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A solid transportation problem for an item with fixed charge, vechicle cost and price discounted varying charge using genetic algorithm
Applied Soft Computing 10 (2010) 100–110
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A solid transportation problem for an item with fixed charge, vechicle cost and price discounted varying charge using genetic algorithm A. Ojha a, B. Das b,*, S. Mondal a, M. Maiti a a b
Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore 721102, India Department of Mathematics, Jhargram Raj College, Midnapore 721507, India
A R T I C L E I N F O
A B S T R A C T
Article history: Received 14 January 2008 Received in revised form 29 January 2009 Accepted 25 June 2009 Available online 14 July 2009
In this paper, discount in transportation cost on the basis of transportated amount is extended to a solid transportation problem. In a transportation model, the available discount is normally offered on items/ criteria, etc., in the form AUD (all unit discount) or IQD (incremental quantity discount) or combination of these two. Here transportation model is considered with fixed charges and vechicle costs where AUD, IQD or combination of AUD and IQD on the price depending upon the amount is offered and varies on the choice of origin, destination and conveyance. To solve the problem, genetic algorithm (GA) based on Roulette wheel selection, arithmetic crossover and uniform mutation has been suitably developed and applied. To illustrate the models, numerical examples have been presented. Here, different types of constraints are introduced and the corresponding results are obtained. To have better customer service, the entropy function is considered and it is displayed by a numerical example. To exhibit the efficiency of GA, another method—weighted average method for multi-objective is presented, executed on a multiobjective problem and the results of these two methods are compared. Crown Copyright ß 2009 Published by Elsevier B.V. All rights reserved.
Keywords: All unit and incremental quantity discount Genetic algorithm Solid transportation problem Fuzzy resource and demand Entropy
1. Introduction Hitchcock [8] defined the transportation problems (TPs) as one of the most common special type linear programming problems involving constraints. In TP, the total transportation cost from some wholesalers to some consumers is minimized and bounds are given on two features—availability of the sources, requirement of the destinations. The solid transportation problem (STP) is a generalization of the well-known transportation problem (TP) in which three-dimensional properties are taken into account in the objective and constraint set instead of source and destination. The STP was first stated by Shell [17]. In many industrial problems, a homogeneous product is delivered from an origin to a destination by means of different modes of transport called conveyances, such as trucks, cargo flights, goods trains, ships, etc. These conveyance are taken as the third dimension. A solid transportation problem can be converted to a classical transportation problem by considering only a single type of conveyance. In many practical applications, it is realistic to assume that the amount which can be sent on any particular route bears a fixed charge for that route. Further, when a route is altogether excluded, this can be expressed by limiting its capacity to zero. Such fixed
* Corresponding author. E-mail address: [email protected] (A. Ojha).
transportation costs may also be applied to some productionplanning models (cf. [4]). But, till now, none has considered solid transportation problem with fixed cost. In today’s highly competitive market the pressure on organizations to find the better ways to create and deliver values to customers is increasing more and more. How and when to send the products to the customers in the quantities they want in a costeffective manner has become more challenging in terms of price discount (AUD and/or IQD) on the unit transportation cost. There is a similarity to economic ordered quantity model for such a discount (cf. [14]). But no researcher considered such a price discount in STP models. Jimcncz and Verdegay [9] investigated interval multi-objective solid transportation problem via Genetic Algorithms. Based on Chanas and Kuchta [3], Das et al. [5] converted the interval number transportation problems into deterministic multi-objective problems. Bit et al. [1,2], Jimenez and Verdegay [10], Li and Lai [12] and Waiel [23] presented the fuzzy compromise programming approach to multi-objective transportation problems. Grzegorzewski [7] approximated the fuzzy number to its nearest interval. Shaocheng [19] discussed about the interval number linear programming. Omar and Samir [16] and Chanas and Kuchta [3] discussed the solution algorithm for solving the transportation problem in fuzzy environment. In the transportation model, the entropy acts as a measure of dispersal of trips between origin and destinations. So it will be more practical, if we could have minimum transportation penalties along with maximum entropy amount. Usefulness of entropy optimization
1568-4946/$ – see front matter . Crown Copyright ß 2009 Published by Elsevier B.V. All rights reserved. doi:10.1016/j.asoc.2009.06.016
A. Ojha et al. / Applied Soft Computing 10 (2010) 100–110
in transportation models as well as other models are discussed in Kapur and Kesavan [11]. The purpose of introducing entropy in transportation problem is-to get better customer service. For the solution of decision-making problems, there are some inherent difficulties in the traditional direct and gradient-based optimization techniques used for this purpose. Normally, these methods (i) are initial solution dependent, (ii) get stuck to a suboptimal solution, (iii) are not efficient in handing problems having discrete variables, (iv) cannot be efficiently used on parallel machines and (v) are not universal, rather problem dependent. To overcome these difficulties, recently genetic algorithms (GAs) are used for optimization of decision making problems. GAs [6,21] are adaptive computational procedures modelled on the mechanics of natural genetic systems. They exploit the historical information to speculate on new offspring with expected improved performance. These are executed iteratively on a set of coded solutions (called population) with three operators—selection/reproduction, crossover and mutation. One set of these three operators is known as a generation in the parlance of GA. Since a GA works simultaneously on a set of solutions, it has very little chance to get suck at local optimum. Here, the resolution of the possible search space is increased by operating on potential solutions and not on the solutions themselves. Further, this search space need not be continuous. Recently, GAs have been applied in different areas like neural network, travelling salesman, scheduling, numerical optimization Michalewicz [21], inventory [13,14], etc. The mathematical formulation of transportation problem is almost of similar form as the mathematical representation of production models with linear production cost (cf. [24,20]). As all these problems are linear programming (LP) problems with linear production cost and linear constraints, discounted LP approach may be used to solve the problems. In this paper, a solid transportation problem with discounted costs, fixed charges and vechicle costs are formulated as a liner programming problem. The discounted costs are in the form of AUD, IQD and combination of AUD and IQD. Here AUD, IQD or combination of these two discounts on purchasing price with two price breaks are considered. The transportation costs are deterministic but the resources and demands at various centres, capacities of different modes of transport between origins and destinations are crisp for problem-I and imprecise for problem-II. These solid transportation problems are solved using genetic algorithm (GA). Here, different types of constraints are introduced and the corresponding results are obtained. To bring all cells under better allotment scheme, i.e. to facilitate better customers’ service, entropy of the transported amounts is considered as an additional
C i jk
8 p > > h 1i jk > > > > > > h p1i jk R1 þ > > > > < p1i jk R1 þ ¼ ......... > > > h. . . . . . . . . > > > > p1i jk R1 þ > > h > > > : p R þ 1i jk 1
2. Assumptions and notation 2.1. Notation In this solid transportation problem the following notation (i) m = no of sources of the transportation problem. (ii) n = no of demands of the transportation problem. (iii) K = no of conveyances, i.e., different modes of the transportation problem. (iv) Oi = origins of the transportation problem. (v) D j = destination of the transportation problem. (vi) Ek = conveyances of the transportation problem. (vii) ai = amount of a homogeneous product available at ith origin. (viii) b j = demand at jth destination. (viii) ek = amount of product which can be carried by k th conveyance. (ix) f i jk = the fixed charge of the transportation problem. (x) C i jk = unit transportation cost. (xi) hðxÞ = the probability that X is in the state x. 2.2. Assumption In this solid transportation problem, the following assumptions are made. (i) The total vechicle cost sV ifsV c ¼ xi jk Gðxi jk Þ ¼ ðs þ 1ÞV Otherwise
(1)
(ii) s ¼ ½xi jk =V c , V c ¼ vechicle capacity and V = vechicle cost. (iii) The unit transformation cost under AUD scheme: 8 p1i jk if0 < xi jk < R1 > > > > p2i jk ifR1 xi jk < R2 > > <......... (2) C i jk ¼ ......... > > > > p ifR x < R > t i jk > : ti jk ðt1Þ pðtþ1Þi jk ifRt xi jk here p1i jk > p2i jk > p3i jk > > pti jk > pðtþ1Þi jk are all unit cost. (iv) Incremental quantity discount (IQD): The unit-transportation cost under incremental quantity discount (IQD) system is p1i jk for 0 < xi jk < R1 ; p2i jk for an additional quantity over R1 but less then R2 and so on, lastly pðtþ1Þi jk for any additional quantity over Rt . Thus unit transportation cost becomes i f 0 < xi jk R1
i p2i jk ðxi jk R1 Þ =xi jk
i p2i jk ðR2 R1 Þ þ p3i jk ðxi jk R2 Þ =xi jk i p2i jk ðR2 R1 Þ þ þ pti jk ðxi jk Rt1 Þ =xi jk i p2i jk ðR2 R1 Þ þ þ pðtþ1Þi jk ðxi jk Rt Þ =xi jk
objective function for two particular models. The presence of various real-life complexities, like—three dimensional nature, price discount, imprecise nature of availability and requirement of STP, various constraints entropy, etc., has been taken into account successfully by introducing GA. A numerical experiment is also performed to illustrate the use of entropy function. To exhibit the efficiency of GA, another method—weighted average method for multi-objective is presented, executed on a multi-objective problem and the results of these two methods are compared.
101
ifR1 < xi jk R2 ifR2 < xi jk R3 (3) ifRt1 < xi jk Rt ifRt < xi jk
here p1i jk > p2i jk > p3i jk > > pti jk > pðtþ1Þi jk are all unit cost. 3. Formulation of a solid transportation problem with discount cost, fixed charge and vechicle cost We assume that m origins (or sources) Oi (i = 1, 2,. . ., m), n destinations (i.e. demands) D j ð j ¼ 1; 2; . . . ; nÞ and K conveyances Ek ðk ¼ 1; 2; . . . ; KÞ. K conveyances, i.e., different modes of transport may be trucks, cargo flights, goods trains, ships, etc. Let ai be the
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A. Ojha et al. / Applied Soft Computing 10 (2010) 100–110
amount of a homogeneous product available at ith origin, b j be the demand at jth destination and ek represents the amount of product which can be carried by k th conveyance. C i jk be the cost under AUD, IQD or combination of these two systems associated with transportation of a unit product from i th source to j th destination by means of the k th conveyance. The variable xi jk represents the unknown quantity to be transported from origin Oi to destination D j by means of k th conveyance. A solid transportation problem with resources, demands, conveyances under all unit discounts(AUD), incremental quantity discount (IQD) and combination of these two systems can be represented as Problem-I (case-1):
subject to n X K X xi jk ¼ a˜ i ;
i ¼ 1; 2; 3; . . . ; m
(12)
j ¼ 1; 2; 3; . . . ; n
(13)
k ¼ 1; 2; 3; . . . ; K
(14)
j¼1 k¼1
m X K X xi jk ¼ b˜i ; i¼1 k¼1 m X n X xi jk ¼ e˜i ; i¼1 j¼1
3.1. Entropy function m X n X K X Min ZðXÞ ¼ ½C i jk xi jk þ f i jk þ Gðxi jk Þ
(4)
i¼1 j¼1 k¼1
subject to n X K X xi jk ¼ ai ;
i ¼ 1; 2; 3; . . . ; m
(5)
j¼1 k¼1
P Pn PK Let T be the transported amount, i.e., T ¼ m i¼1 j¼1 k¼1 xi jk . Consider a function FðXÞ which represents the number of possible assignment for the state X ¼ ðxi jk Þ. The (Shannon) entropy of a variable X is defined as FðXÞ= the number of ways selecting x111 from T, multiplied by the number of ways selecting x112 from T x111 ; . . ., multiplied by the number of ways selecting xmnk from T x111 x112 xmnk1 .
