case in which the fuzziness affects only in the constraint set and a fuzzy solution required. And they improve this paper in Jimenez and Verdegay (1999). Also in (1998), they analyzed two uncertain models for the STP in the names of Interval and Fuzzy STP. Both models are extensions of Interval and Fuzzy TP respectively in which three item properties are assumed. Liu (2006) develops a method that is able to derive the fuzzy
ACCEPTED MANUSCRIPT objective value of the FSTP. Based on the extension principle, the FSTP is transformed into a pair of mathematical programs that is employed to calculate the lower and upper bounds of the fuzzy total transportation cost at possibility level . From different values of , the membership function of the objective value is approximated. In Ojha et al. (2009), a multiobjective solid transportation problem is considered with generalised fuzzy transportation costs. In the proposed problem the objective functions are expressed in fuzzy equality sense through a possibility measure and the entropy function was considered as an additional
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objective function. Ojha et al. (2010)1 considered a STP for an item with fixed charge, vehicle cost and price discounted varying charge. To solve the problem, genetic algorithm which is based on roulette wheel selection, arithmetic crossover and uniform mutation was suitably developed and applied. Ojha et al. (2010)2 investigates the best optimal policy for a multicriteria solid transportation problem with nested discounts in transportation costs. Using multi
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objective genetic algorithm, first a set of pareto optimal solutions is obtained and then the best one solution is chosen using AHP.
In this paper, we focus on the solution procedure of the Solid Transportation Problem with fuzzy parameters, i.e. fuzzy cost coefficients, fuzzy supply, fuzzy demand and fuzzy
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conveyance quantities. Because of its fuzzy parameters, this transportation problem is very complicated and also due to the fuzziness in the costs it has a non-linear structure. To
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overcome these difficulties, we give an approach that generates all possible solutions of FSTP with regard to cuts of fuzzy parameters. Also, we note that our method works only for single
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objective function case and the fuzzy parameters in given triangular forms. In the first stage of our method, the feasibility range based on fuzzy supply-demand quantities is obtained. Then,
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breaking points of cost coefficients (i.e. the possible values of cost-satisfaction level that can change the optimal solution set) are found by intersecting the membership functions of cost coefficient. Finally, considering all the breaking points and the feasibility range, the optimal
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solutions of FSTP is constructed by some proposed auxiliary programs which are based on parametric programming techniques. A numerical example has been provided to illustrate our solution procedure.
1
Ojha, A., Das, B., Mondal, S.K., Maiti, M., (2010), “A solid transportation problem for an item with fixed charge, vechicle cost and price discounted varying charge using genetic algorithm”, Applied Soft Computing, 10: 100–110. 2 Ojha, A., Das, B., Mondal, S.K., Maiti, M., (2010) ,“A stochastic discounted multi-objective solid transportation problem for breakable items using analytical hierarchy process”, Applied Mathematical Modeling, 34: 2256–2271.
ACCEPTED MANUSCRIPT This paper is organized as follows. After having presented brief information about fuzzy mathematics in the next section, the mathematical model of FSTP is given in Section 3. Section 4 introduces the proposed procedure with three stages. Section 5 gives an illustrative numerical example. Finally, Section 6 includes some results. 2. Preliminaries In this paper, we assumed that the parameters of FSTP are expressed as triangular fuzzy
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numbers. In this section, brief information about the fuzzy numbers especially triangular fuzzy numbers are presented. For more detailed information, the reader should check Zimmermann (1993).
Definition 2.1: A fuzzy number a is an upper semi-continuous normal and convex fuzzy
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subset of the real line R .
Definition 2.2: A fuzzy number a a1 , a2 , a3 is said to be a Triangular Fuzzy Number (TFN) if its membership function is given by
, x a1
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, a1 x a2
(1)
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, x a2
, a2 x a3
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0 x a1 a2 a1 a ( x) 1 a x 3 a3 a2 0
, x a3
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where a1 , a2 , a3 R and a1 a2 a3 . The figure of the fuzzy number a is given in Figure 1.
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In this paper, we called these ordered elements as characteristic points of a .
Figure 1- The membership function of a .
ACCEPTED MANUSCRIPT Some algebraic operations on TFNs that will use in this paper are defined as follows: Let a a1 , a2 , a3 and b b1 , b2 , b3 be TFNs. (i) Addition:
a b a1 b1 , a2 b2 , a3 b3
(2)
(ii) Multiplication with a positive crisp number k 0 :
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k a ka1 , ka2 , ka3
(3)
Definition 2.3: The set of elements that belong to the fuzzy number a at least to the degree
is called the cut of a .
L R a x X a x a , a .
