Volume 4 Issue 5
International Journal of Trend in Scientific Research and Development Volume 5 Issue 4, May-June @ IJTSRD | Unique Paper ID – IJTSRD4 Solution of Multi Objective Transportation Problem 1Sheth C. D. Barfiwala College ABSTRACT The transportation problem is one of the earliest applications of the linear programming problems. The basic transportation problem was originally developed by Hitchcock. Efficient methods of solution derived from the simplex algorithm were developed in 19 can be modeled as a standard linear programming problem, which can then be solved by the simplex method. The objective of traditional transportation is to determine the optimal transportation pattern of a certain goods from supplier to demand customer so that the transportation cost become minimum and for this purpose we have different method for getting initial and optimal solution. We can get an initial basic feasible solution for the transportation problem by using the Nort rule, Row minima, Column minima, Matrix minima, or the Vogel Approximation Method (VAM). To get an optimal solution for the transportation problem, we use the MODI method (Modified Distribution Method). In this paper we have developed an algo programming technique to solve multi objective transportation problem. We have also compared the result with raw maxima and EMV and show how the developed approach is more effective than other approaches. KEYWORDS: Multi- objective, Transportation, Fuzzy Programming, Cost, Time 1. INTRODUCTION Two types of research work is done for transportation problem one is formulation of simple and multi objective transportation problem and second is developed a solution approach for simple and multi objective transportation problem. This chapter includes the work done on multi objective transportation problems as well as research objective. The transportation prob formalized by the French Mathematician (Gaspard Monge, 1781). Major advances were made in the field during World War two by the Sovi- et/Russian mathematician and economist Leonid Vitaliyevich Kantorovich. Kantorovich is regarded as the founder of programming. Consequently, the problem as it is stated is sometimes known as the Monge transportation problem. In 2014, Sudipta Midya and Sankar Kumar Roy  solved single sink, fixed-charge, multiobjective, multi stochastic transportation problem by using fuzzy programming approach. A utility function approach to solve multi objective transportation problem was given by Gurupada Maity and Sankar Kumar Roy  in 2014. in 2016 Gurupada Maity, Sankar Kumar Roy, and José Luis Verdegay  gave concept of cost reliability in the transportation cost and they considered supply and demand as uncertain variables. Sheema Sadia, Neha Gupta and Qazi M. Ali  presented their study on multi objective capacitated fractional transportatio 2016. They considered mixed linear constraints. Solution of multi objective transportation problem with non linear cost and multi choice demand was given by Gurupada Maity and Sankar Kumar  Roy in 2016. Mohammad 2021 Available Online: www.ijtsrd.com e-ISSN: 2456 3607 | Volume – 5 | Issue – 4 | May-June 202 Sanjay R. Ahir1, H. M. Tandel2 of Commerce, Surat, Gujarat, India 2Rofel College Vapi, Gujarat, India 47. The transportation problem h-West corner rithm by fuzzy How to cite this paper H. M. Tandel "Solu Transportation Problem" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456 6470, Volume Issue-4, June 2021, pp.1331 www.ijtsrd.com/papers/ijtsrd43607.pdf Copyright © 20 International Journal Scientific Research and Development Journal. This is an Open Access article distributed under the terms Creative Commons Attribution License (http: //creativecommons.org/licenses/by/4.0 lem was linear –Kantorovich -index Then, n problem in Asim Nomani, Irfan Ali an algorithm of proposed method in 2017. In proposed method they have used weighted sum method based on goal programming. In 2017 only, Sankar Kumar Roy, Gurupada Maity, Gerhard Wilhelm Weber and Sirma Zeynep Alparslan Gök  solved transportation problem by using conic scalarization approach with interval goal. Then, by using utility approach with goals Sankar Kumar Roy, Gurupada Maity and Gerhard-Wilhelm Weber  solved multi objective two stage grey transportation inspired from Zimmermann’s fuzzy programming and the neutrosophic set terminology recently in 2018, Rizk M. Rizk-Allah, Aboul Ella