R E S E A R C H A R T I C L E
A Fuzzy Logic Based Approach to Solve Interval Multiobjective
Nonlinear Transportation Problem
Hasan Dalman1 • Mustafa Sivri2
Received: 6 July 2015 / Revised: 28 October 2016 / Accepted: 11 November 2017 / Published online: 6 March 2018 Ó The National Academy of Sciences, India 2018
Abstract This paper presents the solution procedure of multiobjective nonlinear transportation problem (MNOTP) where the cost coefficients of the objective functions, and the supply and the unknown demand parameters have been formulated as interval numbers by the decision maker. This problem has been converted into a conventional MNOTP where to minimize the interval nonlinear objective func-tions, the order relations that define the choice between intervals have been determined by the interval arithmetic. Also, the constraints with interval supply and unknown demand parameters have been transformed into its deter-ministic forms. Then the deterdeter-ministic problems have been solved by two compromise programming methods. Finally, a numerical example is presented to illustrate the efficiency of the proposed procedure as well.
Keywords Fuzzy programming Interval numbers Transportation problem Multiobjective programming Nonlinear programming
1 Introduction
Transportation problem (TP) aims to find the best way to fulfill the demand of n demand points using the capacities of m supply points and has wide practical applications in logistic systems, human resources management, inventory
control, production planning, etc. The parameters of the transportation problem are unit costs, supply and demand quantities. The traditional TP in real life deals with various issues such as selection of sources and delivery routes given the destinations; handling and packing; financing and insurance; duty and taxes, and is related to other operations such as selection of production place and capacity, decision on outsourcing of production, hiring human capital etc.
Mathematical formulation of the transportation cost flow problem was formulated by Hitchcock [1]. TP uniformly follows a special mathematical structure in its constraints. Source parameters may be production units, facilities, basic warehouses, etc., while destination parameters may include common or different warehouses, sales outlets, etc. Penalty factors or coefficients of objective functions may represent transportation cost, average delivery time of goods, unsatisfied demand. Most of the real world problems are naturally expressed by multiple and conflicting aspects of evaluation. Consequently, transportation problems are often formulated as multiobjective problems. Isermann [2] introduced a method, for solving a linear multiobjective TPs, by which the set of all efficient solutions was identified.
Moreover, when discussing real-world problems, usu-ally the parameters are imprecise quantities due to various uncontrollable factors. Interval numbers are very appro-priate for modeling these conditions. The essential idea of this paper is to present a multiobjective nonlinear trans-portation problem (MNOTP) with uncertain coefficients and variables. Uncertainty can occur because the infor-mation are lacking or the data are approximate. For this type of uncertainty usually fuzzy and interval numbers are used.
After developing the idea of fuzzy set theory by Zadeh [3], Bellman and Zadeh [4] and Zimmermann [5] used the & Hasan Dalman
hsandalman@gmail.com
1 Department of Computer Engineering, Istanbul Gelisim University, Avcilar, Istanbul, Turkey
2 Department of Mathematical Engineering, Yildiz Technical University, Esenler, Istanbul, Turkey
fuzzy programming method with some related membership functions to solve multiobjective linear programming problems. By using linear membership function, Bit et al. [6] employed the fuzzy programming method to solve the multiobjective transportation problem.
The interval programming based on interval arithmetic was introduced to design the uncertainty in uncertain pro-gramming problems in which the bounds of the uncertain parameters are only needed, not necessarily knowing the probability distributions or membership functions. Moore [7] introduced the interval arithmetic and Moore [8] rede-fined pointing to some concepts and notations. He devel-oped two transitive order relations between intervals to obtain deterministic sub problems. One of them is depen-dent just on the bounds of the intervals and the other is based on the set inclusion property of intervals.
Inuiguchi and Kume [9] and Inuiguchi and Sakawa [10] studied two different methods to investigate an interval objective function: the satisfying method and the opti-mizing method. In the satisfying method, each interval objective function is transformed into one or several objective functions in order to determine compromise solution. However, the optimizing method increases the concept of efficiency used in traditional multiobjective programming problem to the interval objective function event e.g. Bitran, [11]; Steuer, [12]; Wang and Wang, [13]). Ishibuchi and Tanaka [14] defined more suit-able ordering relation for decision makers. Rommelfanger [15] considered the linear programming problem with interval parameters in the objective function. Chanas and Kuchta [16] proposed a different ordering based on the cut off interval number to convert the linear programming problem with uncertainty into a deterministic programming problem. Chen et al. [17] also introduced some methods to solve interval programming problems. Urli and Nadeau [18] took multiobjective programming models with inter-val coefficients in the whole model, but the results did not provide the decision-maker to take into account the worst-case and the best-worst-case scenarios in order to recognize the risk at the stage. A new method based on fuzzy sets is proposed for interval transportation problems by consid-ering the right bound and midpoint of the interval [19]. Sengupta et al. [20] considered interval linear program-ming problems in which coefficients of the objective function and inequality constraints are all interval numbers. They proposed the theory of acceptability index and pro-vided a solution procedure for interval linear programming. Yu et al. [21] proposed an interactive method to determine the preferred compromise solution for the multiobjective transportation problem where the coefficients of the objective functions and the source and destination param-eters have been considered as interval numbers to handle the uncertainties in the parameters. Hladı´k [22, 23]
investigated the interval linear programming problem and introduced some concepts of intervals.