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According to this definition, cut of a fuzzy number is a closed interval.
(4)
Definition 2.4: Let a fuzzy number a be positive if its membership function is such that
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a x 0, x 0 .
3. Fuzzy Solid Transportation Problem
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Assume that there are m sources (supply points), n destinations and K conveyances. At each source, let ai be the amount of homogenous products which are transported to n
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destinations to satisfy the demand for b j units of the product there. And let ek be the units of this product which can be carried by K conveyances. A cost (penalty) cijk is associated
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with transportation of a unit of the product from source i to destination j by means of the kth conveyance for the objective function F ( x) . Then, the mathematical model of STP can
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be given as follows: m
min F (x) n
K
j 1
k 1
m
K
i 1
k 1
K
j 1
k 1
c
i 1
s.t.
n
x
ijk
x
ijk
x ,
(5)
ijk ijk
ai ,
i 1, 2,
,m
bj ,
j 1, 2,
,n
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n
i 1
j 1
x
ijk
ek ,
xijk 0 , i 1, 2,
k 1, 2,
,K
, m ; j 1, 2,
, n ; k 1, 2,
,K
where xijk is the decision variable which refers to product quantity that transported from source i to destination j by means of the k-th conveyance. Here we pointed out that the constraints of STP do not always have to be in the form of equality. Each constraint
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(supply-demand-conveyance) may be in the form of inequality " " or " " . Through this diversity of constraint types, the variants of STP arise. These variants have been discussed by several authors [Patel and Tripathy(1989), Bit et al.(1993)]. This paper only aims to present a procedure to generate all optimal solutions of STP with fuzzy parameters, so we will not deal with these variants.
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In practice, the parameters of STP are not always exactly known and stable. Thus, assuming the parameters as fuzzy numbers enables to consider tolerances for them. Therefore, STP with fuzzy parameters seems to be more realistic and reliable. The solution procedure of fuzzy transportation problem / fuzzy solid transportation problem (which has fuzzy supply, demand and/or conveyance quantities) can be classified into two groups: In the first group,
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additively to the parameters, decision variables are assumed as fuzzy numbers (triangular, trapezoidal, LR type, generalized fuzzy numbers or fuzzy intervals) and accordingly a fuzzy
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solution is obtained (Ammar and Youness, (2005); Kaur and Kumar (2013)). In the second group, decision variables are assumed as crisp numbers and so a crisp solution is found for
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fuzzy supply, demand and/or conveyance quantities (Chanas, and Kuchta (1996); Liu, and Kao (2004)). The aim this group can be interpreted as finding crisp solution of problem in a
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fuzzy environment. In real life applications, being fuzzy of supply, demand and conveyance quantities are more applicable, however a DM wants to obtain a crisp solution to behave. So
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in our paper, we begin with assumptions of the second group and give a simple method to generate all optimal solutions of the problem with regard to cuts of fuzzy parameters which are in the following triangular forms: ai (ai1 , ai2 , ai3 ) ,
b j (b1j , b2j , b3j ) ,
ek (e1k , ek2 , ek3 ) ,
2 3 cijk (, cijk , cijk ).
The membership functions of ai , b j , ek can be constructed similar to (1). The membership function of the fuzzy cost cijk and its figure are given in (6) and Figure 2, respectively.
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parameters in given triangular forms. 4. Generating all optimal solutions of FSTP
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Figure 2- The membership function of the fuzzy cost cijk . Also, we note that our method works only for single objective function case and the fuzzy
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In this section, we will describe a procedure with three stages to generate all optimal solutions of FSTP. The first stage aims to determine maximal feasibility level based on fuzzy supply-demand-conveyance quantities. The second stage is obtaining the breaking points (i.e. the possible values of cost-satisfaction level that can change the optimal solution set) by intersecting the membership functions of cost coefficients. The last stage constructs
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the optimal solutions of FSTP by means of parametric programming techniques.
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4.1. Feasibility range based on fuzzy supply-demand-conveyance quantities The first stage of our procedure is to determine the feasibility range based on fuzzy supplydemand-conveyance quantities by means of these fuzzy quantities’ cuts. Using (4), the
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cut of the fuzzy supply-demand and conveyance quantities are:
ai ai , ai ai1 ai2 ai1
b
, b
bj
L
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j
, ai3 ai3 ai2 ,
(6.a)
b1 b2 b1 , b3 b3 b 2 , j j j j j j
(6.b)
R
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L
j
R
ek ek , ek e1j e2j e1j L
R
, e3j e3j e2j .
