Fixed Charge Capacitated Non-Linear Transportation Problem

Size: px

Start display at page:

Download "Fixed Charge Capacitated Non-Linear Transportation Problem"

Brandon Barber
5 years ago
Views:

1 Journal of Engineering, Computers & Applied Sciences (JEC&AS) ISSN No: Volume 2, No.12, December 213 Fixed Charge Capacitated Non-Linear Transportation Problem Atanu Das, Department of Mathematics; University of Kalyani; Kalyani (WB.) India, Manjusri Basu, Department of Mathematics; University of Kalyani; Kalyani (WB.) India. Debiprasad Acharya, Department of Mathematics; N. V. College; Nabadwip ( WB.) India. Abstract The fixed charge (fixed cost) may present the cost of renting a vehicle, landing fees in an airport, setup cost for machines in a manufacturing environment, etc. In this paper, we discuss fixed charge capacitated in a non-linear transportation problem. Thereby, we establish local optimum condition of this problem. Next we establish an algorithm for solving this transportation problem. Also, we illustrate a numerical example to support this algorithm. Keywords: Transportation Problem, Fixed Charge, Capacitated Restriction. 21 Subject Classi_cation: 9B6, 9C8 1. Introduction The basic linear transportation problem is one of the extension of linear programmingproblem. The simplex method of a transportation problem was first introduced by Hitchcook[13]. Charnes and Cooper [9] in 1954, developed an alternative way of determining the simplex method information by using the stepping stone method. In 1974, Klingman and Russel [17] considered the transportation problem whose constraints are mixed. Basuet. al. [7] obtained all the paradoxical pairs in a linear transportation problem. Capacitated transportation problem is one of the important area of classical transportation problem. Wolsey [21] introduced submodularity and valid inequalities in capacitated fixed charge networks. Khurana and Arora [16] gave the solution procedure of the sum of linear and linear fractional transportation problem with restricted and enhanced flow. Dahiya and Verma [11] gave the solution procedure of the capacitated transportation problem. In many situations, the cost function of the transportation problems may not be linear innature e.g. quantity discount, price breaks etc. Dinkalbach [12] gave the solution procedure on non-linear fractional programming. Basu and Acharya [6] considered on quadratic fractional generalized solid bi-criterion transportation problem. In, Dahiya and Verma [1] considered the paradox in a non-linear capacitated transportation problem. In 213, Acharya et al. [1] considered discounted generalized transportation problem. The fixed charge transportation problem(fctp) is an extension of the classical transportation problem, where fixed costs are incurred at every origin. In the early day, fixedcharge transportation problem was first calculated by Hirsch and Dantzig [14]. Barr et. al.[5], Sandrock [19], Basuet. al. [8], Thirwani [2], Arora and Ahuja [4], Khurana and Arora[15], Kowalski and Lev [], Adlakhaet. al. [3] considered different types of the fixed chargetransportation problems. But till date, there existseveral open problems applicable in reallife in this area. In 213, Acharya et al. [2] developed a modified method for fixed charge transportation problem. In this paper, we consider fixed charge capacitated transportation problem with non-linear objective functions and linear constraints. Thereby, we present a local optimumcondition of this problem with an algorithm for solving this transportation problem. Also,we solve a numerical problem by applying this algorithm. 2. Problem Formulation And Mathematical Development We consider the following cost minimizing non-linear fixed charge capacitated transportation problem: : (2.1) Blue Ocean Research Journals 49

2 Journal of Engineering, Computers & Applied Sciences (JEC&AS) ISSN No: Volume 2, No.12, December 213 subject to the constraints 1,2,3,.. (2.2) 1,2,3,.. (2.3) and, (2.4) where be the capacitated restriction for the origin to the destination. Weconsider has k- number of steps so that for Where 1, if for 1,2,3,.. and. =, otherwise. So, ( ) are all constant and, are fixed costs. Where ; ; ;. = the amount of product transported from the origin to the destination, = the cost involved in transporting per unit product from the origin to the destination, = per unit depreciation in transporting quantities from the origin to the destination, = per unit profit earned in transporting quantities from the origin to the destination, = the number of units available at the origin, = the number of units required at the destination. Introducing slack variables, ( ) with zero cost, the problem can be rewrittenas: 2.5 subject to the constraints, 1,2,3, ,2,3,.., And, The tabular form of this problem is in Table 1 Table 1 D 1 D 2... D n a i O 1 c 11 p 11 q 11 O 2 c 21 p O m q 21 c m1 p m1 q m1 c 12 p 12 q 12 c 22 p 22 q c m2 p m2 q m c 1n p 1n q 1n c 2n p 2n q 2n..... c mn p mn q mn b j b 1 b 2 b n and,. a 1 a 2. First we solve the problem P 1 with the restriction, instead of the capacitated restriction,. So we consider the problem: a m 2.8 subject to the constraints 2.6, 2.7 and. Let,, satisfying the constraints 2.6 and 2.7, is a feasiblesolution of the problem P 1. Also let be the initial basic feasible solution and B be the basis of theproblem obtained by any one of the well known method and,,. Now we consider the dual variables,,,, for, such that and From the equations 2.9; 2.1 and 2.11 we calculate the values of, ;, and,. Then for non basic cells we set Blue Ocean Research Journals 5