¼ T Cx111 :ðTx111 Cx112 :ðTx111 x112 Cx113 ðT x111 x112 xmnk1 CxmnK ¼
T! m Y n Y K Y
xi jk !
i¼1 j¼1k¼1
m X K X xi jk ¼ b j ;
j ¼ 1; 2; 3; . . . ; n
(6)
lnðFðXÞÞ
¼ lnðT!Þ
i¼1 k¼1 T
m X n X xi jk ¼ ek ;
¼ lnðe k ¼ 1; 2; 3; . . . ; K
T
i¼1 j¼1 k¼1 m X n X K X
x
i jk lnðexi jk xi jk Þ
T Þ
i¼1 j¼1 k¼1
(7)
i¼1 j¼1
m X n X K X lnðxi jk !Þ
[By using Stirlings approximation formula]
where C i jk ’s are given by (2) for AUD, by (3) for IQD and by (2)and (3) for combination of AUD, IQD. Problem-I (case-2): The equality constraints (5)–(7) can be imposed as ‘less than equality’ constraints in some cases. In that case, the objective function (4) remains the same but the changed constraints are n X K X xi jk ai ;
¼ T:lnðTÞ
m X n X K X xi jk lnðxi jk Þ i¼1 j¼1 k¼1
Therefore ðlnðFðXÞÞ=TÞ ¼ lnðTÞ ð1=TÞ
Pm Pn i¼1
j¼1
PK
k¼1
xi jk lnðxi jk Þ
Here the entropy function En ðxÞ ¼ ðlnðFðXÞÞ=TÞ. The function (Shannon) entropy can be expressed as
i ¼ 1; 2; 3; . . . ; m
(8) EnðXÞ ¼
j¼1 k¼1
X
f ðxÞ
x m X K X xi jk b j ;
where j ¼ 1; 2; 3; . . . ; n
(9) (
i¼1 k¼1
f ðxÞ ¼ m X n X xi jk ek ;
k ¼ 1; 2; 3; . . . ; K
(10)
i¼1 j¼1
Problem-II: In this case, resources, demands and capacities of transportation modes are imprecise. The formulation of this modal is the same as the problem-I except that all constraints becomes fuzzy. Thus the problem is expressed in the form: m X n X K X MinZðXÞ ¼ ½C i jk xi jk þ f i jk þ Gðxi jk Þ i¼1 j¼1 k¼1
(11)
hðxÞlnhðxÞ
ifhðxÞ 6¼ 0
0
ifhðxÞ ¼ 0
hðxÞ being the probability that X is in the state x. In transportation problem, normalizing the trip number xi jk by P Pn PK dividing the total number of trips Tð m i¼1 j¼1 k¼1 xi jk Þ, a probability distribution, hi jk ¼ xi jk =T is formulated. Therefore EnðXÞ
m X n X K X ðxi jk =TÞlnðxi jk =TÞ ¼ i¼1 j¼1 k¼1 m X n X K X
¼ lnðTÞ
1 T
xi jk lnðxi jk Þ
i¼1 j¼1 k¼1
(15)
A. Ojha et al. / Applied Soft Computing 10 (2010) 100–110
In transportation problem, this entropy function acts as a measure of dispersal of trips among origins, destinations and conveyances. It becomes more useful, if we would like to have minimum transportation costs as well as maximum entropy amount. Taking entropy function as an additional objective function, the Problem-I takes the following form:
103
˜ which Given A˜ is a fuzzy number. We find a closed interval C d ðAÞ is nearest to A˜ with respect to metric d. It can be done since each interval is also a fuzzy number with constant a-cut for all a 2 ½0; 1. ˜ Hence ðC d ðAÞÞ a ¼ ½C L ; C R . Now we have to minimize sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z 1 Z 1 ˜ ¼ ˜ C d ðAÞÞ ðAL ðaÞ C L ðaÞÞ2 da þ ðAR ðaÞ C R ðaÞÞ2 da dðA; 0
Minimize½ZðXÞ and Maximize EnðXÞ: subject to the constraintsð5Þ ð7Þ:
(15a)
and Minimize½ZðXÞ and Maximize EnðXÞ: subject to the constraintsð8Þ ð10Þ:
(15b)
0
˜ C d ðAÞÞ, ˜ it is with respect to C L and C R . In order to minimize dðA; 2 ˜ ˜ sufficient to minimize the function DðC L ; C R Þ ¼ d ðA; C d ðAÞÞ. The first partial derivatives are
dDðC L ; C R Þ ¼ 2 dC L ¼ 2
Z Z
1
AL ðaÞda þ 2C L and 0
dDðC L ; C R Þ dC R
1
AR ðaÞda þ 2C R 0
4. Fuzzy number and its interval approximation Definition: Fuzzy number: A fuzzy subset A of the real line R with membership function mA : R ! ½0; 1 is called a fuzzy number if
(i) A is normal, i.e.the supremum of mA ðxÞ is 1 8 x 2 S and (ii) A is fuzzy convex, i.e. mA ðlx1 þ ð1 lÞx2 Þ mA ðx1 Þ ^ mA ðx2 Þ 8 x1 ; x2 2 R, for l 2 ½0; 1. a-Cut of a fuzzy number: A a-cut of a fuzzy number A˜ is defined as crisp set Aa ¼ fx : mA˜ ðxÞ a; x 2 Xgwherea 2 ½0; 1 Aa is a non-empty bounded closed interval contained in X and it can be denoted by Aa ¼ ½AL ðaÞ; AR ðaÞ. AL ðaÞ and AR ðaÞ are the lower and upper bounds of the closed interval respectively. A fuzzy number A˜ with a1 ; a2 -cut Aa1 ¼ ½AL ða1 Þ; AR ða1 Þ; Aa2 ¼ ½AL ða2 Þ; AR ða2 Þ and if a2 a1 , then AL ða2 Þ AL ða1 Þ and AR ða1 Þ AR ða2 Þ. Different types of fuzzy numbers: Fuzzy numbers are represented by two types of membership functions: (a) Linear membership functions, e.g. triangular fuzzy number (TFN), trapezoidal fuzzy number, piecewise linear fuzzy number, etc. (b) Non-linear membership functions, e.g. parabolic fuzzy number (PFN), exponential fuzzy number and other non-linear fuzzy number. Trapezoidal fuzzy number: Trapezoidal fuzzy number is the fuzzy number with the membership function mA˜ ðxÞ, a continuous mapping: mA˜ ðxÞ : R ! ½0; 1 8 0 > > > > x a1 > > > > < a2 a1 mA˜ ðxÞ ¼ 1 > > a4 x > > > > > a4 a3 > : 0
for 1 < x < a1 fora1 x < a2
Solving
dDðC L ; C R Þ ¼ 0 and dC L
dDðC L ; C R Þ ¼0 dC R
R1 R1 we get, CL ¼ 0 ðAL ðaÞdaÞ and CR ¼ 0 ðAR ðaÞdaÞ. Again since,
dD2 ðCL ; CR Þ dD2 ðCL ; CR Þ ¼ 2 > 0; ¼ 2 > 0 and 2 dCL dCR2 !2
dD2 ðCL ; CR Þ dD2 ðCL ; CR Þ dD2 ðCL ; CR Þ : 2 2 dC L :dC R dC L dC R
¼ 4>0
˜ C d ðAÞÞ ˜ is global minimum. Therefore the So DðC L ; C R Þ, i.e., dðA; interval
˜ ¼ C d ðAÞ
"Z
1
AL ðaÞda;
Z
0
#
1
AR ðaÞda 0
is nearest interval approximation of fuzzy number A˜ with respect to metric d. Let A˜ ¼ ða1 ; a2 ; a3 ; a4 Þ be a fuzzy number. The a-level ˜ ¼ ½AL ðaÞ; AR ðaÞ. When A˜ is a interval of A˜ is defined as ðAÞ a trapezoidal fuzzy numbers then AL ðaÞ ¼ a1 þ aða2 a1 Þ and AR ðaÞ ¼ a4 aða4 a3 Þ; 0 < a 1. By nearest interval approximation method lower and upper limits of the interval are respectively
CL ¼
Z
1
AL da ¼ 0
CR ¼
Z
1
½a1 þ aða2 a1 Þda ¼ 0
Z
1
AR da ¼ 0
Z
1 ða2 þ a1 Þ 2
1
½a4 aða4 a3 Þda ¼
0
and
1 ða3 þ a4 Þ 2
Therefore, the nearest crisp interval number considering A˜ as a trapezoidal fuzzy number is ½ða1 þ a2 Þ=2; ða3 þ a4 Þ=2. 5. Reduced crisp model
fora2 x a3 fora3 < x a4 fora4 < x < 1
The nearest interval approximation of a fuzzy number: Here a fuzzy number is approximated by a corresponding crisp interval. ˜ two fuzzy numbers with a-cuts are ½AL ðaÞ; AR ðaÞ Suppose A˜ and B, and ½BL ðaÞ; BR ðaÞ respectively. Then the distance between A˜ and B˜ is
Let the fuzzy numbers a˜ i ; b˜ j , and e˜k are approximated to ½aLi ; aRi ; ½bLj ; bRj , and ½eLk ; eRk respectively. Then the earlier transportation model (Problem-II) takes the following form: Problem-IIa: MinimizeZðXÞ ¼
m X n X K X ½C i jk xi jk þ f i jk þ Gðxi jk Þ
(16)
i¼1 j¼1 k¼1
subject to
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z 1 Z 1 ˜ BÞ ˜ ¼ ðAL ðaÞ BL ðaÞÞ2 da þ ðAR ðaÞ BR ðaÞÞ2 da dðA; 0
0
n X K X xi jk ¼ ½aLi ; aRi j¼1 k¼1
(17)
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A. Ojha et al. / Applied Soft Computing 10 (2010) 100–110
m X K X xi jk ¼ ½bLj ; bRj
(18)
i¼1 k¼1 m X n X xi jk ¼ ½eLk ; eRk
(19)
i¼1 j¼1
The constraints (17)–(19) can be written in the following form: aLi
n X K X xi jk aRi
(20)
j¼1 k¼1
bLj
m X K X xi jk bRj
(21)
6.1.3. Mutation The mutation operation is needed after the crossover operation to maintain population diversity and recover possible loss of some good characteristics. It is also consist of two steps: (i) Selection for mutation: For each solution of P 1 ðTÞ generate a random number r from the range [0, . . . ;1]. If r < pm then the solution is taken for mutation, where pm is the probability of mutation. (ii) Mutation process: To mutate a solution X ¼ ðx1 ; x2 ; . . . ; xK Þ select a random integer r in the range ½1; . . . ; K. Then replace xr by randomly generated value within the boundary of r th component of X.
i¼1 k¼1 m X n X xi jk eRk
6.2. Implementation of GA for multi-objective (22)
i¼1 j¼1
6. Solution procedures 6.1. GA for single objective In this paper the proposed solid transportation problem, (4) with (5)–(7) or (16) with (20)–(22) is solved by genetic algorithm. A genetic algorithm is a heuristic search process for optimization that resembles natural selection. GAs has been applied successfully in different areas. Genetic Algorithm for the linear and nonlinear transportation problem develop by Vignauz and Michalewicz [21]. As the name suggests, GA is originated from the analogy of biological evolution. GAs consider a population of individuals. Using the terminology of genetics, a population is a set of feasible solutions of a problem. A member of the population is called a genotype, a chromosome, a string or a permutation. A genetic algorithm contains three operators—reproduction, crossover and mutation. 6.1.1. Reproduction Parents are selected at random with selection chances biased in relation to chromosome evaluations. Next to initialize the population, we first determine the independent and dependent variables from all (here 12) variables and then their boundaries. This problem x221 ; x231 ; x122 ; x132 ; x212 ; x222 and x232 are independent variables and x111 ; x121 ; x131 ; x211 ; x112 are the dependent variables. The independent variables x221 eð0; minða2 ; b2 ; e1 ÞÞ; x231 eð0; minða2 ; b3 ; e1 ÞÞ; x122 eð0; minða1 ; b2 ; e2 ÞÞ; x132 eð0; minða1 ; b3 ; e2 ÞÞ; x212 eð0; minða2 ; b1 ; e2 ÞÞ; x222 eð0; minða2 ; b2 ; e2 ÞÞ and x232 e ð0; minða2 ; b3 ; e2 ÞÞ:
This paper, the proposed transportation problem with entropy function reduces to a multi-objective problem (15a) with constraints (5), (6), (7) and (15b) with (8)–(10). This multiobjective problem is solved with the multi-objective genetic algorithm (MOGA) developed for this purpose. The MOGA is illustrated as follows. We assume that there are M objective functions. In order to cover both minimization and maximization of objective functions, we use the operator between two solutions and as to denote that solution is better than solution on a particular objective. Similarly, for a particular objective implies that solution is worse than solution on this objective. For example, if an objective function is to be minimized, the operator would mean the < operator, whereas if the objective function is to be maximized, the operator would mean the operator. The following definition covers mixed problems with minimization of some objective functions and maximization of the rest of them. Definition: A solution X 1 is said to dominate the other solution X 2 , if the following both conditions (i) and (ii) are true: (i) The solution X 1 is no worse than X 2 in all objectives, or for all j ¼ 1; 2; . . . ; M: (ii) The solution X 1 is strictly better than X 2 in at least one objective, or for at least one j ¼ 1; 2; ::;M. If any of the above condition is violated, the solutions X 1 does not dominate the solution X 2 . If X 1 dominates the solution X 2 , it is also customary to write any of the following: ?
eLk
(i) X 2 is dominated by X 1 ; (ii) X i non-dominated by X 2 ; (iii) X i is non-inferior to X 2 .