(6.c)
We note that value represents the possibility level of fuzzy supply-demand-conveyance quantities. Using (6.a), supply constraints can be rearranged as follows:
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K
j 1 k 1
ai xijk ai1 ai2 ai1 , ai3 ai3 ai2 , i 1, 2, n
x
ijk
K
j 1 k 1
ai1 ai2 ai1
n
K
x j 1 k 1
ijk
ai3 ai3 ai2 , i 1, 2,
, m 0,1 , m 0,1
Similarly, demand-conveyance constraints can be constituted using (6.b) and (6.c). Thus,
K
j 1
k 1
c
i 1
n
n
K
x
s.t.
j 1
k 1
m
K
ijk
L R ai , ai ,
i 1, 2,
, m , 0,1
, b
j 1, 2,
, n , 0,1
b ijk j
x i 1
k 1
m
n
x i 1
j 1
ijk
x ,
ijk ijk
L
j
R
,
L R ek , ek , k 1, 2,
xijk 0 , i 1, 2,
, K , 0,1
, n ; k 1, 2,
,K .
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, m ; j 1, 2,
(7)
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m
min F (x)
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the mathematical formulation of FSTP can be written as:
Considering the rearranged form of equations (6.a), (6.b) and (6.c), the maximal feasibility
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level based on fuzzy supply-demand-conveyance quantities is determined with the following problem:
(8)
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max
ai1 ai2 ai1
n
ijk
ai3 ai3 ai2 ,
i 1, 2,
,m
ijk
b3j b3j b2j ,
j 1, 2,
,n
e3j e3j e2j ,
k 1, 2,
,K
K
x j 1 k 1
x
e1j e2j e1j
m
x
xijk 0 , i 1, 2,
, m ; j 1, 2,
AC
CE
b1j b2j b1j
m
K
i 1 k 1
n
i 1 j 1
ijk
, n ; k 1, 2,
,K
0,1 .
The constraints of (8) represent the feasibility range of the transported quantities
x
ijk
j
x
ijk
i
k
and
x
ijk
i
,
k
, i.e. supply-demand-conveyance quantities, respectively. Thus (8)
j
determines the maximal feasibility level. Let the optimal solution set of (8) is , xijk . Here,
ACCEPTED MANUSCRIPT is the maximal feasibility level based on fuzzy supply-demand-conveyance quantities. If is chosen in the interval 0, , then a feasible solution can be obtained. If is chosen in the interval ,1 , then it is impossible to find a feasible solution due to some negative values of supply-demand-conveyance quantities. If costs coefficients of (7)’s objective function are crisp, for all 0, , the feasible
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solutions of (8) correspond to the cut of the fuzzy set of feasible solutions of (7). Using the optimal xijk values, the maximal value of fuzzy supply-demand-conveyance quantities are calculated as ai xijk , b j xijk , ek xijk . j
i
k
k
j
k
(9)
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As a result of lying in the interval 0, , values of ai , b j , ek will vary depending on . Therefore, they should be written in terms of as follows:
x
ai2 ai ai1 ai2 ai1 , i 1, 2,
,m
(10.a)
x
ai2 ai ai3 ai3 ai2 , i 1, 2,
,m
(10.b)
x
ai2 ai ai2 , i 1, 2,
ijk
k
ijk
j
k
ijk
k
,m
(10.c)
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j
M
j
Similarly, b j and ek can be written in terms of . Let the parametric forms of ai , b j , ek
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denoted by ai , b j and ek . Before solving FSTP, it is needed providing a balance between total supply, demand and conveyance capacities. By means of adding dummy supply,
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demand or conveyance point, the balance equality
a b e i
j
j
AC
i
k
(11)
k
holds for FSTP. 4.2. Breaking points of fuzzy cost coefficients
The membership function of cijk is monotone decreasing in the interval 0,1 . So, there exists at least one parameter 0,1 satisfying
c (cijk ) ijk
cijk cijk1 .
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By introducing the membership function of cijk , we get the following parametric form of
cijk i 1, 2, , m; j 1, 2, , n; k 1, 2, , K :
3 cijk cijk
3 3 2 cijk cijk cijk cijk .
c c 3 ijk
2 ijk
(12)
For 0,1 , the order of costs ( cijk ) and accordingly the optimal solution set of FSTP would
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possibly change. To that end, breaking points of the costs will be determined through intersection of cijk ’s membership functions. Here breaking points are defined as the values of
0,1 that change the order of costs. We note that each breaking point will not change certainly the optimal solution set.
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Let BPC denotes the set of costs’ breaking points. Being empty of the set BPC means that there is a unique optimal solution set regardless of the costs or the value of . According to the definition of breaking points, for the interval form of BPC ’s consecutive
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elements q , q 1 , the optimal solution of FSTP would remain the same. ( q is index of the set BPC ).