The linear functions are the most useful and extensively used in operational research. Also, quadratic functions and quadratic problems are the least complex ones to handle out of all nonlinear programming problems. A fair number of functional relationships occurring in the real world are actually quadratic. For example, the kinetic energy carried by a rocket or an atomic particle is proportional to the square of its velocity, in statistics, the variance of a given sample of observations is a quadratic function of the values that constitute the sample. So there are countless other non-linear relationships occurring in nature, capable of being approximated by quadratic functions.
Ju-Long [24] presented the grey system theory and Ju-Long [25] introduced grey decision-making systems. This theory has a wide application in the field of decision making (Huang et al. [26]; Chen and Huang [27]; Kong et al. [28]). Huang et al. [26] introduced two interval quadratic programming methods of intervals and fuzzy numbers inside quadratic structures. They employed them to the planning for solid waste management problems. Chen and Huang [27] developed a series of solution algorithms for solving interval quadratic problems and applied them to environmental management under uncer-tainty. Hajiagha et al. [29] developed a method to solve the interval programming problems. This method transformed an interval linear programming method into two equivalent models for its lower bound and upper bound. They devel-oped a fuzzy programming method for solving interval multiobjective problems [30] and then, their method based on membership degrees is itself an interval linear pro-gramming which can be solved by known methods. Sivri et al. [31] and Gu¨zel et al. [32] concerned with the multi-objective version of the transportation problem. They proposed a solution procedure based on Taylor series expansion. Dalman et al. [33] proposed a solution proce-dure based on Taylor series expansion for solving interval quadratic transportation problem. Some nonlinear interval programming methods are studied by Jiang et al. [34], Liu and Wang [35], Li and Tian [36].
This paper dealt with the interval MNOTP in which all the parameters are expressed as intervals. Expressing the parameters as interval makes decision maker (DM) more flexible and this enables to consider tolerances for the model parameters in a more natural and direct way. Therefore, interval MNOTP seems to be more realistic and reliable according to crisp numbers. In this paper, we introduce a new fuzzy programming method based on interval numbers as applied to generalized MNOTP. In the first step, the interval MNOTP is transformed into a crisp one using an order relation technique. To determine the best and worst solutions of each interval objective function,
compromise programming model is used. In the second step, the method attempts to reach the better compromise solution which simultaneously satisfied another objective based on an interval fuzzy programming method.
2 Preliminaries and Problem Formulation
In this paper, we assumed that the parameters of MOSTP are expressed as interval numbers. In this section, brief information about the interval numbers are presented. 2.1 Interval Numbers
An interval number is a number whose exact value is unknown, but a range within which the value lies is known (Huang et al. [26]). Interval number is a number with both lower and upper bounds X2 x; ½ x where x x. The main arithmetic operations can be defined on interval numbers. Let ~x1¼ x½ 1; x1 and ~x2 ¼ x½ 2; x2 be a closed interval
numbers. The following notations can be satisfied (Moore et al. [7]): ~ x1þ ~x2¼ x½ 1þ x2; x1þ x2 ~ x1 ~x2¼ x½ 1 x2; x1 x2 ~ x1 ~x2¼ min x½ ð 1x2; x1x2; x1x2; x1x2Þ; max xð 1x2; x1x2; x1x2; x1x2Þ ~ x1 ~x2¼ x½ 1; x1 1 x2; x1 ½
when X2 x; ½ x is an interval number, its absolute value is the maximum of the absolute value of its endpoints:
x
j j ¼ max xðj j; j jxÞ.
The center, xcand xw of a grey number of X2 x; ½ x is
defined as follows: xc¼ 1 2½xþ x xw¼ 1 2½x x 8 > < > : ð1Þ
It is easily verifiable that x¼ xcþ xwand x¼ xc xw.
However Ishibuchi and Tanaka [14] defined the order relations between intervals.
Definition 1 Let ~x¼ x; ½ x and ~y¼ y; h yiare two closed interval numbers and then the order relationLRis defined
as: ~
xLRy~,¼ x y and x y ð2Þ
~
x\LRy~,¼ x LRy and x6¼ y ð3Þ
Definition 2 The order relation CW between two grey
numbers ~x¼ x; ½ x and ~y¼ y; h yiis defined as: ~
xCW y~, xC yC and xW yW ð4Þ
~
x\CWy~, ~xCWy~ and x6¼ y ð5Þ
The order relations CW and LR never conflicts with
each other. Similarly, Ishibuchi and Tanaka [14] defined the order relations
LR and CW for minimization
problems.
Definition 3 The order relation
LR between two
inter-val numbers ~x¼ x; ½ x and ~y¼ y; h yiis defined as ~ xLR ~y, x y and x y ð6Þ ~ x\LRy~, ~x LRy~ and ~x6¼ ~y ð7Þ
Definition 4 The order relation
CW between two
interval numbers ~x¼ x; ½ x and ~y¼ y; h yiis defined as: ~ x CW y~, xC yC and xW yW ð8Þ ~ x\CWy~, ~x CWy~ and x~6¼ ~y ð9Þ
2.2 Interval Multiobjective Programming Problem An interval multiobjective programming problem can be formulated as follows: minðmaxÞflðxÞ ¼ Xn j¼1 cljxj; l¼ 1; 2; . . .; k: s:t: P j2kþaijxj ¼ 0 B B @ 1 C C Ab; i ¼ 1; 2; . . .; m xi 0; j¼ 1; 2; . . .; n 8 > > > > < > > > > : : ð10Þ
where all coefficients c1j; aij and variables xj are the
intervals.