6.1.2. Crossover Crossover is a key operator in the GA and is used to exchange the main characteristics of parent individuals and pass them on the children. It consist of two steps:
It is intuitive that if a solution X i dominates another solution X 2 , the solution X i is better than X 2 in the parlance of multi-objective optimization. Since the concept of domination allows a way to compare solutions with multiple objectives, most multi-objective optimization methods use this domination concept to search for non-dominated solution.
(i) Selection for crossover: For each solution of P 1 ðTÞ generate a random number r from the range [0, . . . ;1]. If r < pc then the solution is taken for crossover, where pc is the probability of crossover. (ii) Crossover process: Crossover taken place on the selected solutions. For each pair of coupled solutions Y 1 , Y 2 a random number c is generated from the range [0, . . . ;1] and Y 1 , Y 2 are replaced by their offspring’s Y 11 and Y 21 respectively where Y 11 ¼ cY 1 þ ð1 cÞY 2 , Y 21 ¼ cY 2 þ ð1 cÞY 1 , provided Y 11 , Y 21 satisfied the constraints of the problem.
6.2.1. Crowding distance Crowding distance of a solution is measured using the following rule. Step-1: Sort the population set according to every objective function values in ascending order of magnitude. Step-2: For each objective function, the boundary solutions are assigned an infinite distance value. All other intermediate solutions are assigned a distance value equal to the absolute normalized difference in the function values of two adjacent solutions. This calculation is continued with other objective functions.
A. Ojha et al. / Applied Soft Computing 10 (2010) 100–110
Step-3: The overall crowding distance value is calculated as the sum of the individual distance values corresponding to each objective. Each objective function is normalized before calculating the crowding distance. Following algorithm is used for this purpose.
Here, F½im refers to the m th objective function value of F½i. fmmax and fmmin are the maximum and minimum values of the m th objective function. 6.2.2. Non-dominated sorting of a population In this case, first, for each solution we calculate two entities: (1) domination count n p , the number of solutions which dominate the solution p, and (2) S p , a set of solutions that the solution p dominates. All solutions in the first non-dominated front will have their domination count as zero. Now, for each solution p with n p ¼ 0, we visit each member(q) of its set S p and reduce its domination count by one. In doing so, if for any member q the domination count becomes zero, we put it in a separate list Q. These members belong to the second non-dominated front. Now, the above procedure is continued with each member of Q and the third front is identified. This process continues until all fronts are identified. 6.2.3. Parameters Firstly, we set the different parameters on which this GA depends. These are the number of generation ðMAXGENÞ, population size ðPOPSIZEÞ, probability of crossover ðPXOVERÞ, probability of mutation ðPMUÞ. There is no clear indication as to how large a population should be. If the population is too large, there may be difficulty in storing the data, but if the population is too small, there may not be enough string for good crossovers. In our problem, POPSIZE ¼ 100, PXOVER ¼ 0:7, PMU ¼ 0:3 and MAXGEN ¼ 5000. 6.2.4. Chromosome representation An important issue in applying a GA is to design an appropriate chromosome representation of solutions of the problem together with genetic operators. Traditional binary vectors used to represent the chromosome are not effective in many non-linear physical problems. Since the proposed problem is non-linear, hence to overcome this difficulty, a real-number representation is used in this problem. 6.2.5. Evaluation Evaluation function plays the same role in GA as that which the environment plays in natural evolution. To this problem, the evaluation function is eðV i Þ ¼ ob jecti function
105
By Roulette wheel selection method the batter chromosome are selected from the population to generate the next the improved chromosomes. Now new chromosomes are produced by arithmetic crossover and uniform mutation. The general outline of the algorithm is following:
Procedure of MOGA Step-1: Step-2: Step-3: Step-4: Step-5: Step-6: Step-7: Step-8:
Step-9: Step-10: Step-11: Step-12: Step-13: Step-14: Step-15: Step-16: Step-17: Step-18:
Generate initial population P 1 of size N. i 1 [i represent the number of current generation.] Select solution from Pi for crossover. Made crossover on selected solution to get child set C1. Select solution from Pi for mutation. Made mutation on selected solution to get solution set C 2 . S S Set Pi0 ¼ P i C 1 C 2 Partition Pi0 into subsets F 1 ; F 2 ; . . . ; F k , such that each subset contains non-dominated solutions of Pi0 and every solutions of F i dominates every solu.s of F iþ1 for i ¼ 1; 2; . . . ; k 1. Select largest possible integer l, so that no of solu.s in the S S S set F 1 F 2 F l N: S S S Set P iþ1 ¼ F 1 F 2 F l . Sort F lþ1 in decreasing order by crowding distance. Set M = number of solutions in Piþ1 . Select first N M solutions from set F lþ1 . Insert these solution in solution set Piþ1 . Set i i þ 1. If termination condition does not hold, goto step-3. Output P i . End.
6.3. Weighted average method for multi-objectives The weighted average method scalarizes a set of objectives into a single objective by multiplying each objective with user’s supplied weights. The weights of an objective are usually chosen in proportion to the objective’s relative importance in the problem. However setting up an appropriate weight vector depends on the scaling of each objective function. It is likely that different objectives take different orders of magnitude. When such objectives are weighted to form a composite objective function, it would be better to scale them appropriately so that each objective possesses more or less the same order of magnitude. This process is called normalization of objectives. After the objectives are normalized, a composite objective function Z¯ can be formed by summing the weighted normalized objectives and the MONLP given in Eq. (15b) is
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then converted to a single-objective optimization problem as follows: k X wi f i ðxÞ
MinimizeZ¯ ¼
i¼1 k X wi
;
wi 2 ½0; 1; x 2 X
(23)
The optimum solutions are obtained using the GA presented in Section 6.2 and presented below. 7.3. Prob.-I (case-II): All constraint are less than equality type (without entropy function) In this case,the following optimum solutions are obtained by GA (Section 6.1)
i¼1
Here, wi is the weight of the i-th objective function. Since the minimum of the above problem does not change if all the weights are multiplied by a constant, it is the usual practice to choose k X weights such that their sum is one, i.e., wi ¼ 1. i¼1
7. Numerical experiment To illustrate the problems, we consider a solid transportation problem with two origins, three destinations and two types of conveyances. So m ¼ 2, n ¼ 3, K ¼ 3 and l ¼ 1; 2; 3. Values of the corresponding origins (i.e. resources), destination (i.e. demands), maximum amount to be transported by a particular conveyances, total vechicle costs and fixed charges are assumed as follows. Input data:
7.4. Prob.-I (case-II, with entropy): All constraint are less than equality type (with entropy function) In this case, all constraints are less than and equality type and we use entropy function as another objective. The optimum solutions are obtained using the GA presented in Section 6.2 and presented below. 7.5. Problem-II (case-III): All constraint are fuzzy Here all constraints, i.e., resources, demands and conveyance are expressed by fuzzy numbers. Let these fuzzy parameters are given by trapezoidal fuzzy numbers, such as a1 ¼ ð45; 48;
a1 ¼ 50; a2 ¼ 30; b1 ¼ 30; b2 ¼ 25; b3 ¼ 25; e1 ¼ 35; e2 ¼ 45; V ¼ 5; V c ¼ 7; f 111 ¼ 10; f 121 ¼ 12; f 131 ¼ 13; f 211 ¼ 11; f 221 ¼ 13; f 231 ¼ 14; f 112 ¼ 13; f 122 ¼ 15; f 132 ¼ 17; f 212 ¼ 18; f 222 ¼ 17; f 232 ¼ 15: P Hence total fixed charge= f i jk ¼ 168. The unit transportation cost under AUD, IQD and combination of these two discount systems are as follows.
52; 55Þ; a2 ¼ ð23; 25; 34; 38Þb1 ¼ ð25; 27; 33; 35Þ; b2 ¼ ð20; 24; 27; 29Þ, b3 ¼ ð18; 24; 28; 30Þ e1 ¼ ð38; 43; 47; 52Þ; e2 ¼ ð27; 32; 39; 42Þ.
C i jk
pli jk
AUD
IQD
C i jk
pli jk
AUD
IQD
C 111
5 4 2
0 < xi jk < 10 10 xi jk < 20 xi jk 20
0 < xi jk 10 10 < xi jk 20 xi jk > 20
C 112
8 7 4
0 < xi jk < 10 10 xi jk < 20 xi jk 20
0 < xi jk 10 10 < xi jk 20 xi jk > 20
C 121
6 5 3
0 < xi jk < 10 10 xi jk < 20 xi jk 20
0 < xi jk 10 10 < xi jk 20 xi jk > 20
C 122
7 6 4
0 < xi jk < 10 10 xi jk < 20 xi jk 20
0 < xi jk 10 10 < xi jk 20 xi jk > 20
C 131
9 8 6
0 < xi jk < 10 10 xi jk < 20 xi jk 20
0 < xi jk 10 10 < xi jk 20 xi jk > 20
C 132
10 8 6
0 < xi jk < 10 10 xi jk < 20 xi jk 20
0 < xi jk 10 10 < xi jk 20 xi jk > 20
C 211
8 7 5
0 < xi jk < 10 10 xi jk < 20 xi jk 20
0 < xi jk 10 10 < xi jk 20 xi jk > 20
C 212
9 8 5
0 < xi jk < 10 10 xi jk < 20 xi jk 20
0 < xi jk 10 10 < xi jk 20 xi jk > 20
C 221
5 3 2
0 < xi jk < 10 10 xi jk < 20 xi jk 20
0 < xi jk 10 10 < xi jk 20 xi jk > 20
C 222
7 6 3
0 < xi jk < 10 10 xi jk < 20 xi jk 20
0 < xi jk 10 10 < xi jk 20 xi jk > 20
C 231
7 5 3
0 < xi jk < 10 10 xi jk < 20 xi jk 20
0 < xi jk 10 10 < xi jk 20 xi jk > 20
C 232
8 6 4
0 < xi jk < 10 10 xi jk < 20 xi jk 20
0 < xi jk 10 10 < xi jk 20 xi jk > 20
With the above input data, we solve the problem-I (case-I and case-II) and problem-II as stated earlier using above mentioned GAs. The optimum results are presented below: 7.1. Prob.-I (case-I): All constraint are of equality sign (without entropy function) Here the optimum solutions are obtained using GA presented in Section 6.1 and given below. 7.2. Prob.-I (case-I, with entropy): All constraint are of equality sign and with entropy function In this case, all constraints are of equal type and we use entropy function as another objective.
Following (12)–(14), the fuzzy number are converted to intervals and the constraints to inequalities as (20)–(22). After that, the problem is formulated as a single objective and solved by GA (Section 6.1). The optimum results are in Table 5. 7.6. Efficiency of GA To illustrate the effectiveness and powerfulness of the proposed method i.e. GA, the problem-I (case-I) with entropy and AUD discount system has been solved by weighted average technique. Weighted average formula converts a multi-objective to the single objective function. Here objective function is Z¯ ¼ ðw1 Z þ w2 ðEnÞ=w1 þ w2 Þ; w1 ¼ 1=2 ¼ w2 . The optimum results for the AUD discount system is given below in Table 6.
A. Ojha et al. / Applied Soft Computing 10 (2010) 100–110
107
Fig. 3. Probability of crossover vs costs. Fig. 1. Generations vs costs.
Fig. 4. Probability of mutation vs costs.
Fig. 2. Population vs costs.
It may be noted that the optimum value of Z, i.e., minimum cost for problem-7.2 is less than the present minimum cost. The convergeness of the method, GA is illustrated with the variations in population size, generation (i.e. iteration) number, probability of crossover and mutation. The results are presented in Table 7 and pictocally in Figs. 1–4. The results indicate that the method GA for the present formulation converges with present input data.
