Inter national Journal of Pure and Applied Mathematics Volume 113 No. 9 2017, 130 138 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu ijpam.eu SOLVING TRANSPORTATION PROBLEMS WITH MIXED CONSTRAINTS IN ROUGH ENVIRONMENT A. Akilbasha 1, G. Natarajan 2 and P. Pandian 3 1,2,3 Department of Mathematics, SAS, VIT University, Vellore-14, Tamil Nadu, India. E-mail: bashaakil@gmail.com Abstract A new method namely, rough slice-sum method for solving fully rough integer interval transportation problems is proposed. The solution obtained by the proposed method provides a plan having minimum unit shipping rough cost for the given problem. The values of decision rough variables and rough objective function for the given problem obtained by the rough slice-sum method are rough integer intervals. Numerical example is presented to understand the solution procedure of the proposed method. AMS Subject Classification: 90B06, 90C08. Key Words and Phrases:Transportation problem, Rough integer intervals, Unit shipping cost, Rough slice-sum method. 1 Introduction Transportation problem is one of the popular and most important applications of the linear programming problem. Many efficient algorithms [3, 5] have been developed for solving transportation problems having deterministic parameters. In many real life situations, some or all parameters of the transportation problem are ijpam.eu 130 2017
not deterministic always, but they are uncertain. Many researchers have studied the transportation problem in various uncertain environments such as fuzzy, random etc.. Pawlak [9] initiated the rough set theory. Then, many researchers have developed the rough set theory both in theoretical and applied. The concept of rough variable which is a measurable function from rough space to the set of real numbers was proposed by Liu [4]. A rough programming problem considering the decision set as a rough set was introduced and solved by Youness [11]. Subhakanta Dash and Mohanty [10] have proposed a compromise solution method for transportation problems considering the unit cost of transportation from a source to a destination as a rough integer interval. Pandian and Natarajan [6] presented a method namely, separation method for solving integer interval transportation problems. Recently, Akilbasha et al.[1] have proposed a new method namely, split and separation method for finding an optimal solution for integer transportation problems with rough nature. Pandian [7] have developed a new method namely, slice-sum method for solving an optimal solution for fully rough interval integer transportation problems. More recently, Pandian [8] a new method namely, less-more method is proposed for solving the transportation problem with mixed constraints based on its unit shipping cost more economically. 2 Rough intervals Now, we need the following definitions of the basic arithmetic operators and partial ordering on a set of all rough intervals which can be found in Hongwei Lu et al.[2] and Pandian et al.[7]. Let D denote the set of all rough intervals on the real line R. That is, D = {[[b, c], [a, d]], a b c d where a, b, c, d R}. Note that if a=b and c=d in D, then D becomes the set of all real intervals and if a=b=c=d in D, then D becomes the set of all real numbers. Definition 1. Let A = [[a 2, a 3 ], [a 1, a 4 ]] and B = [[b 2, b 3 ], [b 1, b 4 ]] be in D. Then, ijpam.eu 131 2017
A B = [[a 2 + b 2, a 3 + b 3 ], [a 1 + b 1, a 4 + b 4 ]] ka = [[ka 2, ka 3 ], [ka 1, ka 4 ]] if k is a positive real interval and A B = [[a 2, a 3 ][b 2, b 3 ], [a 1, a 4 ][b 1, b 4 ]]. 3 Transportation problem with mixed constraints We need the following result which is used in the proposed method which can be found in Pandian [8]. For the purpose of understanding, all results and the less-more method are presented here. Consider the following a transportation problem with mixed constraints (P) in the form of a mathematical model as given below: (P) Minimize z = c ijx ij Subject to x ij a i, i Q; x ij a i, i T ; x ij = a i, i S (1) x ij b j, j U; x ij b j, j V ; x ij = b j, j W (2) x ij 0, i = 1, 2,...m and j = 1, 2,...n and integers (3) where c ij is the cost of shipping one unit from supply point i to the demand point j; x ij is the number of units shipped from supply point i to demand point j; a i is the supply at supply point i; b j is the demand at the demand point j; {Q, T and S} are partitions of I = {1, 2, 3,..., m} and {U, V and W } are partitions of J = {1, 2, 3,..., n}. An allotment {x ij : i I, j J} to the problem (P) satisfies the conditions (1) to (3) is known as a feasible solution to the problem (P). A feasible solution {x ij : i I, j J} to the problem (P) which minimizes the objective function of the problem (P) is called an optimal solution of the problem (P). ijpam.eu 132 2017