3 Journal of Engineering, Computers & Applied Sciences (JEC&AS) ISSN No: Volume 2, No.12, December To establish the optimum condition we express in terms of non basic variables onlyby using the constraints 2.6 and 2.7 and equations Now, from equations and using , we have (2.15) Where and, Similarly,, 2.17 And, 2. Where 2.19 and 2.2 Using (2.15, 2.17, 2.), the objective function Z is given by,,, For any non-basic cell,,, Denote,,.,,,. Now let, where is the change in cost for introducing a non-basic(i, j) cell with value (for, ) into the basis by making re-allocation. Calculate, where is the fixed-cost involved for introducing the variable with value (for, ) into the current basis to form a new basis.. For any non-basic variable if all, the basic feasible solution of the problem is optimum, otherwise let, Then(s, t) cell enters into the basis. We continue this process until all. 3. Algorithm Step 1: Set s = 1, where s is the number of iteration. Step 2: Solve the problem. Step 3: If for all occupied cells in the problem then this solution isoptimum solution ofp and goto step 6, otherwise goto step 4. Step 4: If for the cell (g, h), then we take and,,, where M is a very large positive number. Step 5: Set s = s + 1 and goto step 2. Step 6: Let this process terminated after step and be the solution of.then the solution of is. Where is the union taken over the cell(g, h). Step 7: Write the local optimum value for the optimum solution. Step 8: End. Note: The number of basis 1. Then the value of is found out as,where is the total fixed cost of the basic feasible solution of problem. Blue Ocean Research Journals 51

4 Journal of Engineering, Computers & Applied Sciences (JEC&AS) ISSN No: Volume 2, No.12, December Numerical Example We consider the problem given in Table 2 5 O 1 O O 3 Table 2 D 1 D 2 D 3 D 4 A i B j The fixed costs are 2, 3, 6, 5, 25, 2, 5, 6, 3, 3, 5, 4, Where 1, if for 1,2,3., otherwise; Where 1, if 2 for 1,2,3., otherwise; Where 1, if 35 for 1,2,3., otherwise; Where 1, if 6 for 1,2,3., otherwise; Here 1, 2, 3, 4has four steps. The capacitated Restrictions Are 3, 5, 4, 5, 5, 4, 6, 3, 4, 5, 4, 5, We solve the above problem by applying algorithm in section 3 Applying step 1, set s 1. Applying step 2, solve the problem and we obtain the local optimum solution givenin Table 3 Where for 1,2,3. Applying step 3, we see that in the basis cell. So, goto step 4. Applying step 4, we take 5 and 17, 9,,where M is a very large positive number. Applying step 5, set s s 1 and write the problem, tabulated in Table 4. Applying step 2, solve the problem and we obtain the local optimum solution intable 5. Blue Ocean Research Journals 52