8. Discussion From Tables 1–5, it is observed that AUD system gives minimum transportation cost whereas other systems provide more costs, of which IQD is the highest. Moreover, when the constraints are less than equality type (case-II, Table 3), then for all systems, costs reduce considerably. This is because in this case,
Table 1 Optimum transported amounts and min. cost for Prob.-I (case-I, without entropy). Discount system
AUD IQD AUD IQD IQD AUD
Transported amounts x111
x121
x131
x211
x221
x231
x112
x122
x132
x212
x222
x232
26.38 0.34 26.28 1.57 26.16 1.12 27.20 2.46
3.72 11.40 11.50 6.83 0.22 15.31 9.20 6.87
4.88 3.26 0.35 3.46 0.54 6.65 1.20 3.05
0.59 2.69 0.56 1.59 0.48 2.25 0.22 0.11
4.34 5.53 1.75 4.91 5.21 4.26 4.34 4.58
5.08 11.77 4.56 16.63 12.39 5.41 2.82 17.92
Min. cost
Total vechicle cost
671.39
85
739.84
90
679.02
90
710.63
90
Table 2 Near optimum transported amounts and min. cost for Prob.-I (case-I, with entropy). Discount system
AUD
Transported amounts x111
x121
x131
x211
x221
x231
x112
x122
x132
x212
x222
x232
11.53 7.23
9.62 5.93
8.47 7.22
7.27 3.97
4.03 5.42
4.0755 5.24
10.9 7.68 10.31 9.18
7.65 6.77 8.17 6.29
9.41 7.59 9.62 6.42
6.49 4.94 7.02 3.48
5.95 4.62 5.14 5.39
4.59 3.40 4.73 4.23
Min. cost
Max. entropy
Total vechicle cost
832.0
2.43
90
829.79
2.41
85
829.53
2.07
85
A. Ojha et al. / Applied Soft Computing 10 (2010) 100–110
108 Table 2 (Continued ) Discount system
Transported amounts x111
x121
x131
x211
x221
x231
x112
x122
x132
x212
x222
x232
8.56 7.91 8.78 5.87 8.09 6.12
9.46 6.9 8.54 9.37 7.66 9.26
9.83 7.33 10.16 7.28 11.63 7.25
6.62 6.91 10.23 5.12 9.60 6.19
5.10 3.53 2.61 4.48 4.31 3.76
5.42 2.41 4.67 2.89 3.71 2.42
AUD IQD
7.16 7.14 8.53 7.14 9.23 9.59
6.3 9.84 10.63 8.18 9.07 4.65
12.71 6.85 8.16 7.36 9.0 8.46
11.67 4.02 10.19 4.14 7.45 3.72
4.56 4.29 2.4 3.8 4.78 6.5
2.6 2.84 5.1 4.38 5.46 2.08
IQD AUD
9.57 8.37 10.21 7.3 4.39 11.29
8.9 7.76 5.98 8.49 13.44 3.80
8.84 6.56 11.31 6.71 10.05 7.03
7.64 4.42 9.38 3.11 9.0 5.32
4.58 3.76 5.88 4.65 4.36 3.4
5.47 4.12 2.25 4.73 3.76 4.16
IQD
Min. cost
Max. entropy
Total vechicle cost
848.64
2.43
85
857.66
2.40
90
859.44
2.40
90
834.54
2.37
85
838.12
2.41
95
850.85
2.41
90
847.13
2.44
90
843.71
2.40
85
843.55
2.37
85
Table 3 Optimum transported amounts and min. cost for Prob.-I (case-II, without entropy). Discount system
AUD IQD AUD IQD IQD AUD
Transported amounts
Min. cost
x111
x121
x131
x211
x221
x231
x112
x122
x132
x212
x222
x232
7.06 7.08 6.82 3.59 3.59 2.35 5.60 4.6
0.40 5.52 6.07 5.55 5.3 5.15 1.82 4.32
2.54 0.8 3.81 0.14 9.2 2.46 3.67 0.07
2.73 0.03 6.08 0.42 0.54 0.45 3.2 0.71
0.59 2.86 3.29 2.0 2.74 0.67 4.63 4.69
3.0 1.51 0.17 0.74 2.34 0.1 5.27 0.84
Total vechicle cost
Un-exhausted resources
Unfulfilled demand
a1
a2
b1
b2
b3
480.05
70
26.6
19.28
13.1
15.63
17.15
491.68
60
24.02
17.3
13.09
8.09
20.14
489.07
65
21.94
23.16
23.07
11.14
10.89
499.4
60
29.92
10.66
15.89
9.54
15.15
Table 4 Near optimum transported amounts and min. cost for Prob.-I (case-II, with entropy). Discount system
AUD
IQD
AUD IQD
IQD AUD
Transported amounts
Min. cost
x111
x121
x131
x211
x221
x231
x112
x122
x132
x212
x222
x232
4.84 8.38 3.4 8.50 3.83 8.13
2.09 4.61 3.24 2.76 3.36 3.57
1.63 8.4 2.50 8.39 3.18 8.4
4.45 1.20 6.44 1.20 7.19 1.20
8.56 4.88 6.01 4.88 5.38 , 4.88
1.55 2.35 1.55 2.35 1.55 2.35
8.28 8.56 5.47 3.4 4.19 6.25
3.87 5.02 3.51 3.03 5.89 7.48
7.85 5.87 10.78 5.87 7.13 5.87
5.2 1.57 8.45 1.57 8.93 1.57
5.91 4.64 5.68 4.64 2.24 4.64
1.51 2.74 1.51 2.74 1.51 2.74
4.6 10.37 6.03 6.05 6.82 4.4
3.0 3.08 2.92 3.39 3.48 3.38
7.44 8.4 7.27 8.4 6.0 8.4
5.38 1.20 6.84 1.20 9.35 1.2
3.44 4.88 5.01 4.88 5.15 4.88
1.55 2.35 1.55 2.35 1.55 2.35
6.85 2.81 5.76 3.92 5.43 2.9
4.12 7.24 3.19 6.17 4.34 7.09
8.8 8.4 9.49 8.4 8.74 8.4
5.03 1.2 5.28 1.2 4.66 1.2
3.34 4.88 6.97 4.88 6.88 4.88
1.55 2.35 1.55 2.35 1.55 2.35
Max. entropy
Total vechicle cost
Un-exhausted resources
Unfulfilled demand
a1
a2
b1
b2
b3
631.11
2.33
75
20.06
13.92
11.08
4.86
11.08
624.43
2.36
70
21.51
7.57
10.46
8.41
10.21
644.02
2.34
75
19.54
7.45
9.65
7.81
9.53
690.93
2.37
75
10.56
8.43
6.39
5.57
7.03
665.01
2.34
70
17.94
5.41
11.11
8.14
4.1
686.43
2.36
75
13.19
8.37
9.06
4.75
7.75
673.86
2.32
75
13.13
661.60
2.36
70
15.95
665.03
2.34
70
669.83
2.34
681.68 678.0
11.2
8.45
10.6
5.28
8.17
9.88
8.8
5.44
17.53
5.52
8.23
8.11
6.71
75
11.79
11.65
14.11
5.42
3.91
2.35
70
13.08
7.77
13.84
3.79
3.22
2.35
75
13.11
8.48
15.81
1.81
3.97
A. Ojha et al. / Applied Soft Computing 10 (2010) 100–110
109
Table 5 Optimum transported amounts and min. cost for Prob.-II. Discount system
AUD IQD AUD IQD IQD AUD
Transported amounts
Min. cost
x111
x121
x131
x211
x221
x231
x112
x122
x132
x212
x222
x232
21.8 2.66 19.77 1.74 20.38 5.18 15.65 9.54
3.36 5.5 5.89 10.94 13.18 0.9 12.51 5.7
8.74 9.79 6.94 4.56 2.78 9.15 0.28 6.83
1.06 2.95 2.15 4.13 0.24 2.49 0.66 3.63
4.85 10.36 1.22 5.97 0.48 12.30 0.95 5.3
0.73 2.92 8.02 4.89 9.09 1.72 13.24 1.93
Total vechicle cost
Transported resources
Fulfilled demand
a1
a2
b1
b2
b3
708.05
90
51.86
25.87
29.47
24.07
24.18
767.57
80
49.84
26.37
27.79
24.02
24.4
716.98
90
51.57
26.32
28.29
26.87
22.74
766.87
85
50.54
25.71
29.5
24.46
22.3
Table 6 Using weighted average formula in Table 2. Discount system
Transported amounts
AUD
x111
x121
x131
x211
x221
x231
x112
x122
x132
x212
x222
x232
9.84 8.1
9.59 6.23
9.84 6.4
6.76 5.3
5.55 3.62
3.42 5.34
Table 7 Optimum cost for Prob.-7.1 for different values of population, generation and probability of crossover and mutation with AUD discount system. Papulation
Generation
Probability crossover
Probability mutation
Minimum cost
100 100 100 100 100
1000 800 600 400 200
0.7 0.7 0.7 0.7 0.7
0.3 0.3 0.3 0.3 0.3
671.39 671.39 675.36 679.54 682.25
117 115 110 100 90 80 70 60
1000 1000 1000 1000 1000 1000 1000 1000
0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7
0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3
670.10 670.10 670.10 671.39 676.62 673.52 672.48 677.52
100 100 100 100 100 100 100
1000 1000 1000 1000 1000 1000 1000
0.27 0.3 0.4 0.5 0.6 0.7 0.8
0.3 0.3 0.3 0.3 0.3 0.3 0.3
671.0 662.52 668.33 674.03 675.83 671.39 674.29
100 100 100 100 100
1000 1000 1000 1000 1000
0.7 0.7 0.7 0.7 0.7
0.22 0.3 0.4 0.5 0.62
678.79 671.39 671.5 672.02 672.79
some resources are not completely exhausted and demands are not satisfied fully. But when the constraints are less than equality type and introducing entropy (case-II, Tables 2 and 4) in AUD system is introduced, then transported amounts to destinations are readjusted and the total minimum costs increases and are more than the corresponding case without entropy (cf. Table 3). As this is a multi-objective optimization problem, some compromise solutions are presented. It is observed that resources at different cells are better distributed than the case-I and case-II (without entropy). In these cases, un-exhausted resources also reduce considerably and demands are better satisfied.
Min. cost
Optimum Entropy
837.32
2.43
When the resources, demands and conveyance charge are fuzzy, the behavior of objective function remains same and the allotment behavior is the same as in the case-I. The cost values are more than the corresponding values of the case-I, as the constraints (22)–(25) are satisfied for the values which are higher than the constant resource values of case-I (cf. Table 4). In Table 6, the efficiency of GA is establised and the corresponding minimum cost 832.0 (highest amongst the set of values) in Table 2 is less than the cost in Table 6. From Table 7 and Figs. 1–4 sword the convergence of the GA against generation number, population size, crossover probability and mutation probability. 9. Conclusions and future research work In this paper, we analyzed a multi-item transportation problem with AUD, IQD and combination of these two price breaks, linear cost function, fixed charge and vechicle cost and solved by genetic algorithm. Here we discussed four types of the problem. In case-I, all constraints are equal type with and without entropy function, in case-II, constraints are less then equal type with and without entropy, case-III is with equal constraints where resources, demands, conveyances are triangular fuzzy numbers. The entropy function is constructed from the concept of Shannon’s measure of entropy. It acts as a measure of dispersal of trips among the origins, destinations and conveyances of the transportation model and is used as a second objective. The concept of entropy presented here is quite general in nature and can be extended to other fields of operation research like supply chain model, market research, etc. Different losses like breakability damage of the items during the transportation, etc., can be incorporated in the proposed STP and this may be the future research work. References [1] A.K. Bit, M.P. Biswal, S.S. Alam, Fuzzy programming approach to multi-objective solid transportation problem, Fuzzy Sets and Systems 57 (1993) 183–194. [2] A.K. Bit, M.P. Biswal, S.S. Alam, An additive fuzzy programming model for multi-objective transportation problem, Fuzzy Sets and Systems 57 (1993) 313–319. [3] S. Chanas, D. Kuchta, A concept of the optimal solution of the transportation problem with fuzzy cost coefficients, Fuzzy Sets and System 82 (1996) 299–305. [4] V. Damodaran, R.S. Lashkari, N. Singh, A production planning model for cellular manufacturing systems with refixturing considerations, International Journal of production research 30 (7) (1992) 1603–1615.