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4.3 Proposed Auxilary programs
In the first two stage (Subsection 4.1 and 4.2), the maximal feasibility level based on fuzzy
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supply-demand-conveyance quantities
and the set of breaking points
BPC
are
determined. To construct the optimal solutions of FSTP, in each interval form of BPC ’s
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consecutive elements
q
, q 1 , some auxiliary programs based on parametric
programming are solved for the interval 0, . Obtained results will depend on the parameter
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of supply-demand-conveyance quantities’ satisfactory parameter owing to the original constraints of (7). While variations of change the optimal solution, non-negativity constraints may be violated. The 0, values that violated the feasibility of optimal solution are called as breaking points of supply-demand-conveyance quantities. If a breaking point of are found, relevant variable is removed from the basis and a new variable enters into the basis by means of auxiliary programs. Thus, a new optimal solution is determined.
ACCEPTED MANUSCRIPT These operations will be continued until the maximal feasibility level based on fuzzy supplydemand-conveyance quantities. The procedure of obtaining all optimal solutions can be given with following algorithm: Step 0: (Initialization) Set q 0 . Step 1: If q 1, STOP. Else choose a representative point from the interval q , q 1 .
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Set q 0 . Solve the crisp STP for , q . Let denote the optimal solution of this crisp q* problem as xijk . Determine variable index sets BV and NBV . The set BV is the index set
of non-negative variables in the optimal solution and NBV V BV where V is the set of all indexes of the variables xijk .
We note that, since the optimal solution set doesn’t change between consecutive
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breaking points of costs, a representative point can be chosen arbitrarily for each interval. Substituting this value in costs, the fuzzy costs of FSTP are converted to crisp form for the relevant interval.
Step 2: (AP1) Solve the following auxiliary program: m
n
K
n
K
j 1
k 1
m
K
x
s.t.
ijk
x i 1
k 1
m
n
x
i 1, 2,
,m
bj ,
j 1, 2,
,n
j 1
ijk
ek ,
k 1, 2,
,K
CE
i 1
(AP1)
ai ,
PT
ijk
ED
i 1 j 1 k 1
M
min F (x) cijk xijk
is free , i, j, k BV xijk
AC
0 , i, j, k NBV xijk
where ai
db dai de , b j j , ek k . d d d
We note that AP1 aims to find the parametric optimal solution. Let denote the optimal q * . Determine the sets BV and NBV according to xijk q * . Update the solution of AP1 as xijk * overall optimal solution as xijk xijkq* xijk q* .
ACCEPTED MANUSCRIPT * Step 3: (Feasibility Control) Check the optimal solution xijk , i, j, k whether it
satisfies the feasibility condition for q , . * If xijk satisfies the feasibility condition, q q 1 and go to Step 1, else determine the
breaking point of and update q with the breaking point of . Let the index of variable
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which determines the breaking point of be i , j , k . Step 4: (AP2) If q , than q q 1 and go to Step 1. Else, solve the following auxiliary program for the relevant variable of q : m
n
K
min F (x) cijk xijk
(AP2)
n
K
j 1
k 1
m
K
i 1
k 1
m
n
i 1
j 1
x
ijk
x
ijk
x
ijk
ai ,
i 1, 2,
,m
b j ,
j 1, 2,
,n
ek ,
k 1, 2,
,K
M
s.t.
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i 1 j 1 k 1
is free , i, j, k BV xijk
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0 , i, j, k NBV xijk
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xi j k 0 (Leaving variable)
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0 kk 0 ,i i 0 ,j j where ai , bj , ek . 1 , i i 1 , j j 1 k k We note that AP2 aims to find the entering variable not to deteriorate the optimality
AC
* condition. Determine the new optimal solution xijk within the optimal solution xijk q* of
AP2. Go to Step 3. The flow diagram of Stage 3 can be summarized in Figure 3.
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q=0
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α=0
AP1
q =q +1
YES
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Feasiblity Control ?
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NO
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AP2
Figure 3- The flow chart of Stage 3.
