Transportation Problems

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Stephen Lester
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1 Transportation Problems Pongsa Pornchaiwiseskul Faculty of Economics Chulalongkorn University 1 Drug Distribution Drugco has three drug factories that supply the need for four provinces. Each drug factory can supply the following numbers of tons of drug per month: factory A, 35; factory B, 50; factory C, 40. The drug demands in these provinces are as follows (in tons per month): province 1, 45; province 2, 20; province 3, 30; province 4, 30. The costs of distribution from factory to province depend on the distance as given in table below. 2 Chulalongkorn University 1

2 To Province Supply From tons/mo Factory A $8 $6 $10 $9 35 Factory B $9 $12 $13 $7 50 Factory C $14 $9 $16 $5 40 Demand tons/mo Graphical Representation Factory Province 35 A B C Chulalongkorn University 2

3 Define LP Formulation x ij = tons/month of drug shipped from factory i to province j i = A, B, C j = 1, 2, 3, 4 5 LP Problem for Balanced Transportation Problem MIN 8x A1 + 6x A2 + 10x A3 + 9x A4 + 9x B1 + 12x B2 + 13x B3 + 7x B4 + 14x C1 + 9x C2 + 16x c3 + 5x C4 subject to x A1 + x A2 + x A3 + x A4 = 35 x B1 + x B2 + x B3 + x B4 = 50 x C1 + x C2 + x c3 + x C4 = 40 6 Chulalongkorn University 3

4 LP Problem (cont d) for Balanced Transportation Problem x A1 + x B1 + x C1 = 45 x A2 + x B2 + x C2 = 20 x A3 + x B3 + x C3 = 30 x A4 + x B4 + x C4 = 30 x ij >= 0 for all i,j 7 LINDO inputs for Balanced Transportation Problem min 8xa1+6xa2+10xa3+9xa4 +9xb1+12xb2+13xb3+7xb4 +14xc1+9xc2+16xc3+5xc4 subject to 2) xa1+xa2+xa3+xa4=35 3) xb1+xb2+xb3+xb4=50 4) xc1+xc2+xc3+xc4=40 8 Chulalongkorn University 4

5 LINDO inputs (cont d) for Balanced Transportation Problem 5) xa1+xb1+xc1=45 6) xa2+xb2+xc2=20 7) xa3+xb3+xc3=30 9 Factory Factory B 50 Factory C 40 Demand Chulalongkorn University 5

6 Factory Factory B Factory C 0 40 Demand Factory Factory B Factory C Demand Chulalongkorn University 6

7 Factory Factory B Factory C Demand Factory Factory B Factory C Demand Chulalongkorn University 7

8 Factory Factory B Factory C Demand Factory Factory B p 20+p 0 50 Factory C 0 p 10-p Demand Chulalongkorn University 8

9 Factory A 35-p 0+p Factory B 10+p 10-p Factory C Demand Factory A Factory B Factory C Demand Chulalongkorn University 9

10 Factory A 25-p 10 0+p 0 35 Factory B 20+p 0 30-p 0 50 Factory C Demand Factory A Factory B Factory C Demand Chulalongkorn University 10

11 Unbalanced Transportation Problem To Province Supply From tons/mo Factory A $8 $6 $10 $9 45 Factory B $9 $12 $13 $7 50 Factory C $14 $9 $16 $5 40 Demand tons/mo \ LP Problem for Unbalanced Transportation Problem Min 8x A1 + 6x A2 + 10x A3 + 9x A4 + 9x B1 + 12x B2 + 13x B3 + 7x B4 + 14x C1 + 9x C2 + 16x c3 + 5x C4 subject to x A1 + x A2 + x A3 + x A4 +d A = 45 x B1 + x B2 + x B3 + x B4 +d B = 50 x C1 + x C2 + x c3 + x C4 +d C = Chulalongkorn University 11

12 LP Problem (cont d) for UnBalanced Transportation Problem x A1 + x B1 + x C1 = 45 x A2 + x B2 + x C2 = 20 x A3 + x B3 + x C3 = 30 x A4 + x B4 + x C4 = 30 x ij,d i >= 0 for all i,j 23 Unbalanced Transportation Problem To Province Supply From dummy tons/mo Factory A $8 $6 $10 $9 $0 45 Factory B $9 $12 $13 $7 $0 50 Factory C $14 $9 $16 $5 $0 40 Demand tons/mo Chulalongkorn University 12

06IP/IM74 OPERATIONS RESEARCH. UNIT - 3: Transportation Problem

06IP/IM74 OPERATIONS RESEARCH. UNIT - 3: Transportation Problem 06IP/IM74 OPERATIONS RESEARCH UNIT - 3: Transportation Problem Introduction: The objective of the transportation problem is to transport various quantities of a single homogenous commodity, which are initially