The unit shipping cost of the problem (P) for its feasible solution X = {x ij : i I, j J}, S(X) is given below: S(X) = c ijx ij x ij Now, we construct a problem from the problem (P) namely, least bound (LB) problem for the problem (P) as follows: (LB) Minimize z = c ijx ij Subject to x ij = a i, i Q; x ij = b j, j U; x ij = 0, i T ; x ij = 0, j V ; x ij = a i, i S x ij = b j, j W x ij 0, i I and j J and integers. Note that a feasible solution to the LB problem is a feasible solution to the problem (P). Now, the USC problem for the problem (P) is constructed as follows:(up) Minimize S = c ijx ij x subject to (1) to (3) are satisfied ij Note that a feasible solution to the problem (P) is a feasible solution to the problem (UP) and vice-versa. Before proposing the new method namely, rough slice-sum method for solving USC problem, we establish the following theorems whose proofs are similar to the proofs of the corresponding theorems in pandian [8] which are used in the proposed method. Theorem 2. Let X = {x ij, i I and j J} be a feasible solution to the problem (P) with the unit shipping cost S. If c rt S, for some r and t, then the allotment set X 1 = {x ij, i I and j J, i r and j t} {x rt + p rt} is a feasible solution to the problem (P) with unit shipping cost S 1 S where p rt is the maximum number of items added to the cell (r, t) such that x rt+p rt satisfies the r th row and t th column conditions. ijpam.eu 133 2017
Theorem 3. Let X = {x ij, i I and j J} be a feasible solution to the problem (P) with the unit shipping cost S. If each cell (i, j) in the transportation table for the problem (P) satisfies atleast one of the following conditions: (i) The cost of the cell (i, j), c ij > S. (ii) The cost of the cell (i, j), c ij S and its allotment should not be increased. Then, X = {x ij, i I and j J} is an optimal solution of the problem (UP). 4 Transportation problem with mixed constraints in rough nature Now, we consider the following fully rough interval integer transportation problem with mixed constraints in mathematical model as given below: (RP) Minimize [[z 2, z 3 ], [z 1, z 4 ]] = [[c2 ij, c 3 ij], [c 1 ij, c 4 ij]] [[x 2 ij, x 3 ij], [x 1 ij, x 4 ij]] Subject to n [[x2 ij, x 3 ij], [x 1 ij, x 4 ij]] [[a 2 i, a 3 i ], [a 1 i, a 4 ij]], i Q [[x2 ij, x 3 ij], [x 1 ij, x 4 ij]] [[a 2 i, a 3 i ], [a 1 i, a 4 ij]], i T [[x2 ij, x 3 ij], [x 1 ij, x 4 ij]] = [[a 2 i, a 3 i ], [a 1 i, a 4 ij]], i S [[x2 ij, x 3 ij], [x 1 ij, x 4 ij]] [[b 2 j, b 3 j], [b 1 j, b 4 j]], j U m [[x2 ij, x 3 ij], [x 1 ij, x 4 ij]] [[b 2 j, b 3 j], [b 1 j, b 4 j]], j V m [[x2 ij, x 3 ij], [x 1 ij, x 4 ij]] = [[b 2 j, b 3 j], [b 1 j, b 4 j]], j W x 1 ij, x 2 ij, x 3 ij, x 4 ij 0, i = 1, 2,..., m and j = 1, 2,..., n and integers where c 1 ij, c 2 ij, c 3 ij, c 4 ij are positive integers for all i I and j J, a 1 i, a 2 i, a 3 i & a 4 i are positive integers for all i I and b 1 j, b 2 j, b 3 j&b 4 j are positive integers for all j J. {Q, T & S} are partitions of I = {1, 2,..., m} and {U, V, & W } are partitions of J = {1, 2,..., n}. Now, we separate the above problem (RP) into four sub crisp problems namely, upper approximation of lower bound integer transportation (U1) problem, lower approximation of lower bound integer transportation (L1) problem, lower approximation of upper bound integer transportation (L2) Problem and upper approximation of upper bound integer transportation (U2) problem. Mathematically, they are given as follows: (U2) Minimize z 4 = c4 ijx 4 ij subject to n x4 ij a 4 i, i Q; x4 ij b 4 j, j U; x4 ij b 4 j, j V ; x4 ij a 4 i, i T ; x4 ij = a 4 i, i S x4 ij = b 4 j, j W ijpam.eu 134 2017
x 4 ij 0, i = 1, 2,..., m and j = 1, 2,..., n and integers; similarly we can form the mathematical form of L2, L1, & U1 problems. Now, let {x 4 ij, for all i I & j J} be a feasible solution to the U2 problem with the transportation cost Z4 and the minimum unit shipping cost S4; let {x 3 ij, for all i I & j J} be a feasible solution to the L2 problem with x 3 ij x 4 ij, having the transportation cost Z3 and the minimum unit shipping cost S3; Similarly we can form for L1 and U1 problems. Then, using the arithmetic operations and partial order relation of rough real intervals, we conclude that [[S2, S3], [S1, S4]] is the minimum rough unit shipping cost for the fully rough integer interval transportation problem with objective value [[Z2, Z3], [Z1, Z4]]. 5 Rough Slice-Sum method We, now propose a new method namely, rough slice-sum method for solving the fully rough integer interval transportation problem with mixed constraints based on less-more method developed by Pandian [8] and the slice-sum method proposed by Pandian et al.