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5. Illustrative Example:
Let us consider a FSTP with the following characteristics:
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Supplies: a1 20, 24,30 ; a2 6,8,12 ; a3 23, 27,30
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Demands: b1 35, 40, 45 ; b2 14,17, 22 ; Conveyances capacities: e1 15,16, 20 ; e2 20, 26,32 ; e3 10,15,17
b1
a1 a2 a3
,3,5 ,7,9 ,9,13
Table 1: Penalties of the objective cijk
b2
,6,7 ,9,15 ,9,16 Conveyance 1 e1
b1
,13,18 ,10,18 ,9,10
b2
,5, 7 ,8,10 , 6,8 Conveyance 2 e2
b1
,17,18 ,1,3 ,5,10
b2
,3, 4 ,9,11 ,5, 6 Conveyance 3 e3
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Stage 1: Using (6.a), (6.b) and (6.c), cuts of the fuzzy supply-demand and conveyance (conv.) quantities are constructed and thus the problem corresponding to (8) are obtained as follows: max
(13) 3
1 jk
j 1 k 1 2
3
x
23 4
3 jk
j 1 k 1 3
30 6 ,
6 2
30 3 ,
3
14 3
i 1 k 1
3
2
x i 1 j 1
10 5
3
ij1
x i 1 j 1
ij 3
3
x i 1 k 1
20 4 ,
2
3
20 6
3
2
xijk 0 , i 1, 2,3 ; j 1, 2 ; k 1, 2,3 .
0,1 .
i 2k
x i 1 j 1
17 2
2 jk
12 4 ,
ij 2
22 5 ,
32 6 ,
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15
3
j 1 k 1
xi1k 45 5 ,
35 5
2
x
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2
x
20 4
By solving (13), the maximal feasibility level based on fuzzy supply-demand-conveyance
15 0 0 0 (Conv. 3) 0 0
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8.6 0 0.5 7.3 (Conv. 2), 0 10.2
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0 0 0 0 (Conv. 1), 16.4 0
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quantities are obtained as 0.9 , and the optimal xijk values:
Using the optimal xijk values and (9), the maximal value of fuzzy supply-demand-conveyance
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quantities are calculated as ai 23.6 7.8 26.6 , b j 40.5 17.5 , ek 16.4 26.6 15 . With (9.a)-(9.c), a1 can be constructed as follows:
AC
20 a1 23.6 24 a1 20 4 .
After the other supply, demand and conveyance capacities are written in terms of in a similar way, it is needed to constitute total quantities:
a 49 10 , b 67 10 , e 67 10 i
i
k
j
j
k
To satisfy the balance equality (11), these summations must be made equal for 0,0.9 . To this end, a dummy supply point with the following quantity is added to problem:
ACCEPTED MANUSCRIPT
e a 67 10 49 10 18 20 a k
Thus
4
i
k
18 20
i
a 49 10 18 20 67 10 i
is obtained and (11) holds.
i
Stage 2: Parametric forms of costs that are constructed using (12) are given in Table 2. Table 2: The parametric form of the fuzzy costs. b1 b2 b2
b1
b1
b2
5 2
7
18 5
7 2
18
4
a2
9 2
15 6
18 8
10 2
3 2
11 2
a3
13 4
16 7
10
8 2
10 5
6
0
0
a4
0
0
e1
0
0
e3
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e2
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a1
The breaking point set of costs are constructed by the twice intersection of cijk ’s membership functions as follows:
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1 2 3 BPC 0, , , ,1 , q 0,1, 2,3, 4 . 3 3 4
Stage 3: We now have the maximal feasibility level and the breaking point set of costs. So
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our algorithm can be applied to this balanced problem.
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q 0 , 0 0 . Let us choose the representative point as the arithmetic mean of the boundaries 1 1 1 of interval 0 , 1 0, : 0 2 . 3 6 3
0 0 (Conv. 1), 0 0
AC
20 0 0 0
CE
1 0 0 . The optimal solution xijk0* of the crisp STP for , 0 0 is: 6
0 0 0 0 (Conv. 2), 1 13 18 0
0 6 0 0
0 0 (Conv. 3). 9 0
1 Also the objective function value is F , 0 271.33 . Variable index sets are 6
BV 1,1,1 , 3,1, 2 , 3, 2, 2 , 4,1, 2 , 2,1,3 , 3, 2,3 and NBV V BV . The auxiliary program AP1 are constructed as follows:
ACCEPTED MANUSCRIPT 4 2 3 1 min F (x) cijk xijk 6 i 1 j 1 k 1 3
j 1
k 1
4
3
i 1
k 1
4
2
i 1
j 1
ijk
x
ijk
x
ijk
ai ,
i 1, 2,3, 4
bj ,
j 1, 2
ek ,
k 1, 2,3
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2
x
s.t.