5 Journal of Engineering, Computers & Applied Sciences (JEC&AS) ISSN No: Volume 2, No.12, December 213 Table 5 D 1 D 2 D 3 D 4 D 5 a i F B O 1 5 O M M M O b j In Table 5 We See That,, So This Is An Optimum Table For Theproblem. Goto Step 6. Applying Step 6, We Obtain The Local Optimum Solution Of The Problem P, given in Table6. Table 6 D 1 D 2 D 3 D 4 D 5 a i F B O O O b j Applying step 7, we have the local optimum value for the optimum solution 5, 17, 41, 53, 1, 9, 31, given in Table 6. Note: The number of basis is 8 instead of Conclusion We show an algorithm for obtaining the optimum value of the objective function of fixed charge non-linear capacitated transportation problem, optimum solution is not global due to the presence of fixed charge. The interesting part of this problem is that the number of solution in the basis may exceed the number of origin (m) + the number of destination (n)1. These types of problems have application in many industry, trading company, etc. References: [1] Acharya, D., Basu, M. and Das, A., Discounted Generalized Transportation Problem, International Journal of Scientific and Research Publications, vol. 3(213), Issue 7. [2] Acharya, D., Basu, M. and Das, A., Modified method for fixed charge transportation problem, International journal of engineering inventions, vol. 3 (213), p.p [3] V. Adlakha, K. Kowalski and B. Lev, A branching Method for the fixed charge transportation problem, Omega, vol. 38,(21), p.p [4] A. Ahuja and S. R. Arora, Multi-index Bi-criterion Fixed Charge Transportation Problem, Indian Blue Ocean Research Journals 53

6 Journal of Engineering, Computers & Applied Sciences (JEC&AS) ISSN No: Volume 2, No.12, December 213 Journal of Pure and Applied Mathematics, Vol. ; (21), p.p [5] R. S. Barr, F. Glover and D. Klingman, A new optimization method for large scale fixed charge transportation problems, Operations Research, 1981, 29: [6] M. Basu and D. Acharya, On quadratic fractional generalized solid bi-criterion transportation problem, An International Journal of Applied Mathematics & Computing, 22, Vol. 1, No 1-2, [7] M. Basu, D. Acharya and A. Das, The algorithm of finding all paradoxical pair in a linear transportation problem, Discrete Mathematics; Algorithm and Application, Vol. 4, No. 3 (212) (9pages), World Scientific Publising Company, DOI:1.1142/S [8] M. Basu, B. B. Pal, and A. Kundu, An algorithm for the optimal time-cost trade-offin fixed charge bicriterion transportation problem, Optimization, vol. 3 (1994), [9] A. Charnes and W. W. Cooper, The stepping stone method of explaning linear programming calculations in transportation problems, Management Science, 1954, vol.1, p.p [1] K. Dahiya and V. Verma, Paradox in a non linear capacitated transportation problem, Yugoslav journal of Operational Research, vol. 16(2),(), p.p [11] K. Dahiya and V. Verma, Capacitated transportation problem with bounds on rimconditions, Europeon journal of Operational Research, (), vol. 178, p.p [12] W. Dinkelbach, Onnon linear fractional programming, Management Science; vol. 13.No. 7,(1967), p.p [13] F. L. Hitchcock, The distribution of a product from several resources to numerouslocalities, Journal of Mathematical Physics, Vol. 2 (1941), pp [14] W. M. Hirsch and G.B. Dantzig, Notes on linear Programming: Part XIX, The fixed charge problem. Rand Research Memorandum No. 1383, Santa Monica; California,1954. [15] A. Khurana and S. R. Arora, Three Dimensional Fixed Charge Bi-criterion Indefinite Quadratic Transportation Problem, Yugoslav Journal of Operations Research, Vol.14, (24), p.p [16] A. Khurana and S. R. Arora, The sum of a linear and linear fractional transportation problem with restricted and enhanced flow, Journal of Interdisciplinary Mathematics, vol. 9; (), p.p [17] D. Klingman and R. Russel, The transportation problem with mixed constraints, Operational Research Quaterly, Vol. 25 (1974), No. 3; p.p [] K. Kowalski and B. Lev, On step fixed-charge transportation problem, OMEGA : The International Journal of Management Science, 28, vol.36, p.p [19] K. Sandrock, A simple algorithm for solving small fixed charge transportation problems, Journal of the Operational Research Society, 1988, vol. 39, p.p [2] D. Thirwani, A note on fixed charge bi-criterion transportation problem with enhanced flow, Indian Journal of Pure and Applied Mathematics, vol 29(5); (1998), p.p [21] L. A.Wolsey, Submodularity and valid inequalities in capacitated fixed charge networks, Oper. Res. Lett., 1989, vol. 8, p.p Blue Ocean Research Journals 54

An algorithm for solving cost time trade-off pair in quadratic fractional transportation problem with impurity restriction

$An algorithm for solving cost time trade-off pair in quadratic fractional transportation problem with impurity restriction$ International Research Journal of Engineering and Technology (IRJET) e-issn: 9- Volume: Issue: Apr-1 wwwirjetnet p-issn: 9-7 An algorithm for solving cost time trade-off pair in quadratic fractional transportation