[7]. The proposed method proceeds as follows. Step1 : Construct four crisp transportation problems with mixed constraints namely, L1, L2, U1 and U2 from the problem (P). Step2 : Solve the U2 problem using the less-more method. Let {x 4 ij, for all i I and j J} be a feasible solution to the U2 problem with the transportation cost Z4 and the minimum unit shipping cost S4. Step3 : Solve the L2 problem with x 3 ij x 4 ij, for all i I and j J using the less-more method. Let{x 3 ij, for all i I and j J} be a feasible solution to the L2 problem with the transportation cost Z3 and the minimum unit shipping cost S3. Step4 : Solve the L1 problem with x 2 ij x 3 ij, for all i I and j J using the less-more method. Let {x 2 ij, for all i I and j J} be a feasible solution to the L1 problem with the transportation cost Z2 and the minimum unit shipping cost S2. Step5 : Solve the U1 problem with x 1 ij x 2 ij, for all i I and j J using the less-more method. Let {x 1 ij, for all i I and j J} be a feasible solution to the L1 problem with the transportation cost Z1 and the minimum unit shipping cost S1. ijpam.eu 135 2017
Step6 : The minimum shipping cost solution to given problem (P) is {[[x 2 ij, x 3 ij ], [x 1 ij, x 4 ij ]], for all i I and j J} with the transportation cost [[Z2, Z3], [Z1, Z4]] and the minimum unit shipping cost [[S2, S3], [S1, S4]]. Example 1 : Consider the following 3 3 Transportation Problem Solution: Now, using the Step1., the given problem is separated into four crisp transportation problems with mixed constraints namely, U1, L1, L2 and U2. After using the above proposed method we can get the minimum unit shipping cost solution to the given problem is given as follows: [[x 2 11, x 3 11], [x 1 11, x 4 11]] = [[6, 7], [5, 8]]; [[x 2 21, x 3 21], [x 1 21, x 4 21]] = [[3, 3], [3, 3]]; [[x 2 22, x 3 22], [x 1 22, x 4 22]] = [[12, 14], [10, 16]]; [[x 2 23, x 3 23], [x 1 23, x 4 23]] = [[7, 9], [5, 11]]; with objective value [[Z2, Z3], [Z1, Z4]] = [[120, 195], [63, 288]] and shipping cost [[S2, S3], [S1, S4]] = [[4.286, 5.909], [2.739, 7.579]]. 6 Conclusion A transportation problem with mixed constraints having all parameters as rough integer intervals is considered in this paper. A new method namely, rough slice-sum method is proposed to solve fully rough integer interval transportation problems with mixed constraints such that its shipping cost is minimum. The proposed method is a systematic procedure, both easy to understand and to apply and also, it is a crisp method and provides an exact appropriate solution to the given problem. The solution procedure of the proposed method is illustrated with numerical example. The rough slice-sum method can be served an important tool for the decision makers when they are handling various types of logistic models for real life situations in rough nature. ijpam.eu 136 2017
References [1] A. Akilbasha, G. Natarajan and P. Pandian, Finding an optimal solution of the interval integer transportation problems with rough nature by split and separation method, In. J. of Pure & Applied Math., 106 (2016), 1-8. [2] Hongwei Lu, Guohe Huang and Li He, An inexact roughinterval fuzzy linear programming method for generating conjunctive water-allocation strategies to agricultural irrigation systems, Applied Math. Modelling, 35 (2011), 4330-4340. [3] H.S.Kasana and K.D. Kumar, Introductory O.R: Theory and Applications, Springer In. Edition, New Delhi (2005). [4] B. Liu, Theory and Practice of Uncertain Programming, Physical-Verlag, Heidelberg, 2012. [5] P. Pandian and G. Natarajan, A new method for finding an optimal solution for transportation problems, In. J. of Math. Sci. & Engg. Appls., 4 (2010), 59-65. [6] P. Pandian and G. Natarajan, A new method for finding an optimal solution of fully interval integer transportation problems, Applied Mathematical Sciences, 4 (2010), 1819-1830. [7] P. Pandian, G. Natarajan and A. Akilbasha, Fully rough integer interval transportation problems, In. J. of Pharmacy & Technology, 8(2) (2016), 13866-13876. [8] P. Pandian, Solving transportation problems with mixed constraints based on unit shipping cost, Research J. of Pharmacy and Technology, (accepted for publication in 2017). [9] Z. Pawlak, Rough sets, In. J. of Inform. and Comp. Science, 11 (1982), 341-356. [10] Subhakanta Dash and S.P. Mohanty, Transportation programming under uncertain environment, In. J. of Eng. Research and Development, 7 (2013), 22-28. [11] E. Youness, Characterizing solutions of rough programming problems, European J. of O.R, 168 (2006), 1019-1029. ijpam.eu 137 2017
138