Hajiagha et al. [30] suggested a different method to optimize this type problems. The method transformed an interval linear programming method into two equivalent models for its lower bound and upper bound. Suppose that Kþincludes those variables that their objective coefficients have both positive lower and upper bound, K includes those variables that their objective coefficients have both negative lower and upper bound and K0 includes those
variables that their objective coefficients have different sign and contain zero in their intervals. Then, the objective function fl can be formulated as follows:
flð Þ ¼x X j2kþ cþl j x þ j þ X j2k cl j x j þ X j2k0 c0l j x 0 j X j2kþ cþl j x þ j þ X j2k cl j x j þ X j2k0 c0l j x 0 j 2 6 6 6 4 3 7 7 7 5 ð11Þ
Also, the left hand side of each constraint can be written as P j2kþ aijxjwhere aij2 aij; aij i.e.; X j2kþ aþijxþj þX j2k aijxj þX j2k0 a0ijx0j ð12Þ where aþij; aij and a0
ij are the associated coefficients of the
variables. The constraints of a programming problem is an interval number which can be shown as L; U½ where L and U are the lower and upper bounds of constraints. Using the interval numbers. L; U ½ b½ i; bi ) U bi L 2þ L 2 bi 2þ bi 2 8 <
: ðusing the order relation RC;Þ ð13Þ L; U ½ b½ i; bi ) L bi L 2þ L 2 bi 2þ bi 2 8 <
: ðusing the order relation
LC;Þ ð14Þ L; U ½ ¼ b½ i; bi ) U¼ xj L¼ bi ( ð15Þ
2.3 Mathematical Formulation of Nonlinear Transportation Problem with Convex Costs Anholcer [37, 38] analyzed the relation between the reduction ratio and the construction of the optimal solution and some modification of the problem were also studied.
Here, we present the nonlinear transportation problem (NTP), where the transportation costs and the costs that depend on the quantity of goods delivered to the destina-tion points are strictly convex funcdestina-tions. The quantities of goods change during the transportation process. This pat-tern may match, for instance, with a jammed network where the time costs are included. This kind of cost functions seems, for instance, when a jammed network is taken into consideration and the transportation costs include also the cost of time [37,38].
In order to construct the mathematical model for the NTP, some notations and assumptions are listed as follows: xj; the total amount of good delivered to destination j for
j¼ 1; 2; . . .; n: fj xj
; quadratic convex functions in xj:
xij; the total of good transfer from supply i to destination
j for i¼ 1; 2; . . .; m and j ¼ 1; 2; . . .; n:
rij; the respective discount ratio for i¼ 1; 2; . . .; m and
j¼ 1; 2; . . .; n:
rijxij; the amount of good that reaches destination j for
j¼ 1; 2; . . .; n:
cij; the unit transportation costs from supply i to
destination j for i¼ 1; 2; . . .; m and j ¼ 1; 2; . . .; n: ai; the total supply of the product from supply i for
i¼ 1; 2; . . .; m:
bj; the minimal demand of products in destination i for
i¼ 1; 2; . . .; m:
Under the above information, the mathematical model of nonlinear transportation problem can be formulated as follows: min fðxÞ ¼X m i¼1 Xn j¼1 cij xij þX n j¼1 fj xj s:t: Pm i¼1rijxij¼ xj; j¼ 1; 2; . . .; n Pn j¼1xij ai; i¼ 1; 2; . . .; m xij 0; i¼ 1; 2; . . .; m; j¼ 1; 2; . . .; n 8 > < > : : ð16Þ In the above model, the following condition is generally satisfied: Xm ji1 ai Xn j¼1 bj
The opinion about the linearity of costs is not practical. In real applications, the cost cij usually is a nonlinear function in xij; increasing and convex for xij. Furthermore,
some extra costs may seem at destination points. This may be the costs of converting the carried goods at the destination points or the distribution and promotion costs. In the case of unknown demand, one may involve the belief of the deficit and remainder costs. In those cases we can consider that the cost functions are convex. Thus we assume that an increasing convex function is assigned to every destination point j: The restriction bj on the amount of goods produced to every destination j is not covered in the index of constraints, i.e. we assume that the demand is not restricted in a small time horizon, which is also a practical opinion for many goods. We suppose that each of the functions cij and fj is differentiable at all point of its domain. In addition, multiobjective nonlinear transportation problem can be described in mathematical terms as follows: min f1ðxÞ; f2ðxÞ; . . .; flðxÞ s:t: Pm i¼1rijxij¼ xj; j¼ 1; 2; . . .; n Pn j¼1xij ai; i¼ 1; 2; . . .; m xij 0; i¼ 1; 2; . . .; m; j ¼ 1; 2; . . .; n 8 > < > : : ð17Þ where flð Þx is defined as a function by Pm i¼1 Pn j¼1clij xij þPnj¼1fl j xj for l [ 1:
The above model assumes that all variables and coeffi-cients are deterministic. But in the real world, it is difficult for us to estimate these values exactly. If there is poor data of the information about them, we can consider them as interval numbers. However, when we are deficit of history data, or history data is wrong because of sudden situations. In this case, we normally have some field specialist to decide the belief degree that each event occurs. This expert data is just the subject of the uncertainties.
Thus, the mathematical formulation of multiobjective nonlinear transportation problem with interval numbers can be formulated as follows: min flðxÞ ¼ Xm i¼1 Xn j¼1 cl ij; clij h i xij; xij þX n j¼1 fl j; f l j h i xj; xj ; l¼ 1; 2; . . .; k s:t: Pm i¼1 rij; rij xij; xij ¼ xj; xj ; j¼ 1; 2; . . .; n Pn j¼1 xij; xij a½ i; ai; i¼ 1; 2; . . .; m xij 0; xij xij; i¼ 1; 2; . . .; m; j¼ 1; 2; . . .; n 8 > < > : ð18Þ 2.4 Deterministic Equivalences of Interval
Nonlinear Multiobjective Transportation Problem
Theorem 1 In model (18), for the interval numbers’ lower bound is smaller than its upper’s bound. According to definition of interval numbers multiplication, we can easily obtain Theorem 1.