(14)
is free , i, j, k BV xijk 0 , i, j, k NBV xijk
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4 , i 1 4 , k 1 2 , i 2 5 , j 1 where ai , b j , ek 6 , k 2 . 5 , j 2 0 ,k 3 4 , i 3 20 , i 4
0* of (14) is: The optimal solution xijk
0 0 0 0 (Conveyance 2), 17 3 20 0
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0 0 (Conveyance 1), 0 0
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4 0 0 0
0 8 2 0 (Conveyance 3). 0 10 0 0
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* Using the equation xijk xijkq* xijk 0* , the parametric form of the corresponding solution
AC
CE
is given in Table 3.
a1
20 4 a2 6 2 a3
23 4 a4 18 20
Table 3: The parametric solution for 1 6 . b1 b2 b1 b1 b2 b2 45 5 22 5 20 - 4α 8α 7 18 5 7 2 18 4 5 2 6+2α 9 2 15 6 18 8 10 2 3 2 11 2 1+17α 13-3α 9-10α 13 4 16 7 10 10 5 6 8 2 18-20α 0 0 0 0 0 0 e2 32 6 e1 20 4 e3 15
ACCEPTED MANUSCRIPT 1 and the corresponding objective value is F , 271.33 103.17 . The optimal 6 solution given in Table 3 satisfies the feasibility condition for 0,0.9 . Thus this solution 1 is valid for the interval 0 , 1 0, and 0,0.9 . 3 1 2 For the next interval 1 , 2 , , the representative point is obtained as 3 3
1* of the crisp STP for , 1 0 2 . 1 0 . The optimal solution xijk 3 3 2 2 1
2
is: 0 0 (Conv. 1), 0 0
0 0 0 0 (Conv. 2), 0 14 18 0
0 6 1 0
0 0 (Conv. 3) 8 0
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20 0 0 0
1
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1
1 Also the objective function value is F , 0 241.50 . Variable index sets are 2 BV 1,1,1 , 3, 2, 2 , 4,1, 2 , 2,1,3 , 3,1,3 , 3, 2,3 and NBV V BV . The auxiliary
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program AP1 are constructed as follows:
3
j 1
k 1
x 4
ijk
3
x k 1
CE
i 1
4
2
i 1
j 1
ijk
x
AC
ai ,
PT
s.t.
2
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4 2 3 1 min F (x) cijk xijk 2 i 1 j 1 k 1
ijk
i 1, 2,3, 4
bj ,
j 1, 2
ek ,
k 1, 2,3
is free , i, j, k BV xijk
0 , i, j, k NBV xijk
4 , i 1 4 , k 1 2 , i 2 5 , j 1 where ai , b j , ek 6 , k 2 . 5 , j 2 0 ,k 3 4 , i 3 20 , i 4
1* of (15) is: The optimal solution xijk
(15)
ACCEPTED MANUSCRIPT 4 0 0 0
0 0 (Conveyance 1), 0 0
0 0 0 0 (Conveyance 2), 0 14 20 0
8 0 2 0 (Conveyance 3) 17 27 0 0
And the parametric form of the corresponding solution is given in Table 4. Variable index sets are BV 1,1,1 , 3, 2, 2 , 4,1, 2 , 1, 2,3 , 2,1,3 , 3,1,3 , 3, 2, 3 and NBV V BV .
20 4 a2
15 6
9 2
6 2 a3 23 4 a4
13 4
16 7
18 20
0
0
18 8
b2
8α 4
10 2
11 2
14 + 14α 8 2
1 + 17α 10 5
8 -27α 6
0
0
10 18 - 20α 0 0 e2 32 6
e1 20 4
b1
18 6 + 2α 3 2
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a1
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Table 4: The parametric solution for 1 2 . b1 b2 b1 b2 45 5 22 5 20 - 4α 7 18 5 7 2 5 2
e3 15
27
PT
8 27 0 8
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1 1* and the corresponding objective value is F , 241.50 93 . Since x323 8 27 2 does not satisfy the feasibility condition for 0,0.9 , the breaking point of is determined as:
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The optimal solution given in Table 4 is valid for 0,0.2963 . But for 0.2963 , a new optimal solution should be obtained. To this end, AP2 is constructed for the leaving variable
AC
with the index i , j , k 3, 2,3 . In other words, let give an t increment in the leaving variable’s cell and create the optimal cycle with AP2: 4 2 3 1 min F (x) cijk xijk 2 i 1 j 1 k 1
s.t.