Hassanien and Mohamed Elhoseny  proposed a model under neutrosophic environment. In this model for each objective functions, they considered three membership functions namely, truth membership, indeterminacy membership and falsity membership. Srikant Gupta, Irfan Ali and Aquil Ahmed  presented their study on multi objective capaciated transportation problem with uncertain supply and demand. They formulated deterministic form of the problem by using solution procedure of multi choice and fuzzy numbers. Then they used goal programming approach to solve fractional objective function. Many researchers have done tremendous work with this method, which is not mentioned all but some of the research is summarized over here. It is just a brief summary of fuzzy programming technique based optimization to provide comprehensive knowledge of fuzzy optimization and solutions. (IJTSRD) – 6470 1 Page 1331 : Sanjay R. Ahir | tion of Multi Objective - -5 | -1337, URL: 21 by author (s) and of Trend in of the (CC BY 4.0) ) d A. Ahmed  presented multi objective problem in 2017. By In real situation the IJTSRD43607 International Journal of Trend in Scientific Research and Development (IJTSRD) @ www.ijtsrd.com eISSN: 2456-6470 @ IJTSRD | Unique Paper ID – IJTSRD43607 | Volume – 5 | Issue – 4 | May-June 2021 Page 1332 objective parameter are decided according to the requirement of decision maker. Many times he is unable to give such kind of information and to deal with these imprecision the parameter are formulated as fuzzy number, especially as triangular fuzzy numbers. Means, the objective function is fuzzified and leverage is provided to the decision maker to operate. Zadeh  first introduced the concept of fuzzy set theory. Then Zimmermann  first applied the fuzzy set theory concept with some suitable membership functions to solve linear programming problem with several objective functions. He showed that solutions obtained by fuzzy linear programming are always efficient. Bit et al.  applied the fuzzy programming technique with linear membership function to solve the multi-objective transportation problem. In this paper we have find the solution of multi objective transportation problems by fuzzy programming technique using linear membership function. 2. Multi objective Transportation Problem: Let us consider that any company has m production centres, say , , , …… and n warehouses or markets, say ,,, …… . Let supply capacities of each production centers be , , , …… respectively and demand levels of each destinations be , , , …… respectively. The decision maker or manager of company wants to optimize r number of penalties ( , ℎ = 1,2,3……. ) like minimize the transportation cost, maximize the profit, minimize the transportation time, minimize the risk, etc. Now, if be the cost associated with objective to transport a unit product from production center to warehouse and ! be the unknown quantity to be transported from production center to warehouse. Then to solve this type of problem the multi objective transportation problem is defined as shown in model (1.2).  Model: Multi objective transportation problem Minimize: Z* = ++ C-.*X-., r = 1 .2 1,2,3, ………K 4 -2 ; Subject to the constraints + X-. 1 .2 = a-, i = 1,2,3, ………m; +X-. 4 -2 = b., j = 1,2,3, ………n; X-. ≥ 0, ∀ i, j. Where,m = Number of sources; n = Number of destinations; a- = Available supplies at iKL source; b. = Demand level of jKL destination; =Cost associated with rKL objective for transporting a unit of product from iKL source to jKL destination; X-. = The quantity of product to be transported from iKL source to jKL destination. To find optimal solution of any multi objective transportation problem, so many approaches are there like, goal programming, fuzzy approach, genetic algorithm, etc. In most of the solution approaches one general objective function is defined by considering each single objective function as the constraints. Solution given by general objective function may or may not give optimal solution to each objective function but this will give us compromise solution. 3. Fuzzy Programming Technique to Solve Multi- Objective Problems Most of the entrepreneur now a day’s do not have a aim of single objective but they wish to target multi objective i.e.