The center and width of a function is defined as given in Eq. (1). Thus, the following concepts are obtained for the interval transportation cost functions.
Definition 5 The order relation LC between two
inter-val numbers ~x¼ x; ½ x and ~y¼ y; h yiis defined as: ~
xLC~y, x y and x~C ~yC ð19Þ
~
x\LCy~, ~xLCy~ and x~6¼ ~y ð20Þ
Definition 6 If x be feasible solution of multiobjective nonlinear transportation problem (18) with maximization cost function, it is an optimal solution if there are not any feasible solution x0 that flð Þ fx lð Þ for all l ¼ 1; 2; . . .; k:x0
Here, the solution of nonlinear problem (18) with maximization cost function can be constructed as the set of Pareto optimal solutions of the following multi objective problem:
max f
lð Þ; fx lCð Þx
ð21Þ
where flð Þ and fx lCð Þ are the lower bound and center ofx
transportation cost function. If the problem has a mini-mization cost function, then the
RC defined as follows.
Definition 7 The order relation
RC between two
interval numbers ~x¼ x; ½ x and ~y¼ y; h yiis defined as: ~
xRCy~, xC yC and xR yR ð22Þ
~
x\RCy~, ~xRC y~ and x~6¼ ~y ð23Þ
Definition 8 If x be any solution of nonlinear problem (18) with maximization objective function, it is an optimal solution if there are not any feasible solution x0 that flð Þ fx0 lð Þ for all l ¼ 1; 2; . . .; k:x
Here, the solution of model (18) with minimization cost function can be constructed as the set of Pareto optimal solutions of the following multi objective problem:
min fðlCð Þ; x flð ÞxÞ ð24Þ
where flCð Þ and x flð Þ are the center and the upper bound ofx
transportation cost function, respectively.
Constraints of nonlinear transportation model (18), have programming problems and can be composed of one inequalities framework and an equality framework. For each kind of constraints, the order relations as described in the above definitions can be used to convert them into the related form that can be investigated with current methods. The constraints of a programming problem is an interval number which can be shown (from Eqs. (13–15)) as follows: Pm i¼1rijxij¼ xj; j¼ 1; 2; . . .; n Pm i¼1rijxij¼ xj; j¼ 1; 2; . . .; n Pn j¼1xij ai; i¼ 1; 2; . . .; m Pn j¼1 xijþ xij 2 aiþ ai 2 ; i¼ 1; 2; . . .; m xij 0; xij xij i¼ 1; 2; . . .; m; j¼ 1; 2; . . .; n 8 > > > > > < > > > > > : ð25Þ
Thus, we can easily obtain two deterministic multiobjective nonlinear sub problems for nonlinear transportation problem (18) as follows:The above problems are then solved, the ideal objective vector of problem (18) is determined as flopt¼ flC; fl
: Here, the objective function flC has accepted the lower bound value of problem. Then the objective function value f
l can be
3 An Interval Fuzzy Programming Method
to NMTP
In order to construct an interval fuzzy method, we consider the l th interval cost function of model (18). After deter-mining the compromise optimal range flopt ¼ fh l; fli from deterministic models of problem (18), its membership function for the minimization problems is determined as follows: llðxÞ ¼ 1 flð Þ fx l fl flð Þx fl fl flð Þ fx l 8 > < > : ð27Þ
where the decreasing of flð Þ increases the membershipx degree llðxÞ.
Similarly, for maximization type objective, the mem-bership function is determined as follows:
llðxÞ ¼ 1 flð Þ x fl flð Þ fx l fl fl flð Þ x fl 8 > < > : ; ð28Þ
where the increasing of flð Þ increases the membershipx degree llðxÞ.