2
3
j 1
k 1
4
3
x
ijk
x i 1
k 1
ijk
ai ,
i 1, 2,3, 4
b j ,
j 1, 2
(16)
ACCEPTED MANUSCRIPT 4
2
i 1
j 1
x
ijk
ek ,
k 1, 2,3
is free , i, j, k BV xijk
0 , i, j, k NBV xijk
0 (Leaving variable) x323
1* of (16) is: The optimal solution xijk 0 0 (Conveyance 1), 0 0
0 0 0 0 (Conveyance 2), 1 1 0 0
0 0 1 0
0 0 (Conveyance 3) 0 0
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0 0 0 0
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0 ,i 3 0 ,j2 0 k 3 where ai , bj , ek . 1 , j 2 1 , i 3 1 k 3
1* is given in Table 5. The optimal cycle that constructed within xijk
6 2 a3 23 4 a4
13 4
15 6
16 7
0 0 e1 20 4
CE
18 20
9 2
7
M
20 4 a2
PT
a1
45 5 20 - 4α 5 2
18 5
7 2
18 8 10 2 +t 14 + 14α - t 10 8 2 18 - 20α 0 0 e2 32 6
ED
b1
Table 5: The optimal cycle for 1 2 . b2 b1 b2 22 5
b1
b2 8α 4
18 6 + 2α 3 2 1 + 17α – t 10 5
11 2 8 -27α + t 6
0
0
e3 15
The value of increment t is determined as follows:
AC
8 27 t 0 t 27 8 .
Thus, new optimal solution is obtained and given in Table 6. And the corresponding objective
1 value for 0.2963,0.9 is F , 241.50 93 . 2
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20 4 a2 6 2 a3
23 4 a4 18 20
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a1
Table 6: New optimal solution for 1 2 , 0.2963 . b1 b2 b1 b1 b2 b2 45 5 22 5 20 - 4α 8α 7 18 5 7 2 18 4 5 2 6 + 2α 9 2 15 6 18 8 10 2 11 2 3 2 27α - 8 22 -13α 9 - 10α . 6 13 4 16 7 10 10 5 8 2 18 - 20α 0 0 0 0 0 0 e2 32 6 e1 20 4 e3 15
The optimal solution given in Table 6 satisfies the feasibility condition for 0.2963,0.9 .
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1 2 Thus this solution is valid for the interval 1 , 2 , and 0.2963,0.9 . 3 3 Our proposed procedure can be continued similarly. The results of the procedure are given in
Table 7.
5.1. Analysing the features of the obtained solutions
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Almost all existing methods to FSTP generate a single or a few crisp or fuzzy compromise solution for both single and multi-objective case. Some of them are Liu and Kao (2004),
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Ammar and Youness (2005), Manjot and Kumar (2013). However, the optimal distributions (solutions) of the proposed algorithm are in terms of which corresponds to the possibility
PT
level of fuzzy supply-demand-conveyance quantities. And also the level of costs (objective function coefficients) are expressed the parameter . Thus, our method generates a detailed
CE
solution table based on the level of and . This enables the DM to make a relationship between the level of quantities and the optimal distribution easily. Owing to the table of the
AC
optimal distributions of the proposed algorithm, the DM can guess without difficulty the possibility level of fuzzy quantities considering such as qualitative factors, market analysis, economic conditions, etc. Finally, it can be stated that our method supports considerably the application of the obtained solution for making real life decisions. 6. Conclusions This paper deals with the Fuzzy Solid Transportation Problem (FSTP) that has fuzzy cost coefficients, fuzzy supplies, fuzzy demands and fuzzy conveyances. FSTP is a wellknown problem in the literature. Existing procedures for analyzing the FSTP fall into two
ACCEPTED MANUSCRIPT general categories. These methods either generate all optimal solutions or they construct a single (or a part) compromise solution. In this paper, we mainly aim to find all optimal solutions of FSTP based on the satisfaction levels. Our procedure with three stages generates all optimal solutions corresponding to all possible values of fuzzy parameters and thus allows a more general perspective to DM. Besides that, our paper has some limitations in terms of including only one objective function and considering the fuzzy parameters in given triangular forms. For future work, we
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try our method to solve multi objective form of STP and to modify for other type fuzzy numbers. References
Ahlatcıoğlu, M., Sivri, M. ve Güzel, N., (2002), “Transportation of the fuzzy amounts using the fuzzy cost”, Journal of Marmara for Pure and Applied Sciences, 18.
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Ammar, E.E. and Youness, E. A., (2005), “Study on multiobjective transportation problem with fuzzy numbers”, Applied Mathematics and Computation, 166 : 241-253. Bit, A.K., Biswal, M.P. and Alam, S.S., (1993), “Fuzzy programming approach to multiobjective solid transportation problem”, Fuzzy Sets and Systems, 57(2): 183-194. Chanas, S. and Kuchta, D., (1996), “A concept of the optimal solution of the transportation problem with fuzzy cost coefficients”, Fuzzy Sets and Systems, 82(3): 299-305.
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Chiang, J., (2005), “The Optimal Solution of the Transportation Problem with Fuzzy Demand and Fuzzy Product”, Journal of Information Science And Engineering, 21: 439-451.