` they not only try to minimize cost but try to minimize some recourse so that their business can grow in best of manner. In competitive world entrepreneur need to be aware of competition and should monopolized business. Their important objective could be to minimize risk using the same set of constraints. Such general multi objective linear programming problem can be defined as under [14,15] Minimize 1 , 1, 2,3, 4....., i n k k i i i z c x k r = = = = ∑ Subject to the constraints, 1 ( , , ) , 1,2,3...... n i i j i a x B j m = ≤ = ≥ = ∑ , 0 i x ≥ . In fuzzy programming technique following procedure applied to solve the multi objective optimization problem : The formulated multi objective linear programming problem first solve by using single objective function and derive optimal solution say 1 1 2 3 ( , , .......... ) n f x x x x for first objective 11 z and then obtain other objective value with the same solution say 21 z 31 z 41 z … 1 kz . Procedure repeats same for 2 ........ r z z objectives. Step 2: Corresponding to above data we can construct a pay off matrix which can give various alternate optimal value. Z1 Z2 ......... Zr 1 1 2 3 ( , , .......... ) n f x x x x Z11 Z21 ......... Zr1 2 1 2 3 ( , , .......... ) n f x x x x Z12 Z22 ......... Zr2 ...... ......... 1 2 3 ( , , .......... ) n n f x x x x Z1n Z2n ......... Zrn Table: 1- Pay-off matrix for MOLPP Here, ki z : indicated optimal solution of ‘kth objective using solution of ‘ith objective, 1, 2,3, 4....., k r = and 1, 2,3...... i n = . Or International Journal of Trend in Scientific Research and Development (IJTSRD) @ www.ijtsrd.com eISSN: 2456-6470 @ IJTSRD | Unique Paper ID – IJTSRD43607 | Volume – 5 | Issue – 4 | May-June 2021 Page 1333 Find out the positive ideal solution (PIS) and negative ideal solution (NIS) for each objective function of the model Now, by using pay-off matrix or positive ideal solution (PIS) and negative ideal solution (NIS) define a membership function S(UV)for the Wobjective function. Here two different membership function are utilized to find efficient solution of this multi-objective resource allocation problem and by using this membership function convert the MOLPP into the following model Model -1: Maximum λ , Subject to the constraints ( ) kZ λ µ ≤ , 1 ( , , ) , 1, 2,3...... m i i j i a x B j n = ≤ = ≥ = ∑ 0 i x ≥ , When we utilize Fuzzy linear membership function  then model structure is as follows Model- 2: Maximum λ , Subject to the constraints ( ) k k k k z U L U λ + − ≤ , 1 ( , , ) , 1, 2,3...... m i i j i a x B j n = ≤ = ≥ = ∑ 0 i x ≥ . Solution of this model will give you an efficient solution 4. Algorithm to solve Multi-Objective Linear Programming Problem Input: Parameters: 1 2 ( , ,..., , ) k Z Z Z n Output: Solution of multi-objective programming problem Solve multi-objective programming problem ( , k Z X ↓ ↑ ) begin read: problem while problem = multi-objective programming problem do for k=1 to m do enter matrix k Z end -| determine pay-off matrix Or -| the positive ideal solution and negative ideal solution for each objective. for k=1 to m do ( )0 PIS min ij i z z = Under given constraints end for k=1 to m do ( )0 NIS max ij i z z = Under given constraints end - find single objective optimization models under given constraints from multi-objective optimization models. fork=1 to m do max λ Subject to the constraints: ( ) ij E Z x λ µ ≤ Under given constraints End |- find the solution SOPs using Lingo software. 5. Numerical Examples This section considers several numerical examples of transportation problem and finds their solution by fuzzy programming technique Numerical Illustration 1: Transportation problem with some demand and supply are given below . D1 D2 D3 D4 Supply S1 1 2 7 7 8 S2 1 9 3 4 19 S3 8 9 4 6 17 Demand 11 3 14 16 Table 2 showing objective function 1 D1 D2 D3 D4 Supply S1 4 4 3 3 8 S2 5 8 9 10 19 S3 6 2 5 1 17 Demand 11 3 14 16 Table 3 showing objective function 2 Mathematical Formulation of this problem can be written as Min Z1 = x11 + 2x12 + 7x13 + 7x14 + x21 + 9x22 + 3x23 + 4x24+ 8x31+ 9x32+ 4x33 + 6x34 Min Z2 = 4x11 + 4x12 + 3x13 + 3x14 + 5x21 + 8x22 + 9x23 + 10x24+ 6x31+ 2x32+ 5x33 + x34 Subject to the constraints x11+x12+x13+x14=8; x21+x22+x23+x24=19; x31+x32+x33+x34=17; x11+x21+x31=11; x12+x22+x32=3; x13+x23+x33=14; x14+x24+x34=16; x11 0; x12 0; x13 0; x14 0; International Journal of Trend in Scientific Research and Development (IJTSRD) @ www.ijtsrd.com eISSN: 2456-6470 @ IJTSRD | Unique Paper ID – IJTSRD43607 | Volume – 5 | Issue – 4 | May-June 2021 Page 1334 x21 0; x22 