After constructing the membership function (27) and/or (28), interval nonlinear transportation model (18) is trans-formed into the following interval fuzzy programming problem: max ll¼ l l; ll h i n o ; l¼ 1; 2; . . .; k s:t: ll¼ l l; ll h i 1 Pm i¼1 rij; rij xij; xij ¼ xj; xj ; j¼ 1; 2; . . .; n Pn j¼1 xij; xij a½ i; ai; i¼ 1; 2; . . .; m xij; xij 0; i¼ 1; 2; . . .; m; j¼ 1; 2; . . .; n 8 > > > > > < > > > > > : ð29Þ This problem is converted into the following equivalent interval problem: max ll¼ ll; ll h i n o ; l¼ 1; 2; . . .; k s:t: fl flð Þ; x flð Þx h i fl fl 1 ) flð Þ; x flð Þx h i fl Pm i¼1 rij; rij xij; xij ¼ xj; xj ; j¼ 1; 2; . . .; n Pn j¼1xij; xij a½ i; ai; i¼ 1; 2; . . .; m xij; xij 0; i¼ 1; 2; . . .; m; j¼ 1; 2; . . .; n 8 > > > > > > > < > > > > > > > : ð30Þ Then, we have the following deterministic equivalent model: maxX k l¼1 l lþ llC s:t: flð Þ fx l flCð Þ fx l Pm i¼1rijxij¼ xj; j¼ 1; 2; . . .; n Pm i¼1rijxij¼ xj; j¼ 1; 2; . . .; n Pn j¼1xij ai; i¼ 1; 2; . . .; m Pn j¼1 xij 2 þ xij 2 ai 2þ ai 2 ; i¼ 1; 2; . . .; m xij; xij 0; i¼ 1; 2; . . .; m; j¼ 1; 2; . . .; n 8 > > > > > > > > > > > > > < > > > > > > > > > > > > > : ð31Þ min flðxÞ ¼P m i¼1 Pn j¼1 cl ij xij þP n j¼1 fl j xj ; l¼ 1; 2; . . .; k min flCðxÞ ¼P m i¼1 Pn j¼1 cl ij xij 2 þ cl ij xij 2 ! þP n j¼1 fl j xj 2 þ fl j xj 2 ! ; l¼ 1; 2; . . .; k s:t: Xm i¼1 rijxij¼ xj; j¼ 1; 2; . . .; n Xm i¼1 rijxij¼ xj; j¼ 1; 2; . . .; n Xn j¼1 xij ai; i¼ 1; 2; . . .; m Xn j¼1 xijþ xij 2 aiþ ai 2 ; i¼ 1; 2; . . .; m xij 0; xij xij i¼ 1; 2; . . .; m; j¼ 1; 2; . . .; n 8 > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > : s:t: Xm i¼1 rijxij¼ xj; j¼ 1; 2; . . .; n Xm i¼1 rijxij¼ xj; j¼ 1; 2; . . .; n Xn j¼1 xij ai; i¼ 1; 2; . . .; m Xn j¼1 xijþ xij 2 aiþ ai 2 ; i¼ 1; 2; . . .; m xij 0; xij xij; i¼ 1; 2; . . .; m; j¼ 1; 2; . . .; n 8 > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > : ð26Þ
It should be noted that if the decision maker has some priority preferences over different objectives, the objective function of the model (30) can be changed by Pk
l¼1wl llþ llC
where wl is the weight of objective function such that wl 0: The concepts of Pareto optimal and fuzzy efficient solution are given as follows.
Definition 9 (Jimenez and Bilbao [39]) Assume that the feasible set of model (30) is X and x0 2 X is an efficient to
the model (30) if there is not any other solution x 2 X such that llð Þ lx0 lð Þ and lx rð Þ [ lx rð Þ at least one index r:x0
Lemma 1 Assume that the feasible set of a model (30) is X and x02 X is an efficient to the model (30). Then x02 X
is a Pareto optimal to interval nonlinear transportation model (18).
Proof According to Definition 9, x02 X is an efficient to
the model (30) if there is not any other solution x 2 X such that llð Þ lx0 lð Þ and lx lð Þ [ lx0 lð Þ at least one index r:x
Definition of Pareto optimal (when the minimization problem) llð Þ lx0 lð Þx is equivalent to say that
flð Þ fx0 lð Þ which is clear from the definition of mem-x
bership functions in Eqs. (27) and (28). Thus, lemma is proved.
3.1 The Global Criterion Method
The global criterion method is generally called a compro-mise programming method. In this method, the distance between some reference point and the feasible region of the objective functions are minimized. The investigator has to decide the reference point and the metric for measuring the distances. All the objective functions are considered to be equally important.
In this method, we considered the global criteria method where the ideal objective vector is employed as a reference point. If the problem has a maximization objective func-tion, the objective function of problem is formulated as follows: min Lp¼ flð Þ fx l f l p þflCð Þ fx lC flC p ( )1=p ð32Þ where f
lð Þ is the lower bound of interval objective func-x
tion (18) and flCð Þ is its center of interval objectivex function. flopt ¼ f
lC; f l
is a reference point obtained from deterministic model (26).
If the problem has a minimization objective function (from Definition 8), the objective function of problem is formulated as follows: min Lp¼ flð Þ x fl fl p þflCð Þ fx lC flC p 1=p ð33Þ
where flð Þ is the upper bound of interval objective func-x tion (18) and flCð Þ is its center of interval objectivex function. flopt ¼ fh l; flCiis a reference point obtained from deterministic model of problem (26).
4 Numerical Example
To illustrate the proposed method we consider the fol-lowing interval NMOTP model.