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Applied Soft Computing journal homepage: www.elsevier.com/locate/asoc
A solid transportation problem for an item with fixed charge, vechicle cost and price discounted varying charge using genetic algorithm A. Ojha a, B. Das b,*, S. Mondal a, M. Maiti a a b
Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore 721102, India Department of Mathematics, Jhargram Raj College, Midnapore 721507, India
A R T I C L E I N F O
A B S T R A C T
Article history: Received 14 January 2008 Received in revised form 29 January 2009 Accepted 25 June 2009 Available online 14 July 2009
In this paper, discount in transportation cost on the basis of transportated amount is extended to a solid transportation problem. In a transportation model, the available discount is normally offered on items/ criteria, etc., in the form AUD (all unit discount) or IQD (incremental quantity discount) or combination of these two. Here transportation model is considered with fixed charges and vechicle costs where AUD, IQD or combination of AUD and IQD on the price depending upon the amount is offered and varies on the choice of origin, destination and conveyance. To solve the problem, genetic algorithm (GA) based on Roulette wheel selection, arithmetic crossover and uniform mutation has been suitably developed and applied. To illustrate the models, numerical examples have been presented. Here, different types of constraints are introduced and the corresponding results are obtained. To have better customer service, the entropy function is considered and it is displayed by a numerical example. To exhibit the efficiency of GA, another method—weighted average method for multi-objective is presented, executed on a multiobjective problem and the results of these two methods are compared. Crown Copyright ß 2009 Published by Elsevier B.V. All rights reserved.
Keywords: All unit and incremental quantity discount Genetic algorithm Solid transportation problem Fuzzy resource and demand Entropy
1. Introduction Hitchcock [8] defined the transportation problems (TPs) as one of the most common special type linear programming problems involving constraints. In TP, the total transportation cost from some wholesalers to some consumers is minimized and bounds are given on two features—availability of the sources, requirement of the destinations. The solid transportation problem (STP) is a generalization of the well-known transportation problem (TP) in which three-dimensional properties are taken into account in the objective and constraint set instead of source and destination. The STP was first stated by Shell [17]. In many industrial problems, a homogeneous product is delivered from an origin to a destination by means of different modes of transport called conveyances, such as trucks, cargo flights, goods trains, ships, etc. These conveyance are taken as the third dimension. A solid transportation problem can be converted to a classical transportation problem by considering only a single type of conveyance. In many practical applications, it is realistic to assume that the amount which can be sent on any particular route bears a fixed charge for that route. Further, when a route is altogether excluded, this can be expressed by limiting its capacity to zero. Such fixed
* Corresponding author. E-mail address: [email protected] (A. Ojha).
transportation costs may also be applied to some productionplanning models (cf. [4]). But, till now, none has considered solid transportation problem with fixed cost. In today’s highly competitive market the pressure on organizations to find the better ways to create and deliver values to customers is increasing more and more. How and when to send the products to the customers in the quantities they want in a costeffective manner has become more challenging in terms of price discount (AUD and/or IQD) on the unit transportation cost. There is a similarity to economic ordered quantity model for such a discount (cf. [14]). But no researcher considered such a price discount in STP models. Jimcncz and Verdegay [9] investigated interval multi-objective solid transportation problem via Genetic Algorithms. Based on Chanas and Kuchta [3], Das et al. [5] converted the interval number transportation problems into deterministic multi-objective problems. Bit et al. [1,2], Jimenez and Verdegay [10], Li and Lai [12] and Waiel [23] presented the fuzzy compromise programming approach to multi-objective transportation problems. Grzegorzewski [7] approximated the fuzzy number to its nearest interval. Shaocheng [19] discussed about the interval number linear programming. Omar and Samir [16] and Chanas and Kuchta [3] discussed the solution algorithm for solving the transportation problem in fuzzy environment. In the transportation model, the entropy acts as a measure of dispersal of trips between origin and destinations. So it will be more practical, if we could have minimum transportation penalties along with maximum entropy amount. Usefulness of entropy optimization
1568-4946/$ – see front matter . Crown Copyright ß 2009 Published by Elsevier B.V. All rights reserved. doi:10.1016/j.asoc.2009.06.016
A. Ojha et al. / Applied Soft Computing 10 (2010) 100–110
in transportation models as well as other models are discussed in Kapur and Kesavan [11]. The purpose of introducing entropy in transportation problem is-to get better customer service. For the solution of decision-making problems, there are some inherent difficulties in the traditional direct and gradient-based optimization techniques used for this purpose. Normally, these methods (i) are initial solution dependent, (ii) get stuck to a suboptimal solution, (iii) are not efficient in handing problems having discrete variables, (iv) cannot be efficiently used on parallel machines and (v) are not universal, rather problem dependent. To overcome these difficulties, recently genetic algorithms (GAs) are used for optimization of decision making problems. GAs [6,21] are adaptive computational procedures modelled on the mechanics of natural genetic systems. They exploit the historical information to speculate on new offspring with expected improved performance. These are executed iteratively on a set of coded solutions (called population) with three operators—selection/reproduction, crossover and mutation. One set of these three operators is known as a generation in the parlance of GA. Since a GA works simultaneously on a set of solutions, it has very little chance to get suck at local optimum. Here, the resolution of the possible search space is increased by operating on potential solutions and not on the solutions themselves. Further, this search space need not be continuous. Recently, GAs have been applied in different areas like neural network, travelling salesman, scheduling, numerical optimization Michalewicz [21], inventory [13,14], etc. The mathematical formulation of transportation problem is almost of similar form as the mathematical representation of production models with linear production cost (cf. [24,20]). As all these problems are linear programming (LP) problems with linear production cost and linear constraints, discounted LP approach may be used to solve the problems. In this paper, a solid transportation problem with discounted costs, fixed charges and vechicle costs are formulated as a liner programming problem. The discounted costs are in the form of AUD, IQD and combination of AUD and IQD. Here AUD, IQD or combination of these two discounts on purchasing price with two price breaks are considered. The transportation costs are deterministic but the resources and demands at various centres, capacities of different modes of transport between origins and destinations are crisp for problem-I and imprecise for problem-II. These solid transportation problems are solved using genetic algorithm (GA). Here, different types of constraints are introduced and the corresponding results are obtained. To bring all cells under better allotment scheme, i.e. to facilitate better customers’ service, entropy of the transported amounts is considered as an additional
C i jk
8 p > > h 1i jk > > > > > > h p1i jk R1 þ > > > > < p1i jk R1 þ ¼ ......... > > > h. . . . . . . . . > > > > p1i jk R1 þ > > h > > > : p R þ 1i jk 1
2. Assumptions and notation 2.1. Notation In this solid transportation problem the following notation (i) m = no of sources of the transportation problem. (ii) n = no of demands of the transportation problem. (iii) K = no of conveyances, i.e., different modes of the transportation problem. (iv) Oi = origins of the transportation problem. (v) D j = destination of the transportation problem. (vi) Ek = conveyances of the transportation problem. (vii) ai = amount of a homogeneous product available at ith origin. (viii) b j = demand at jth destination. (viii) ek = amount of product which can be carried by k th conveyance. (ix) f i jk = the fixed charge of the transportation problem. (x) C i jk = unit transportation cost. (xi) hðxÞ = the probability that X is in the state x. 2.2. Assumption In this solid transportation problem, the following assumptions are made. (i) The total vechicle cost sV ifsV c ¼ xi jk Gðxi jk Þ ¼ ðs þ 1ÞV Otherwise
(1)
(ii) s ¼ ½xi jk =V c , V c ¼ vechicle capacity and V = vechicle cost. (iii) The unit transformation cost under AUD scheme: 8 p1i jk if0 < xi jk < R1 > > > > p2i jk ifR1 xi jk < R2 > > <......... (2) C i jk ¼ ......... > > > > p ifR x < R > t i jk > : ti jk ðt1Þ pðtþ1Þi jk ifRt xi jk here p1i jk > p2i jk > p3i jk > > pti jk > pðtþ1Þi jk are all unit cost. (iv) Incremental quantity discount (IQD): The unit-transportation cost under incremental quantity discount (IQD) system is p1i jk for 0 < xi jk < R1 ; p2i jk for an additional quantity over R1 but less then R2 and so on, lastly pðtþ1Þi jk for any additional quantity over Rt . Thus unit transportation cost becomes i f 0 < xi jk R1
i p2i jk ðxi jk R1 Þ =xi jk
i p2i jk ðR2 R1 Þ þ p3i jk ðxi jk R2 Þ =xi jk i p2i jk ðR2 R1 Þ þ þ pti jk ðxi jk Rt1 Þ =xi jk i p2i jk ðR2 R1 Þ þ þ pðtþ1Þi jk ðxi jk Rt Þ =xi jk
objective function for two particular models. The presence of various real-life complexities, like—three dimensional nature, price discount, imprecise nature of availability and requirement of STP, various constraints entropy, etc., has been taken into account successfully by introducing GA. A numerical experiment is also performed to illustrate the use of entropy function. To exhibit the efficiency of GA, another method—weighted average method for multi-objective is presented, executed on a multi-objective problem and the results of these two methods are compared.
101
ifR1 < xi jk R2 ifR2 < xi jk R3 (3) ifRt1 < xi jk Rt ifRt < xi jk
here p1i jk > p2i jk > p3i jk > > pti jk > pðtþ1Þi jk are all unit cost. 3. Formulation of a solid transportation problem with discount cost, fixed charge and vechicle cost We assume that m origins (or sources) Oi (i = 1, 2,. . ., m), n destinations (i.e. demands) D j ð j ¼ 1; 2; . . . ; nÞ and K conveyances Ek ðk ¼ 1; 2; . . . ; KÞ. K conveyances, i.e., different modes of transport may be trucks, cargo flights, goods trains, ships, etc. Let ai be the
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A. Ojha et al. / Applied Soft Computing 10 (2010) 100–110
amount of a homogeneous product available at ith origin, b j be the demand at jth destination and ek represents the amount of product which can be carried by k th conveyance. C i jk be the cost under AUD, IQD or combination of these two systems associated with transportation of a unit product from i th source to j th destination by means of the k th conveyance. The variable xi jk represents the unknown quantity to be transported from origin Oi to destination D j by means of k th conveyance. A solid transportation problem with resources, demands, conveyances under all unit discounts(AUD), incremental quantity discount (IQD) and combination of these two systems can be represented as Problem-I (case-1):
subject to n X K X xi jk ¼ a˜ i ;
i ¼ 1; 2; 3; . . . ; m
(12)
j ¼ 1; 2; 3; . . . ; n
(13)
k ¼ 1; 2; 3; . . . ; K
(14)
j¼1 k¼1
m X K X xi jk ¼ b˜i ; i¼1 k¼1 m X n X xi jk ¼ e˜i ; i¼1 j¼1
3.1. Entropy function m X n X K X Min ZðXÞ ¼ ½C i jk xi jk þ f i jk þ Gðxi jk Þ
(4)
i¼1 j¼1 k¼1
subject to n X K X xi jk ¼ ai ;
i ¼ 1; 2; 3; . . . ; m
(5)
j¼1 k¼1
P Pn PK Let T be the transported amount, i.e., T ¼ m i¼1 j¼1 k¼1 xi jk . Consider a function FðXÞ which represents the number of possible assignment for the state X ¼ ðxi jk Þ. The (Shannon) entropy of a variable X is defined as FðXÞ= the number of ways selecting x111 from T, multiplied by the number of ways selecting x112 from T x111 ; . . ., multiplied by the number of ways selecting xmnk from T x111 x112 xmnk1 .