0; x23 0; x24 0; x31 0; x32 0; x33 0; x34 0; PIS and NIS value of first objective function is given by PIS = 143, NIS = 265 PIS and NIS value of second objective function is given by PIS = 167, NIS = 310 Hence, U1=265, L1=143, U2= 310, L2=167 U1- L1= 122 U2 - L2 = 143 Applying fuzzy linear membership function, we get the following model When we solve this problem with computational software like LINGO then the solution of the model is as follows: The allocations are, X11 = 3.000000 X21 = 8.000000 X33 = 1.000000 X12 = 3.000000 X23 = 11.00000 X34 = 16.00000 X13 = 2.000000 The values of objective functions are as follows: Z1 = 164.0000, Z2 = 190.0000 Using these allocations we have Z1=164, Z2= 190 with degree of satisfaction=0.8278689 International Journal of Trend in Scientific Research and Development (IJTSRD) @ www.ijtsrd.com eISSN: 2456-6470 @ IJTSRD | Unique Paper ID – IJTSRD43607 | Volume – 5 | Issue – 4 | May-June 2021 Page 1335 The table given below shows the comparison of the given transportation problem with other approaches Method Numerical Example Row maxima method  Z1= 172,Z2=213. EMV method [20 ] Z1=143, Z2=265 fuzzy Multiobjective programming Z1= 164, Z2= 190. Numerical illustration -2: A supplier, supply a product to different destination from different sources. The supplier has to take decision in this TP so that the transportation cost and transportation time should be minimum. The data for the cost and time is as follows: Destination sources↓ D1 D2 D3 Supply S1 16 19 12 14 S2 22 13 19 16 S3 14 28 8 12 Demand 10 15 17 42 Table 4 Data for time Destination sources↓ D1 D2 D3 SUPPLY S1 9 14 12 14 S2 16 10 14 16 S3 8 20 6 12 Demand 10 15 17 42 Table 5 Data for cost The table given below shows the comparison of the given transportation problem with other approaches Method Objectives values Row maxima method  Z1= 518, Z2=374 EMV method  Z1= 518, Z2=374 fuzzy Multi objective programming Z1= 518, Z2=374 Numerical Illustration 3: The data is collected by a person, who supplies product to different companies after taking it from different origins. There are four different suppliers named as A, B, C and D and four demand destinations namely E, F, G and H. How much amount of material is supplied from different origins to all other demand destinations so that total cost of transportation and product impairment is minimum . Supplies: a1 = 21, a2 = 24, a3 = 18, a4 = 30. Demands: b1 = 15, b2 = 22, b3 = 26, b4 = 30. Table 6 Transportation cost for TP E F G H Supply A 24 29 18 23 21 B 33 20 29 32 24 C 21 42 12 20 18 D 25 30 1 24 30 Demand 15 22 26 30 Table 7 Product Impairment for TP E F G H Supply A 24 29 18 23 21 B 33 20 29 32 24 C 21 42 12 20 18 D 25 30 1 24 30 Demand 15 22 26 30 The table given below shows the comparison of the given transportation problem with other approaches Objectives Osuji et. al. (LMF)  Osuji et. al. (HMF)  Jignash G Patel GSDMT Method  Fuzzy Programming Approach Cost 1900 1898 1902 1902 Product impairment 1279 1286 1198 1198 Numerical Illustration 4: International Journal of Trend in Scientific Research and Development (IJTSRD) @ www.ijtsrd.com eISSN: 2456-6470 @ IJTSRD | Unique Paper ID – IJTSRD43607 | Volume – 5 | Issue – 4 | May-June 2021 Page 1336 Illustration 4: A company has four origins A, B, C and D with production capability of 5, 4, 2 and 9 units of manufactured goods, respectively. These units are to be transported to fivewarehouses E, F, G, H and I with necessity of 4, 4, 6, 2 and 4 units, respectively. The transportation cost, risk and product impairment between companies to warehouses are given below . Table 8 Transportation cost for TP E F G H I Supply A 9 12 9 6 9 5 B 7 3 7 7 5 4 C 6 5 9 11 3 2 D 6 8 11 2 2 9 Demand 4 4 6 2 4 Table 9 Transportation cost for TP E F G H I Supply A 2 9 8 1 4 5 B 1 9 9 5 2 4 C 8 1 8 4 5 2 D 2 8 6 9 8 9 Demand 4 4 6 2 4 The table given below shows the comparison of the given transportation problem with other approaches Objectives Abo-Elnaga et. al. . Jignash G Patel GSDMT Method  Fuzzy Programming Approach Cost 144 155 129 Risk 104 90 97 Product impairment 173 80 83 The comparison of four shows that the solution obtained by fuzzy programming technique gives better solution of multi objective transportation problem Conclusion This paper discussed a fuzzy programming technique for solution of multi objective transportation problem. 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