Suppose there are two supply points and two unknown destination points. Let,
a1¼ ½a1; a1 ¼ ½10; 15; a2¼ a1¼ ½a1; a1 ¼ ½12; 17;
Interval unit transportation cost for the first objective c1 ij¼ c1ij; c1ij h i c1ij¼ c ð1Þ 11 c ð1Þ 12 cð1Þ21 cð1Þ22 " # ¼ 0:80 0:75 0:35 0:40 ; c1ij¼ c 1 11 c 1 12 c1 21 c 1 22 ¼ 0:90 0:95 0:65 0:70 ;
Interval unit transportation cost for the second objective c2 ij¼ c2ij; c2ij h i c2ij¼ c ð2Þ 11 c ð2Þ 12 cð2Þ21 cð2Þ22 " # ¼ 0:70 0:15 0:50 0:40 ; c2ij¼ c 2 11 c212 c2 21 c222 ¼ 0:90 0:30 0:70 0:60 ; Interval quadratic functions of demand j:
f1 1ð½x1; x1Þ ¼ ½0:2; 0:5 ½x21; x21 þ ½5; 3 ½x1; x1 þ ½5; 10 f1 2ð½x2; x2Þ ¼ ½0:15; 0:8 ½x22; x22 þ ½7; 4 ½x2; x2 þ ½4; 10 ; f1 2ð½x1; x1Þ ¼ ½0:3; 0:5 ½x 2 1; x 2 1 þ ½5; 4 ½x1; x1 þ ½8; 10 f2 2ð½x2; x2Þ ¼ ½0:7; 0:9 ½x22; x22 þ ½8; 6 ½x2; x2 þ ½4; 9 Interval discount rates
r11 ¼ ½0:1; 0:9; r12¼ ½0:6; 0:8
r21 ¼ ½0:8; 0:9; r12¼ ½0:7; 0:9
Here, by using the above data, the nonlinear transportation model is formulated as follows:
min flðxÞ ¼X 2 i¼1 X2 j¼1 cl ij; clij h i xij; xij þX 2 j¼1 fl j; f l j h i xj; xj ; l¼ 1; 2: s:t: P2 i¼1½ri1; ri1 x½ i1; xi1 ¼ x½ 1; x1; P2 i¼1½ri2; ri2 x½ i2; xi2 ¼ x½ 2; x2 P2 j¼1½xi1; xi1 a½ 1; a1 P2 j¼1½xi2; xi2 a½ 2; a2; xij 0; xij xij; i¼ 1; 2:; j¼ 1; 2: 8 > > > > > > > < > > > > > > > :
Therefore, the above problem is rewritten as follows: min f1ð Þ ¼ ½0:8; 0:9½xx 211; x 2 11 þ ½0:75; 0:95½x 2 12; x 2 12 þ ½0:35; 0:65 ½x2 21; x 2 21 þ ½0:4; 0:7 ½x 2 22; x 2 22 þ ½0:2; 0:5 ½x2 1; x 2 1 þ ½5; 3 ½x1; x1 þ ½5; 10 þ ½0:15; 0:80 ½x2 2; x 2 2 þ ½7; 4 ½x2; x2 þ ½4; 10 min f2ð Þ ¼ ½0:7; 0:9½xx 211; x 2 11 þ ½0:15; 0:3½x 2 12; x 2 12 þ ½0:5; 0:7 ½x2 21; x 2 21 þ ½0:4; 0:6 ½x 2 22; x 2 22 þ ½0:3; 0:5 ½x2 1; x 2 1 þ ½5; 4 ½x1; x1 þ ½8; 10 þ ½0:7; 0:9 ½x2 2; x 2 2 þ ½8; 6 ½x2; x2 þ ½4; 9 ðP1Þ s:t: ½0:1; 0:9 ½x11; x11 þ ½0:8; 0:9 ½x21; x21 ¼ ½x1; x1 ½0:6; 0:8 ½x12; x12 þ ½0:7; 0:9 ½x22; x22 ¼ ½x2; x2 ½x11; x11 þ ½x12; x12 ½10; 15 ½x21; x21 þ ½x22; x22 ½12; 17 xij 0; i ¼ 1; 2:; j ¼ 1; 2: 8 > > > > < > > > > :
The above problem transformed into the following two equivalent problems: min f1ð Þ ¼ ½0:8; 0:9½xx 211; x 2 11 þ ½0:75; 0:95½x 2 12; x 2 12 þ ½0:35; 0:65 ½x2 21; x 2 21 þ ½0:4; 0:7 ½x2 22; x 2 22 þ ½0:2; 0:5 ½x 2 1; x 2 1 þ ½5; 3 ½x1; x1 þ ½5; 10 þ ½0:15; 0:80 ½x 2 2; x 2 2 þ ½7; 4 ½x2; x2 þ ½4; 10 s:t: ½0:1; 0:9 ½x11; x11 þ ½0:8; 0:9 ½x21; x21 ¼ ½x1; x1 ½0:6; 0:8 ½x12; x12 þ ½0:7; 0:9 ½x22; x22 ¼ ½x2; x2 ½x11; x11 þ ½x12; x12 ½10; 15 ½x21; x21 þ ½x22; x22 ½12; 17 xij 0; i ¼ 1; 2:; j ¼ 1; 2: 8 > > > > < > > > > : ðP2Þ min f2ð Þ ¼ ½0:7; 0:9½xx 211; x 2 11 þ ½0:15; 0:3½x 2 12; x 2 12 þ ½0:5; 0:7 ½x2 21; x 2 21 þ ½0:4; 0:6 ½x 2 22; x 2 22 þ ½0:3; 0:5 ½x2 1; x 2 1 þ ½5; 4 ½x1; x1 þ ½8; 10 þ ½0:7; 0:9 ½x22; x 2 2 þ ½8; 6 ½x2; x2 þ ½4; 9 s:t: ½0:1; 0:9 ½x11; x11 þ ½0:8; 0:9 ½x21; x21 ¼ ½x1; x1 ½0:6; 0:8 ½x12; x12 þ ½0:7; 0:9 ½x22; x22 ¼ ½x2; x2 ½x11; x11 þ ½x12; x12 ½10; 15 ½x21; x21 þ ½x22; x22 ½12; 17 xij 0; i ¼ 1; 2:; j ¼ 1; 2: 8 > > > > < > > > > : ðP3Þ At first, objectives of the problem (P2) and (P3) are transformed into its deterministic functions as follows:
f1ð Þ :x ¼ 0:8x2 11þ 0:75x 2 12þ 0:35x 2 21þ 0:4x 2 22þ 0:2x 2 1 5x1 þ 0:15x22 7x2þ 9 f1ð Þ ¼ 0:9x x211þ 0:95x 2 12þ 0:65x 2 21þ 0:7x 2 22þ 0:5x 2 1 3x1 þ 5 þ 0:15x22 4x2þ 5 f1Cð Þ ¼ 0:4xx 211þ 0:9 2 x 2 11þ 0:75x2 12 2 þ 0:95x2 12 2 þ 0:35x2 21 2 þ0:65x 2 21 2 þ 0:2x 2 22þ 0:7 2 x 2 22þ 0:1x 2 1þ 0:5 2 x 2 1 3 2x1 5 2x1þ 0:15 2 x 2 2þ 0:40x 2 2 2x2 7 2x2 þ10 2 þ 9 2 f 2ð Þ :x ¼ 0:7x2 11þ 0:15x 2 12þ 0:5x 2 21þ 0:4x 2 22þ 0:3x 2 1 5x1 þ 8 þ 0:7x2 2 8x2þ 4 f2 : ¼ 0:9x211þ 0:3x212þ 0:7x212 þ 0:6x222þ 0:5x21 4x1þ 8 þ 0:9x22 6x2þ 9 f2Cð Þ :x ¼0:7 2 x 2 11þ 0:15 2 x 2 12þ 0:5 2 x 2 21þ 0:4 2 x 2 22þ 0:3 2 x 2 1 5 2x1þ 0:7 2 x 2 2 7 2x2þ 0:9 2 x 2 11þ 0:3 2 x 2 12 þ0:7 2 x 2 21þ 0:6 2 x 2 22þ 0:5 2 x 2 1 4 2x1þ 0:9 2 x 2 2 6 2x2 þ 6 þ17 2