¼ T Cx111 :ðTx111 Cx112 :ðTx111 x112 Cx113 ðT x111 x112 xmnk1 CxmnK ¼
T! m Y n Y K Y
xi jk !
i¼1 j¼1k¼1
m X K X xi jk ¼ b j ;
j ¼ 1; 2; 3; . . . ; n
(6)
lnðFðXÞÞ
¼ lnðT!Þ
i¼1 k¼1 T
m X n X xi jk ¼ ek ;
¼ lnðe k ¼ 1; 2; 3; . . . ; K
T
i¼1 j¼1 k¼1 m X n X K X
x
i jk lnðexi jk xi jk Þ
T Þ
i¼1 j¼1 k¼1
(7)
i¼1 j¼1
m X n X K X lnðxi jk !Þ
[By using Stirlings approximation formula]
where C i jk ’s are given by (2) for AUD, by (3) for IQD and by (2)and (3) for combination of AUD, IQD. Problem-I (case-2): The equality constraints (5)–(7) can be imposed as ‘less than equality’ constraints in some cases. In that case, the objective function (4) remains the same but the changed constraints are n X K X xi jk ai ;
¼ T:lnðTÞ
m X n X K X xi jk lnðxi jk Þ i¼1 j¼1 k¼1
Therefore ðlnðFðXÞÞ=TÞ ¼ lnðTÞ ð1=TÞ
Pm Pn i¼1
j¼1
PK
k¼1
xi jk lnðxi jk Þ
Here the entropy function En ðxÞ ¼ ðlnðFðXÞÞ=TÞ. The function (Shannon) entropy can be expressed as
i ¼ 1; 2; 3; . . . ; m
(8) EnðXÞ ¼
j¼1 k¼1
X
f ðxÞ
x m X K X xi jk b j ;
where j ¼ 1; 2; 3; . . . ; n
(9) (
i¼1 k¼1
f ðxÞ ¼ m X n X xi jk ek ;
k ¼ 1; 2; 3; . . . ; K
(10)
i¼1 j¼1
Problem-II: In this case, resources, demands and capacities of transportation modes are imprecise. The formulation of this modal is the same as the problem-I except that all constraints becomes fuzzy. Thus the problem is expressed in the form: m X n X K X MinZðXÞ ¼ ½C i jk xi jk þ f i jk þ Gðxi jk Þ i¼1 j¼1 k¼1
(11)
hðxÞlnhðxÞ
ifhðxÞ 6¼ 0
0
ifhðxÞ ¼ 0
hðxÞ being the probability that X is in the state x. In transportation problem, normalizing the trip number xi jk by P Pn PK dividing the total number of trips Tð m i¼1 j¼1 k¼1 xi jk Þ, a probability distribution, hi jk ¼ xi jk =T is formulated. Therefore EnðXÞ
m X n X K X ðxi jk =TÞlnðxi jk =TÞ ¼ i¼1 j¼1 k¼1 m X n X K X
¼ lnðTÞ
1 T
xi jk lnðxi jk Þ
i¼1 j¼1 k¼1
(15)
A. Ojha et al. / Applied Soft Computing 10 (2010) 100–110
In transportation problem, this entropy function acts as a measure of dispersal of trips among origins, destinations and conveyances. It becomes more useful, if we would like to have minimum transportation costs as well as maximum entropy amount. Taking entropy function as an additional objective function, the Problem-I takes the following form:
103
˜ which Given A˜ is a fuzzy number. We find a closed interval C d ðAÞ is nearest to A˜ with respect to metric d. It can be done since each interval is also a fuzzy number with constant a-cut for all a 2 ½0; 1. ˜ Hence ðC d ðAÞÞ a ¼ ½C L ; C R . Now we have to minimize sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z 1 Z 1 ˜ ¼ ˜ C d ðAÞÞ ðAL ðaÞ C L ðaÞÞ2 da þ ðAR ðaÞ C R ðaÞÞ2 da dðA; 0
Minimize½ZðXÞ and Maximize EnðXÞ: subject to the constraintsð5Þ ð7Þ:
(15a)
and Minimize½ZðXÞ and Maximize EnðXÞ: subject to the constraintsð8Þ ð10Þ:
(15b)
0
˜ C d ðAÞÞ, ˜ it is with respect to C L and C R . In order to minimize dðA; 2 ˜ ˜ sufficient to minimize the function DðC L ; C R Þ ¼ d ðA; C d ðAÞÞ. The first partial derivatives are
dDðC L ; C R Þ ¼ 2 dC L ¼ 2
Z Z
1
AL ðaÞda þ 2C L and 0
dDðC L ; C R Þ dC R
1
AR ðaÞda þ 2C R 0
4. Fuzzy number and its interval approximation Definition: Fuzzy number: A fuzzy subset A of the real line R with membership function mA : R ! ½0; 1 is called a fuzzy number if
(i) A is normal, i.e.the supremum of mA ðxÞ is 1 8 x 2 S and (ii) A is fuzzy convex, i.e. mA ðlx1 þ ð1 lÞx2 Þ mA ðx1 Þ ^ mA ðx2 Þ 8 x1 ; x2 2 R, for l 2 ½0; 1. a-Cut of a fuzzy number: A a-cut of a fuzzy number A˜ is defined as crisp set Aa ¼ fx : mA˜ ðxÞ a; x 2 Xgwherea 2 ½0; 1 Aa is a non-empty bounded closed interval contained in X and it can be denoted by Aa ¼ ½AL ðaÞ; AR ðaÞ. AL ðaÞ and AR ðaÞ are the lower and upper bounds of the closed interval respectively. A fuzzy number A˜ with a1 ; a2 -cut Aa1 ¼ ½AL ða1 Þ; AR ða1 Þ; Aa2 ¼ ½AL ða2 Þ; AR ða2 Þ and if a2 a1 , then AL ða2 Þ AL ða1 Þ and AR ða1 Þ AR ða2 Þ. Different types of fuzzy numbers: Fuzzy numbers are represented by two types of membership functions: (a) Linear membership functions, e.g. triangular fuzzy number (TFN), trapezoidal fuzzy number, piecewise linear fuzzy number, etc. (b) Non-linear membership functions, e.g. parabolic fuzzy number (PFN), exponential fuzzy number and other non-linear fuzzy number. Trapezoidal fuzzy number: Trapezoidal fuzzy number is the fuzzy number with the membership function mA˜ ðxÞ, a continuous mapping: mA˜ ðxÞ : R ! ½0; 1 8 0 > > > > x a1 > > > > < a2 a1 mA˜ ðxÞ ¼ 1 > > a4 x > > > > > a4 a3 > : 0
for 1 < x < a1 fora1 x < a2
Solving
dDðC L ; C R Þ ¼ 0 and dC L
dDðC L ; C R Þ ¼0 dC R
R1 R1 we get, CL ¼ 0 ðAL ðaÞdaÞ and CR ¼ 0 ðAR ðaÞdaÞ. Again since,
dD2 ðCL ; CR Þ dD2 ðCL ; CR Þ ¼ 2 > 0; ¼ 2 > 0 and 2 dCL dCR2 !2
dD2 ðCL ; CR Þ dD2 ðCL ; CR Þ dD2 ðCL ; CR Þ : 2 2 dC L :dC R dC L dC R
¼ 4>0
˜ C d ðAÞÞ ˜ is global minimum. Therefore the So DðC L ; C R Þ, i.e., dðA; interval
˜ ¼ C d ðAÞ
"Z
1
AL ðaÞda;
Z
0
#
1
AR ðaÞda 0
is nearest interval approximation of fuzzy number A˜ with respect to metric d. Let A˜ ¼ ða1 ; a2 ; a3 ; a4 Þ be a fuzzy number. The a-level ˜ ¼ ½AL ðaÞ; AR ðaÞ. When A˜ is a interval of A˜ is defined as ðAÞ a trapezoidal fuzzy numbers then AL ðaÞ ¼ a1 þ aða2 a1 Þ and AR ðaÞ ¼ a4 aða4 a3 Þ; 0 < a 1. By nearest interval approximation method lower and upper limits of the interval are respectively
CL ¼
Z
1
AL da ¼ 0
CR ¼
Z
1
½a1 þ aða2 a1 Þda ¼ 0
Z
1
AR da ¼ 0
Z
1 ða2 þ a1 Þ 2
1
½a4 aða4 a3 Þda ¼
0
and
1 ða3 þ a4 Þ 2
Therefore, the nearest crisp interval number considering A˜ as a trapezoidal fuzzy number is ½ða1 þ a2 Þ=2; ða3 þ a4 Þ=2. 5. Reduced crisp model
fora2 x a3 fora3 < x a4 fora4 < x < 1
The nearest interval approximation of a fuzzy number: Here a fuzzy number is approximated by a corresponding crisp interval. ˜ two fuzzy numbers with a-cuts are ½AL ðaÞ; AR ðaÞ Suppose A˜ and B, and ½BL ðaÞ; BR ðaÞ respectively. Then the distance between A˜ and B˜ is
Let the fuzzy numbers a˜ i ; b˜ j , and e˜k are approximated to ½aLi ; aRi ; ½bLj ; bRj , and ½eLk ; eRk respectively. Then the earlier transportation model (Problem-II) takes the following form: Problem-IIa: MinimizeZðXÞ ¼
m X n X K X ½C i jk xi jk þ f i jk þ Gðxi jk Þ
(16)
i¼1 j¼1 k¼1
subject to
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z 1 Z 1 ˜ BÞ ˜ ¼ ðAL ðaÞ BL ðaÞÞ2 da þ ðAR ðaÞ BR ðaÞÞ2 da dðA; 0
0
n X K X xi jk ¼ ½aLi ; aRi j¼1 k¼1
(17)
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A. Ojha et al. / Applied Soft Computing 10 (2010) 100–110
m X K X xi jk ¼ ½bLj ; bRj
(18)
i¼1 k¼1 m X n X xi jk ¼ ½eLk ; eRk
(19)
i¼1 j¼1
The constraints (17)–(19) can be written in the following form: aLi
n X K X xi jk aRi
(20)
j¼1 k¼1
bLj
m X K X xi jk bRj
(21)
6.1.3. Mutation The mutation operation is needed after the crossover operation to maintain population diversity and recover possible loss of some good characteristics. It is also consist of two steps: (i) Selection for mutation: For each solution of P 1 ðTÞ generate a random number r from the range [0, . . . ;1]. If r < pm then the solution is taken for mutation, where pm is the probability of mutation. (ii) Mutation process: To mutate a solution X ¼ ðx1 ; x2 ; . . . ; xK Þ select a random integer r in the range ½1; . . . ; K. Then replace xr by randomly generated value within the boundary of r th component of X.
i¼1 k¼1 m X n X xi jk eRk
6.2. Implementation of GA for multi-objective (22)
i¼1 j¼1
6. Solution procedures 6.1. GA for single objective In this paper the proposed solid transportation problem, (4) with (5)–(7) or (16) with (20)–(22) is solved by genetic algorithm. A genetic algorithm is a heuristic search process for optimization that resembles natural selection. GAs has been applied successfully in different areas. Genetic Algorithm for the linear and nonlinear transportation problem develop by Vignauz and Michalewicz [21]. As the name suggests, GA is originated from the analogy of biological evolution. GAs consider a population of individuals. Using the terminology of genetics, a population is a set of feasible solutions of a problem. A member of the population is called a genotype, a chromosome, a string or a permutation. A genetic algorithm contains three operators—reproduction, crossover and mutation. 6.1.1. Reproduction Parents are selected at random with selection chances biased in relation to chromosome evaluations. Next to initialize the population, we first determine the independent and dependent variables from all (here 12) variables and then their boundaries. This problem x221 ; x231 ; x122 ; x132 ; x212 ; x222 and x232 are independent variables and x111 ; x121 ; x131 ; x211 ; x112 are the dependent variables. The independent variables x221 eð0; minða2 ; b2 ; e1 ÞÞ; x231 eð0; minða2 ; b3 ; e1 ÞÞ; x122 eð0; minða1 ; b2 ; e2 ÞÞ; x132 eð0; minða1 ; b3 ; e2 ÞÞ; x212 eð0; minða2 ; b1 ; e2 ÞÞ; x222 eð0; minða2 ; b2 ; e2 ÞÞ and x232 e ð0; minða2 ; b3 ; e2 ÞÞ:
This paper, the proposed transportation problem with entropy function reduces to a multi-objective problem (15a) with constraints (5), (6), (7) and (15b) with (8)–(10). This multiobjective problem is solved with the multi-objective genetic algorithm (MOGA) developed for this purpose. The MOGA is illustrated as follows. We assume that there are M objective functions. In order to cover both minimization and maximization of objective functions, we use the operator between two solutions and as to denote that solution is better than solution on a particular objective. Similarly, for a particular objective implies that solution is worse than solution on this objective. For example, if an objective function is to be minimized, the operator would mean the < operator, whereas if the objective function is to be maximized, the operator would mean the operator. The following definition covers mixed problems with minimization of some objective functions and maximization of the rest of them. Definition: A solution X 1 is said to dominate the other solution X 2 , if the following both conditions (i) and (ii) are true: (i) The solution X 1 is no worse than X 2 in all objectives, or for all j ¼ 1; 2; . . . ; M: (ii) The solution X 1 is strictly better than X 2 in at least one objective, or for at least one j ¼ 1; 2; ::;M. If any of the above condition is violated, the solutions X 1 does not dominate the solution X 2 . If X 1 dominates the solution X 2 , it is also customary to write any of the following: ?
eLk
(i) X 2 is dominated by X 1 ; (ii) X i non-dominated by X 2 ; (iii) X i is non-inferior to X 2 .