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Das, S.K., Goswami, A. ve Alam, S.S., (1999), “Multiobjective transportation problem with interval cost, source and destination parameters”, European Journal of Operational Research, 117: 100-112.
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Liu, S.T. and Kao, C., (2004) “Solving fuzzy transportation problems based on extension principle”, European Journal of Operational Research 153 (3): 661–674.
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Barough, A. H., (2011), “Fuzzy Cost Analysis in a Fuzzy Transportation System a Study of the Supply Chain Management in a General Contractor Company”, The Journal of Mathematics and Computer Science, 2(1): 186196.
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Jimenez, F. and Verdegay, J.L., (1997), “Obtaining fuzzy solutions to the fuzzy solid transportation problem with genetic algorithms”, Fuzzy Systems, Proceedings of the Sixth IEEE International Conference on , vol 3, pp.1657-1663, 1-5 Jul 1997 (doi: 10.1109/FUZZY.1997.619789) Jimenez, F. and Verdegay, J.L., (1998), “Uncertain solid transportation problems”, Fuzzy Sets and Systems, 100(1-3): 45-57. Jimenez, F. and Verdegay, J.L., (1999), “Solving fuzzy solid transportation problems by an evolutionary algorithm based parametric approach”, European Journal of Operational Research, 117(3): 485-510. Kaur, Manjot, and Amit Kumar. "Method for solving unbalanced fully fuzzy multi-objective solid minimal cost flow problems." Applied Intelligence 38.2 (2013): 239-254. Liu, S.T., (2006), “Fuzzy total transportation cost measures for fuzzy solid transportation problem”, Applied Mathematics and Computation, 174(2): 927-941.
ACCEPTED MANUSCRIPT Patel, G. and Tripathy, J., (1989), “The solid transportation problem and its variants”, International Journal Management and Systems, 5: 17-36. Ojha, A., Das, B., Mondal, S.K., Maiti, M., (2009), “An Entropy based Solid Transportation Problem for General Fuzzy Costs and Time with Fuzzy Equality”, Mathematical and Computer Modelling, 50: 166–178. Ojha, A., Das, B., Mondal, S.K., Maiti, M., (2010), “A solid transportation problem for an item with fixed charge, vechicle cost and price discounted varying charge using genetic algorithm”, Applied Soft Computing, 10: 100–110.
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Ojha, A., Das, B., Mondal, S.K., Maiti, M., (2010) ,“A stochastic discounted multi-objective solid transportation problem for breakable items using analytical hierarchy process”, Applied Mathematical Modeling, 34: 2256– 2271. Ojha, A., Mondal, S.K., Maiti, M., (2011), “Transportation policies for single and multi-objective transportation problem using fuzzy logic”, Mathematical and Computer Modelling, 53: 1637-1646.
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Zimmermann, H.-J.,(1993). Fuzzy Set Theory and its Applications, Second, Revised Edition, Sixth Printing , Kluwer Academic Publishers, Boston/ Dordrecht/ London.
ACCEPTED MANUSCRIPT Table 7: The results of FSTP in terms of ve . α
8 0, 27
x111 20 4 x312 1 17 x322 13 3
x111 20 4 x312 1 17 x322 13 3
x412 18 20 x123 8 x213 6 2 x323 9 10
x412 18 20 x123 8 x213 6 2 x323 9 10
x412 18 20 x123 8 x213 6 2 x323 9 10
x111 20 4 x322 14 14
x111 20 4 x312 27 8 x322 22 13
x111 20 4 x312 27 8 x322 22 13
x412 18 20 x123 8 x213 6 2 x313 9 10
x412 18 20 x123 8 x213 6 2 x313 9 10
x111 20 4 x122 27 8 x322 22 13
x111 20 4
x412 18 20 x123 8 x213 6 2 x313 1 17 x323 8 27
1 2 3 , 3
x111 20 4 x322 14 14
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x412 18 20 x123 8 x213 6 2 x313 1 17 x323 8 27
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2 3 3 , 4
AC
CE
PT
x111 20 4 x322 14 14
3 4 ,1
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1 0, 3
8 9 19 , 10
x111 20 4 x312 1 17 x322 13 3
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γ
8 8 27 , 19
x412 18 20 x123 8 x213 6 2 x313 1 17 x323 8 27
x412 18 20 x123 8 19 x213 6 2 x313 1 17 x111 20 4 x122 27 8 x322 22 13
x412 18 20 x123 8 19 x213 6 2 x313 1 17
x122 8 , x312 19 8 x322 22 13 x412 18 20 x213 6 2 x313 9 2 x111 20 4
x122 8 , x312 19 8 x322 22 13 x412 18 20 x213 6 2 x313 9 2