Constraints of both problems are converted to deterministic constraints as follows: 0:1x11þ 0:8x21¼ x1 x11þ x12 15 0:9x11þ 0:9x21¼ x1 x11 2 þ x11 2 þ x12 2 þ x12 2 10 2 þ 15 2 0:6x12þ 0:7x22¼ x2 x21þ x22 17 0:8x12þ 0:9x22¼ x2 x21 2 þ x21 2 þ x22 2 þ x22 2 12 2 þ 17 2 xij 0; xij xij; i¼ 1; 2:; j ¼ 1; 2; 3: ðP4Þ Thus, the problems (P2) and (P3) are converted into two deterministic multiobjective programming model as follows: f1Cð Þ ¼ 0:4xx 211þ 0:9 2 x 2 11þ 0:75x2 12 2 þ 0:95x2 12 2 þ 0:35x2 21 2 þ0:65x 2 21 2 þ 0:2x 2 22þ 0:7 2 x 2 22þ 0:1x 2 1þ 0:5 2 x 2 1 3 2x1 5 2x1þ 0:15 2 x 2 2þ 0:40x 2 2 2x2 7 2x2 þ10 2 þ 9 2
f1ð Þ ¼ 0:9x x211þ 0:95x212þ 0:65x221þ 0:7x222þ 0:5x21 3x1 þ 5 þ 0:15x22 4x2þ 5 0:1x11þ 0:8x21¼ x1 x11þ x12 15 0:9x11þ 0:9x21 ¼ x1 x11 2 þ x11 2 þ x12 2 þ x12 2 10 2 þ 15 2 s:t: 0:6x12þ 0:7x22¼ x2 x21þ x22 17 0:8x12þ 0:9x22¼ x2 x21 2 þ x21 2 þ x22 2 þ x22 2 12 2 þ 17 2 xij 0; xij xij; i¼ 1; 2:; j ¼ 1; 2; 3: ðP5Þ and f 2Cð Þ :x ¼0:7 2 x 2 11þ 0:15 2 x 2 12þ 0:5 2 x 2 21þ 0:4 2 x 2 22þ 0:3 2 x 2 1 5 2x1þ 0:7 2 x 2 2 7 2x2þ 0:9 2 x 2 11þ 0:3 2 x 2 12 þ0:7 2 x 2 21þ 0:6 2 x 2 22þ 0:5 2 x 2 1 4 2x1þ 0:9 2 x 2 2 6 2x2 þ 6 þ17 2 f2: ¼ 0:9x211þ 0:3x212þ 0:7x212 þ 0:6x222þ 0:5x21 4x1þ 8 þ 0:9x22 6x2þ 9 0:1x11þ 0:8x21¼ x1 x11þ x12 15 0:9x11þ 0:9x21 ¼ x1 x11 2 þ x11 2 þ x12 2 þ x12 2 10 2 þ 15 2 s:t: 0:6x12þ 0:7x22¼ x2 x21þ x22 17 0:8x12þ 0:9x22¼ x2 x21 2 þ x21 2 þ x22 2 þ x22 2 12 2 þ 17 2 xij 0; xij xij; i¼ 1; 2:; j ¼ 1; 2; 3: ðP6Þ Solving the above problem (P5) as single objective programming ignoring other objective function, the optimal range of objective is obtained as follows:
f1C ¼ 2:764 and f1¼ 16:883: Therefore f1¼ 19:216; 16:883½ :
Similarly, the problem (P6) is constructed and its opti-mal range is obtained as follows:
f2¼ 12:290 and f2C¼ 0:682
Then, we have f2¼ 20508; 12:290½ :
Now, the membership functions of (P1) and (P2) are constructed as follows: l1ðxÞ ¼ 1 f1ð Þ 19:216x 16:883 f1ð Þx 16:883 ð19:216Þ; 19:216 f1ð Þx 8 < : ðP7Þ and l2ðxÞ ¼ 1 f2ð Þ 7:3620x 12:290 f2ð Þx 12:290 ð20:508Þ; 0:6748: f2ð Þx 8 < : ðP8Þ
The problem based on the model (29) is transformed into an interval fuzzy nonlinear programming problem as follows: max l1ðxÞ ¼ 16:883 f1ð Þx 16:883 ð19:216Þ;l2ðxÞ ¼ 12:290 f2ð Þx 12:290 ð20:508Þ s:t: l1ðxÞ ¼ 16:883 f1ð Þx 16:883 ð19:216Þ 1; l2ðxÞ ¼ 12:290 f2ð Þx 12:290 ð20:508Þ 1 ½0:1; 0:9 ½x11; x11 þ ½0:8; 0:9 ½x21; x21 ¼ ½x1; x1 ½0:6; 0:8 ½x12; x12 þ ½0:7; 0:9 ½x22; x22 ¼ ½x2; x2 ½x11; x11 þ ½x12; x12 ½10; 15 ½x21; x21 þ ½x22; x22 ½12; 17 xij 0; xij xij; i¼ 1; 2:; j¼ 1; 2: 8 > > > > > > > > > > > > < > > > > > > > > > > > > : ðP9Þ The above problem is rearranged and we have
max l1ðxÞ ¼ 14:6994 f1ð Þx 14.6994 4.5671;l2ðxÞ ¼ 7:3620 f2ð Þx 7.3620 0:6748 s:t: f1ð Þ 19:216;x f2ð Þ 20:508,x ½0:1; 0:9 ½x11; x11 þ ½0:8; 0:9 ½x21; x21 ¼ ½x1; x1 ½0:6; 0:8 ½x12; x12 þ ½0:7; 0:9 ½x22; x22 ¼ ½x2; x2 ½x11; x11 þ ½x12; x12 ½10; 15 ½x21; x21 þ ½x22; x22 ½12; 17 xij 0; xij xij; i¼ 1; 2:; j ¼ 1; 2:: 8 > > > > > > > > > > > < > > > > > > > > > > > : ðP10Þ The problem (P10) is an interval multiobjective programming problem which can be solved by using the maximization model (21). Here, multiobjective problem (P10) based on model (31) is converted to a single objective programming problem with a simple weighted sum model. The optimal solutions is obtained as follows:
½x1; x1 ¼ ½1:528; 2:613; ½x2; x2 ¼ ½2:321; 3:031;
½x11; x11 ¼ ½0:202; 1:020; ½x12; x12 ¼ 1:679;
½x21; x21 ¼ 1:884; ½x22; x22 ¼ 1:876:
f1¼ 19:216; f1¼ 25:286; f2¼ 17:215; f1¼ 17:031:
Now, using Eq. (33), the LP metric function is
constructed to minimize the deviations from individual optimal points, called ideal solution, as follows:
min Lp¼ f1ð Þ 16:883x 16:883 p þf1Cð Þ 2:764Þx 2:764 p þ f2ð Þ 12:286x 12:286 p þf2Cð Þ ð0:682Þx ð0:682Þ p g1=p
0:1x11þ 0:8x21¼ x1 x11þ x12 15 0:9x11þ 0:9x21 ¼ x1 x11 2 þ x11 2 þ x12 2 þ x12 2 10 2 þ 15 2 s:t: 0:6x12þ 0:7x22¼ x2 x21þ x22 17 0:8x12þ 0:9x22¼ x2 x21 2 þ x21 2 þ x22 2 þ x22 2 12 2 þ 17 2 xij 0; xij xij; i¼ 1; 2:; j ¼ 1; 2; 3: ðP11Þ Problem (P11) is solved for values of p¼ 2 and the results are obtained as:
½x1; x1 ¼ ½1:471; 2:127; ½x2; x2 ¼ ½2:006; 2:619;
½x11; x11 ¼ ½0:304; 0:563; ½x12; x12 ¼ 1:376;
½x21; x21 ¼ 1:800; ½x22; x22 ¼ 1:687:
f1 ¼ 18:588; f1 ¼ 25:995; f2 ¼ 17:467; f1 ¼ 16:679:
Hence the achieved values of (P1) appears more satisfying L2 metric problem.
5 Conclusion
The uncertainty is a certain nature of mathematical mod-eling of real world systems. Usually, the decision maker does not have enough information to determine a precise value for the needed conditions in a model. In this condi-tion, the decision maker has to decide these parameters. Interval arithmetic presents a good structure for operating the systems’ knowledge as interval numbers, rather than crisp numbers. In this paper, the problem of interval non-linear multiobjective transportation problem with convex costs is constructed, where all of the model’s variables and coefficients are considered as interval numbers. An interval fuzzy programming method, include a process where a nonlinear multiobjective transportation problem is con-verted into a single objective nonlinear transportation model which maximizes the sum of membership degrees of different interval nonlinear objectives. This single objec-tive problem itself is an interval nonlinear transportation problem which can be solved by converting it into a mul-tiobjective nonlinear transportation model and this model can be solved by two multiobjective programming meth-ods. Also, the efficiency of obtained solution using the interval fuzzy method is verified. Eventually, employment of the interval fuzzy method is analyzed in a numerical example.
The main highlights of this investigation in the field of transportation problem can be defined as follows;
(i) Uncertain muItiobjective nonlinear transportation problem with interval numbers including unit cost, supply, demand, discount rate and time cost is formulated.
(ii) To obtain the Pareto optimal solution of the problem, an interval fuzzy programming and global criteria methods are used.
(iii) Interval uncertainty provide more accurate values than the crisp values. Thus, DM is able to take more consistent preference with the help of present investigation.
Hence, this model may be formulated as a budget constraint, deteriorating item and also one can consider the safety factor and so on. Also, this model may be solved in various environments.
Compliance with Ethical Standards
Conflict of interest The authors confirm that there is no conflict of interest regarding a financial supporter.
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