6.1.2. Crossover Crossover is a key operator in the GA and is used to exchange the main characteristics of parent individuals and pass them on the children. It consist of two steps:
It is intuitive that if a solution X i dominates another solution X 2 , the solution X i is better than X 2 in the parlance of multi-objective optimization. Since the concept of domination allows a way to compare solutions with multiple objectives, most multi-objective optimization methods use this domination concept to search for non-dominated solution.
(i) Selection for crossover: For each solution of P 1 ðTÞ generate a random number r from the range [0, . . . ;1]. If r < pc then the solution is taken for crossover, where pc is the probability of crossover. (ii) Crossover process: Crossover taken place on the selected solutions. For each pair of coupled solutions Y 1 , Y 2 a random number c is generated from the range [0, . . . ;1] and Y 1 , Y 2 are replaced by their offspring’s Y 11 and Y 21 respectively where Y 11 ¼ cY 1 þ ð1 cÞY 2 , Y 21 ¼ cY 2 þ ð1 cÞY 1 , provided Y 11 , Y 21 satisfied the constraints of the problem.
6.2.1. Crowding distance Crowding distance of a solution is measured using the following rule. Step-1: Sort the population set according to every objective function values in ascending order of magnitude. Step-2: For each objective function, the boundary solutions are assigned an infinite distance value. All other intermediate solutions are assigned a distance value equal to the absolute normalized difference in the function values of two adjacent solutions. This calculation is continued with other objective functions.
A. Ojha et al. / Applied Soft Computing 10 (2010) 100–110
Step-3: The overall crowding distance value is calculated as the sum of the individual distance values corresponding to each objective. Each objective function is normalized before calculating the crowding distance. Following algorithm is used for this purpose.
Here, F½im refers to the m th objective function value of F½i. fmmax and fmmin are the maximum and minimum values of the m th objective function. 6.2.2. Non-dominated sorting of a population In this case, first, for each solution we calculate two entities: (1) domination count n p , the number of solutions which dominate the solution p, and (2) S p , a set of solutions that the solution p dominates. All solutions in the first non-dominated front will have their domination count as zero. Now, for each solution p with n p ¼ 0, we visit each member(q) of its set S p and reduce its domination count by one. In doing so, if for any member q the domination count becomes zero, we put it in a separate list Q. These members belong to the second non-dominated front. Now, the above procedure is continued with each member of Q and the third front is identified. This process continues until all fronts are identified. 6.2.3. Parameters Firstly, we set the different parameters on which this GA depends. These are the number of generation ðMAXGENÞ, population size ðPOPSIZEÞ, probability of crossover ðPXOVERÞ, probability of mutation ðPMUÞ. There is no clear indication as to how large a population should be. If the population is too large, there may be difficulty in storing the data, but if the population is too small, there may not be enough string for good crossovers. In our problem, POPSIZE ¼ 100, PXOVER ¼ 0:7, PMU ¼ 0:3 and MAXGEN ¼ 5000. 6.2.4. Chromosome representation An important issue in applying a GA is to design an appropriate chromosome representation of solutions of the problem together with genetic operators. Traditional binary vectors used to represent the chromosome are not effective in many non-linear physical problems. Since the proposed problem is non-linear, hence to overcome this difficulty, a real-number representation is used in this problem. 6.2.5. Evaluation Evaluation function plays the same role in GA as that which the environment plays in natural evolution. To this problem, the evaluation function is eðV i Þ ¼ ob jecti function
105
By Roulette wheel selection method the batter chromosome are selected from the population to generate the next the improved chromosomes. Now new chromosomes are produced by arithmetic crossover and uniform mutation. The general outline of the algorithm is following:
Procedure of MOGA Step-1: Step-2: Step-3: Step-4: Step-5: Step-6: Step-7: Step-8:
Step-9: Step-10: Step-11: Step-12: Step-13: Step-14: Step-15: Step-16: Step-17: Step-18:
Generate initial population P 1 of size N. i 1 [i represent the number of current generation.] Select solution from Pi for crossover. Made crossover on selected solution to get child set C1. Select solution from Pi for mutation. Made mutation on selected solution to get solution set C 2 . S S Set Pi0 ¼ P i C 1 C 2 Partition Pi0 into subsets F 1 ; F 2 ; . . . ; F k , such that each subset contains non-dominated solutions of Pi0 and every solutions of F i dominates every solu.s of F iþ1 for i ¼ 1; 2; . . . ; k 1. Select largest possible integer l, so that no of solu.s in the S S S set F 1 F 2 F l N: S S S Set P iþ1 ¼ F 1 F 2 F l . Sort F lþ1 in decreasing order by crowding distance. Set M = number of solutions in Piþ1 . Select first N M solutions from set F lþ1 . Insert these solution in solution set Piþ1 . Set i i þ 1. If termination condition does not hold, goto step-3. Output P i . End.
6.3. Weighted average method for multi-objectives The weighted average method scalarizes a set of objectives into a single objective by multiplying each objective with user’s supplied weights. The weights of an objective are usually chosen in proportion to the objective’s relative importance in the problem. However setting up an appropriate weight vector depends on the scaling of each objective function. It is likely that different objectives take different orders of magnitude. When such objectives are weighted to form a composite objective function, it would be better to scale them appropriately so that each objective possesses more or less the same order of magnitude. This process is called normalization of objectives. After the objectives are normalized, a composite objective function Z¯ can be formed by summing the weighted normalized objectives and the MONLP given in Eq. (15b) is
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then converted to a single-objective optimization problem as follows: k X wi f i ðxÞ
MinimizeZ¯ ¼
i¼1 k X wi
;
wi 2 ½0; 1; x 2 X
(23)
The optimum solutions are obtained using the GA presented in Section 6.2 and presented below. 7.3. Prob.-I (case-II): All constraint are less than equality type (without entropy function) In this case,the following optimum solutions are obtained by GA (Section 6.1)
i¼1
Here, wi is the weight of the i-th objective function. Since the minimum of the above problem does not change if all the weights are multiplied by a constant, it is the usual practice to choose k X weights such that their sum is one, i.e., wi ¼ 1. i¼1
7. Numerical experiment To illustrate the problems, we consider a solid transportation problem with two origins, three destinations and two types of conveyances. So m ¼ 2, n ¼ 3, K ¼ 3 and l ¼ 1; 2; 3. Values of the corresponding origins (i.e. resources), destination (i.e. demands), maximum amount to be transported by a particular conveyances, total vechicle costs and fixed charges are assumed as follows. Input data:
7.4. Prob.-I (case-II, with entropy): All constraint are less than equality type (with entropy function) In this case, all constraints are less than and equality type and we use entropy function as another objective. The optimum solutions are obtained using the GA presented in Section 6.2 and presented below. 7.5. Problem-II (case-III): All constraint are fuzzy Here all constraints, i.e., resources, demands and conveyance are expressed by fuzzy numbers. Let these fuzzy parameters are given by trapezoidal fuzzy numbers, such as a1 ¼ ð45; 48;
a1 ¼ 50; a2 ¼ 30; b1 ¼ 30; b2 ¼ 25; b3 ¼ 25; e1 ¼ 35; e2 ¼ 45; V ¼ 5; V c ¼ 7; f 111 ¼ 10; f 121 ¼ 12; f 131 ¼ 13; f 211 ¼ 11; f 221 ¼ 13; f 231 ¼ 14; f 112 ¼ 13; f 122 ¼ 15; f 132 ¼ 17; f 212 ¼ 18; f 222 ¼ 17; f 232 ¼ 15: P Hence total fixed charge= f i jk ¼ 168. The unit transportation cost under AUD, IQD and combination of these two discount systems are as follows.
52; 55Þ; a2 ¼ ð23; 25; 34; 38Þb1 ¼ ð25; 27; 33; 35Þ; b2 ¼ ð20; 24; 27; 29Þ, b3 ¼ ð18; 24; 28; 30Þ e1 ¼ ð38; 43; 47; 52Þ; e2 ¼ ð27; 32; 39; 42Þ.
C i jk
pli jk
AUD
IQD
C i jk
pli jk
AUD
IQD
C 111
5 4 2
0 < xi jk < 10 10 xi jk < 20 xi jk 20
0 < xi jk 10 10 < xi jk 20 xi jk > 20
C 112
8 7 4
0 < xi jk < 10 10 xi jk < 20 xi jk 20
0 < xi jk 10 10 < xi jk 20 xi jk > 20
C 121
6 5 3
0 < xi jk < 10 10 xi jk < 20 xi jk 20
0 < xi jk 10 10 < xi jk 20 xi jk > 20
C 122
7 6 4
0 < xi jk < 10 10 xi jk < 20 xi jk 20
0 < xi jk 10 10 < xi jk 20 xi jk > 20
C 131
9 8 6
0 < xi jk < 10 10 xi jk < 20 xi jk 20
0 < xi jk 10 10 < xi jk 20 xi jk > 20
C 132
10 8 6
0 < xi jk < 10 10 xi jk < 20 xi jk 20
0 < xi jk 10 10 < xi jk 20 xi jk > 20
C 211
8 7 5
0 < xi jk < 10 10 xi jk < 20 xi jk 20
0 < xi jk 10 10 < xi jk 20 xi jk > 20
C 212
9 8 5
0 < xi jk < 10 10 xi jk < 20 xi jk 20
0 < xi jk 10 10 < xi jk 20 xi jk > 20
C 221
5 3 2
0 < xi jk < 10 10 xi jk < 20 xi jk 20
0 < xi jk 10 10 < xi jk 20 xi jk > 20
C 222
7 6 3
0 < xi jk < 10 10 xi jk < 20 xi jk 20
0 < xi jk 10 10 < xi jk 20 xi jk > 20
C 231
7 5 3
0 < xi jk < 10 10 xi jk < 20 xi jk 20
0 < xi jk 10 10 < xi jk 20 xi jk > 20
C 232
8 6 4
0 < xi jk < 10 10 xi jk < 20 xi jk 20
0 < xi jk 10 10 < xi jk 20 xi jk > 20
With the above input data, we solve the problem-I (case-I and case-II) and problem-II as stated earlier using above mentioned GAs. The optimum results are presented below: 7.1. Prob.-I (case-I): All constraint are of equality sign (without entropy function) Here the optimum solutions are obtained using GA presented in Section 6.1 and given below. 7.2. Prob.-I (case-I, with entropy): All constraint are of equality sign and with entropy function In this case, all constraints are of equal type and we use entropy function as another objective.
Following (12)–(14), the fuzzy number are converted to intervals and the constraints to inequalities as (20)–(22). After that, the problem is formulated as a single objective and solved by GA (Section 6.1). The optimum results are in Table 5. 7.6. Efficiency of GA To illustrate the effectiveness and powerfulness of the proposed method i.e. GA, the problem-I (case-I) with entropy and AUD discount system has been solved by weighted average technique. Weighted average formula converts a multi-objective to the single objective function. Here objective function is Z¯ ¼ ðw1 Z þ w2 ðEnÞ=w1 þ w2 Þ; w1 ¼ 1=2 ¼ w2 . The optimum results for the AUD discount system is given below in Table 6.
A. Ojha et al. / Applied Soft Computing 10 (2010) 100–110
107
Fig. 3. Probability of crossover vs costs. Fig. 1. Generations vs costs.
Fig. 4. Probability of mutation vs costs.
Fig. 2. Population vs costs.
It may be noted that the optimum value of Z, i.e., minimum cost for problem-7.2 is less than the present minimum cost. The convergeness of the method, GA is illustrated with the variations in population size, generation (i.e. iteration) number, probability of crossover and mutation. The results are presented in Table 7 and pictocally in Figs. 1–4. The results indicate that the method GA for the present formulation converges with present input data.
8. Discussion From Tables 1–5, it is observed that AUD system gives minimum transportation cost whereas other systems provide more costs, of which IQD is the highest. Moreover, when the constraints are less than equality type (case-II, Table 3), then for all systems, costs reduce considerably. This is because in this case,
Table 1 Optimum transported amounts and min. cost for Prob.-I (case-I, without entropy). Discount system
AUD IQD AUD IQD IQD AUD
Transported amounts x111
x121
x131
x211
x221
x231
x112
x122
x132
x212
x222
x232
26.38 0.34 26.28 1.57 26.16 1.12 27.20 2.46
3.72 11.40 11.50 6.83 0.22 15.31 9.20 6.87
4.88 3.26 0.35 3.46 0.54 6.65 1.20 3.05
0.59 2.69 0.56 1.59 0.48 2.25 0.22 0.11
4.34 5.53 1.75 4.91 5.21 4.26 4.34 4.58
5.08 11.77 4.56 16.63 12.39 5.41 2.82 17.92
Min. cost
Total vechicle cost
671.39
85
739.84
90
679.02
90
710.63
90
Table 2 Near optimum transported amounts and min. cost for Prob.-I (case-I, with entropy). Discount system
AUD
Transported amounts x111
x121
x131
x211
x221
x231
x112
x122
x132
x212
x222
x232
11.53 7.23
9.62 5.93
8.47 7.22
7.27 3.97
4.03 5.42
4.0755 5.24
10.9 7.68 10.31 9.18
7.65 6.77 8.17 6.29
9.41 7.59 9.62 6.42
6.49 4.94 7.02 3.48
5.95 4.62 5.14 5.39
4.59 3.40 4.73 4.23
Min. cost
Max. entropy
Total vechicle cost
832.0
2.43
90
829.79
2.41
85
829.53
2.07
85
A. Ojha et al. / Applied Soft Computing 10 (2010) 100–110
108 Table 2 (Continued ) Discount system
Transported amounts x111
x121
x131
x211
x221
x231
x112
x122
x132
x212
x222
x232
8.56 7.91 8.78 5.87 8.09 6.12
9.46 6.9 8.54 9.37 7.66 9.26
9.83 7.33 10.16 7.28 11.63 7.25
6.62 6.91 10.23 5.12 9.60 6.19
5.10 3.53 2.61 4.48 4.31 3.76
5.42 2.41 4.67 2.89 3.71 2.42
AUD IQD
7.16 7.14 8.53 7.14 9.23 9.59
6.3 9.84 10.63 8.18 9.07 4.65
12.71 6.85 8.16 7.36 9.0 8.46
11.67 4.02 10.19 4.14 7.45 3.72
4.56 4.29 2.4 3.8 4.78 6.5
2.6 2.84 5.1 4.38 5.46 2.08
IQD AUD
9.57 8.37 10.21 7.3 4.39 11.29
8.9 7.76 5.98 8.49 13.44 3.80
8.84 6.56 11.31 6.71 10.05 7.03
7.64 4.42 9.38 3.11 9.0 5.32
4.58 3.76 5.88 4.65 4.36 3.4
5.47 4.12 2.25 4.73 3.76 4.16
IQD
Min. cost
Max. entropy
Total vechicle cost
848.64
2.43
85
857.66
2.40
90
859.44
2.40
90
834.54
2.37
85
838.12
2.41
95
850.85
2.41
90
847.13
2.44
90
843.71
2.40
85
843.55
2.37
85
Table 3 Optimum transported amounts and min. cost for Prob.-I (case-II, without entropy). Discount system
AUD IQD AUD IQD IQD AUD
Transported amounts
Min. cost
x111
x121
x131
x211
x221
x231
x112
x122
x132
x212
x222
x232
7.06 7.08 6.82 3.59 3.59 2.35 5.60 4.6
0.40 5.52 6.07 5.55 5.3 5.15 1.82 4.32
2.54 0.8 3.81 0.14 9.2 2.46 3.67 0.07
2.73 0.03 6.08 0.42 0.54 0.45 3.2 0.71
0.59 2.86 3.29 2.0 2.74 0.67 4.63 4.69
3.0 1.51 0.17 0.74 2.34 0.1 5.27 0.84
Total vechicle cost
Un-exhausted resources
Unfulfilled demand
a1
a2
b1
b2
b3
480.05
70
26.6
19.28
13.1
15.63
17.15
491.68
60
24.02
17.3
13.09
8.09
20.14
489.07
65
21.94
23.16
23.07
11.14
10.89
499.4
60
29.92
10.66
15.89
9.54
15.15
Table 4 Near optimum transported amounts and min. cost for Prob.-I (case-II, with entropy). Discount system
AUD
IQD
AUD IQD
IQD AUD
Transported amounts
Min. cost
x111
x121
x131
x211
x221
x231
x112
x122
x132
x212
x222
x232
4.84 8.38 3.4 8.50 3.83 8.13
2.09 4.61 3.24 2.76 3.36 3.57
1.63 8.4 2.50 8.39 3.18 8.4
4.45 1.20 6.44 1.20 7.19 1.20
8.56 4.88 6.01 4.88 5.38 , 4.88
1.55 2.35 1.55 2.35 1.55 2.35
8.28 8.56 5.47 3.4 4.19 6.25
3.87 5.02 3.51 3.03 5.89 7.48
7.85 5.87 10.78 5.87 7.13 5.87
5.2 1.57 8.45 1.57 8.93 1.57
5.91 4.64 5.68 4.64 2.24 4.64
1.51 2.74 1.51 2.74 1.51 2.74
4.6 10.37 6.03 6.05 6.82 4.4
3.0 3.08 2.92 3.39 3.48 3.38
7.44 8.4 7.27 8.4 6.0 8.4
5.38 1.20 6.84 1.20 9.35 1.2
3.44 4.88 5.01 4.88 5.15 4.88
1.55 2.35 1.55 2.35 1.55 2.35
6.85 2.81 5.76 3.92 5.43 2.9
4.12 7.24 3.19 6.17 4.34 7.09
8.8 8.4 9.49 8.4 8.74 8.4
5.03 1.2 5.28 1.2 4.66 1.2
3.34 4.88 6.97 4.88 6.88 4.88
1.55 2.35 1.55 2.35 1.55 2.35
Max. entropy
Total vechicle cost
Un-exhausted resources
Unfulfilled demand
a1
a2
b1
b2
b3
631.11
2.33
75
20.06
13.92
11.08
4.86
11.08
624.43
2.36
70
21.51
7.57
10.46
8.41
10.21
644.02
2.34
75
19.54
7.45
9.65
7.81
9.53
690.93
2.37
75
10.56
8.43
6.39
5.57
7.03
665.01
2.34
70
17.94
5.41
11.11
8.14
4.1
686.43
2.36
75
13.19
8.37
9.06
4.75
7.75
673.86
2.32
75
13.13
661.60
2.36
70
15.95
665.03
2.34
70
669.83
2.34
681.68 678.0
11.2
8.45
10.6
5.28
8.17
9.88
8.8
5.44
17.53
5.52
8.23
8.11
6.71
75
11.79
11.65
14.11
5.42
3.91
2.35
70
13.08
7.77
13.84
3.79
3.22
2.35
75
13.11
8.48
15.81
1.81
3.97
A. Ojha et al. / Applied Soft Computing 10 (2010) 100–110
109
Table 5 Optimum transported amounts and min. cost for Prob.-II. Discount system
AUD IQD AUD IQD IQD AUD
Transported amounts
Min. cost
x111
x121
x131
x211
x221
x231
x112
x122
x132
x212
x222
x232
21.8 2.66 19.77 1.74 20.38 5.18 15.65 9.54
3.36 5.5 5.89 10.94 13.18 0.9 12.51 5.7
8.74 9.79 6.94 4.56 2.78 9.15 0.28 6.83
1.06 2.95 2.15 4.13 0.24 2.49 0.66 3.63
4.85 10.36 1.22 5.97 0.48 12.30 0.95 5.3
0.73 2.92 8.02 4.89 9.09 1.72 13.24 1.93
Total vechicle cost
Transported resources
Fulfilled demand
a1
a2
b1
b2
b3
708.05
90
51.86
25.87
29.47
24.07
24.18
767.57
80
49.84
26.37
27.79
24.02
24.4
716.98
90
51.57
26.32
28.29
26.87
22.74
766.87
85
50.54
25.71
29.5
24.46
22.3
Table 6 Using weighted average formula in Table 2. Discount system
Transported amounts
AUD
x111
x121
x131
x211
x221
x231
x112
x122
x132
x212
x222
x232
9.84 8.1
9.59 6.23
9.84 6.4
6.76 5.3
5.55 3.62
3.42 5.34
Table 7 Optimum cost for Prob.-7.1 for different values of population, generation and probability of crossover and mutation with AUD discount system. Papulation
Generation
Probability crossover
Probability mutation
Minimum cost
100 100 100 100 100
1000 800 600 400 200
0.7 0.7 0.7 0.7 0.7
0.3 0.3 0.3 0.3 0.3
671.39 671.39 675.36 679.54 682.25
117 115 110 100 90 80 70 60
1000 1000 1000 1000 1000 1000 1000 1000
0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7
0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3
670.10 670.10 670.10 671.39 676.62 673.52 672.48 677.52
100 100 100 100 100 100 100
1000 1000 1000 1000 1000 1000 1000
0.27 0.3 0.4 0.5 0.6 0.7 0.8
0.3 0.3 0.3 0.3 0.3 0.3 0.3
671.0 662.52 668.33 674.03 675.83 671.39 674.29
100 100 100 100 100
1000 1000 1000 1000 1000
0.7 0.7 0.7 0.7 0.7
0.22 0.3 0.4 0.5 0.62
678.79 671.39 671.5 672.02 672.79
some resources are not completely exhausted and demands are not satisfied fully. But when the constraints are less than equality type and introducing entropy (case-II, Tables 2 and 4) in AUD system is introduced, then transported amounts to destinations are readjusted and the total minimum costs increases and are more than the corresponding case without entropy (cf. Table 3). As this is a multi-objective optimization problem, some compromise solutions are presented. It is observed that resources at different cells are better distributed than the case-I and case-II (without entropy). In these cases, un-exhausted resources also reduce considerably and demands are better satisfied.
Min. cost
Optimum Entropy
837.32
2.43
When the resources, demands and conveyance charge are fuzzy, the behavior of objective function remains same and the allotment behavior is the same as in the case-I. The cost values are more than the corresponding values of the case-I, as the constraints (22)–(25) are satisfied for the values which are higher than the constant resource values of case-I (cf. Table 4). In Table 6, the efficiency of GA is establised and the corresponding minimum cost 832.0 (highest amongst the set of values) in Table 2 is less than the cost in Table 6. From Table 7 and Figs. 1–4 sword the convergence of the GA against generation number, population size, crossover probability and mutation probability. 9. Conclusions and future research work In this paper, we analyzed a multi-item transportation problem with AUD, IQD and combination of these two price breaks, linear cost function, fixed charge and vechicle cost and solved by genetic algorithm. Here we discussed four types of the problem. In case-I, all constraints are equal type with and without entropy function, in case-II, constraints are less then equal type with and without entropy, case-III is with equal constraints where resources, demands, conveyances are triangular fuzzy numbers. The entropy function is constructed from the concept of Shannon’s measure of entropy. It acts as a measure of dispersal of trips among the origins, destinations and conveyances of the transportation model and is used as a second objective. The concept of entropy presented here is quite general in nature and can be extended to other fields of operation research like supply chain model, market research, etc. Different losses like breakability damage of the items during the transportation, etc., can be incorporated in the proposed STP and this may be the future research work. References [1] A.K. Bit, M.P. Biswal, S.S. Alam, Fuzzy programming approach to multi-objective solid transportation problem, Fuzzy Sets and Systems 57 (1993) 183–194. [2] A.K. Bit, M.P. Biswal, S.S. Alam, An additive fuzzy programming model for multi-objective transportation problem, Fuzzy Sets and Systems 57 (1993) 313–319. [3] S. Chanas, D. Kuchta, A concept of the optimal solution of the transportation problem with fuzzy cost coefficients, Fuzzy Sets and System 82 (1996) 299–305. [4] V. Damodaran, R.S. Lashkari, N. Singh, A production planning model for cellular manufacturing systems with refixturing considerations, International Journal of production research 30 (7) (1992) 1603–1615.
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