Or2016 s07 Transportation Problem Pt1 Handout

8/15/2019 Or2016 s07 Transportation Problem Pt1 Handout

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Outline Introduction Linear Programming Model References

Outline

Outline

IntroductionThe Transportation ProblemIllustration: The Foster Generators Problem

Linear Programming ModelLP Model for Foster Generators Transportation ProblemComputer Solution

References

IFC6503C PSTI UNSRAT


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Acknowledgement

The explanation about transportation problemand the Foster Generators example are adapted

from [1].



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The Transportation Problem

Introduction

Transportation problem belongs to a special class of linearprogramming called Network Flow problems.

The transportation problem arises frequently in planning forthe distribution of goods and services from several supplylocations to several demand locations.

Typically the goods quantity at each supply location (origin )is limited, and the quantity of goods needed at each of several

locations (destinations ) is known. The common objective in transportation problem is tominimize the cost of shipping (transporting) goods from theorigin to the destinations.



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Illustration

IllustrationThe Foster Generators Problem

Foster Generators is a generator manufacturer. It has three plants, and four distribution centers. It operates plants in Cleveland, Ohio; Bedford, Indiana; and

York, Pennsylvania. Production capacities over the next three-month planningperiod for one particular type of generator are as follows:

Table 1 : Production capacities for next three-month period.

Origin Plant Prod. Capacity (units)1 Cleveland 5,0002 Bedford 6,0003 York 2,500

Total: 13,500



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Illustration


The rm distributes its generators through four regionaldistribution centers located in Boston, Chicago, St. Louis, andLexington.

The three-month forecast of demand for the distributioncenters is as follows:Table 2 : Production capacities for next three-month period.

Destination Distribution Demand Forecast

Center (units)1 Boston 6,0002 Chicago 4,0003 St. Louis 2,0004 Lexington 1,500

Total: 13,500



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O tli I t d ti Li P i M d l R f


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Illustration


5000 1Cleveland

6000 2Bedford

2500 3

York

1Boston

2Chicago

3St. Louis

4Lexington

6000

2500

1500

4000

3

27

6

75

23

2

54

5

Supplies Distribution Routes(arcs)

Demands

Figure 1 : Network Representation




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Illustration


The circles are referred to as nodes . The lines connecting the nodes are arcs .

Each origin and destination is represented by a node, and eachpossible shipping route represented by an arc. The amount of supply is written next to each destinationnode, and the amount of demand is written next to eachdestination node.

The goods shipped from the origins to destinations representthe ow in the network, the direction indicated by the arrows.

The cost for each unit shipped on each route is shown inTable 3.




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Illustration


Table 3 : Transportation cost per unit.

DestinationOrigin Boston Chicago St. Louis Lexington

Cleveland 3 2 7 6Bedford 7 5 2 3

York 2 5 4 5




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Illustration


A linear programming model can be used to solve thistransportation problem.

We use double-subscripted variables as the decision variables.For m origins and n destinations the decision variables arewritten asx ij = number of units shipped from origin i to destination j ,where i = 1 , 2, . . . , m and j = 1 , 2, . . . , n

Example: x11 is the number of units shipped from origin 1(Cleveland) to destination 1 (Boston).Example: x12 is the number of units shipped from origin 1(Cleveland) to destination 2 (Chicago).




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Illustration


Remember that the objective of the transportation problems isto minimize the total transportation cost

Therefore we can use the cost data in Table 3Transportation costs for units shipped from Cleveland =3x 11 + 2 x 12 + 7 x 13 + 6 x 14Transportation costs for units shipped from Bedford =7x 21 + 5 x 22 + 2 x 23 + 3 x 24Transportation costs for units shipped from York =2x 31 + 5 x 32 + 4 x 33 + 5 x 34

The sum of these expressions provides the objective functionshowing the total transportation cost for Foster Generators.

3x 11 +2 x 12 +7 x 13 +6 x 14 +7 x 21 +5 x 22 +2 x 23 +3 x 24 +2 x 31 +5 x 32 +4 x 33 +5 x 34




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g g

Illustration


Transportation problems need constraints because each originhas a limited supply and each destination has a demandrequirement.

With three origins (plants), the Foster transportation problemhas three supply constraints.

The supply constraints are:x 11 + x 12 + x 13 + x 14 ≤ 5, 000 Cleveland supply.x 21 + x 22 + x 23 + x 24 ≤ 6, 000 Bedford supply.x 31 + x 32 + x 33 + x 34 ≤ 2, 500 York supply.




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g g

Illustration


With four distribution centers (destinations), four demandconstraints are needed.

The demand constraints are:x 11 + x 21 + x 31 = 6 , 000 Boston demand.x 12 + x 22 + x 32 = 4 , 000 Chicago demand.x 13 + x 23 + x 33 = 2 , 000 St. Louis demand.x 14 + x 24 + x 34 = 1 , 500 Lexington demand.




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Illustration


Min.

3x 11 +2 x 12 +7 x 13 +6 x 14 +7 x 21 +5 x 22 +2 x 23 +3 x 24 +2 x 31 +5 x 32 +4 x 33 +5 x 34

s.tx 11 + x 12 + x 13 + x 14 ≤ 5, 000

x 21 + x 22 + x 23 + x 24 ≤ 6, 000

x 31 + x 32 + x 33 + x 34 ≤ 2, 500

x 11 + x 21 + x 31 = 6 , 000x 12 + x 22 + x 32 = 4 , 000

x 13 + x 23 + x 33 = 2 , 000

x 14 + x 24 + x 34 = 1 , 500

x ij ≥ 0 for i = 1 , 2, 3 and j = 1 , 2, 3, 4IFC6503C PSTI UNSRAT



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Illustration

Transportation ProblemA General Linear Programming Model

Minimizem

i =1

n

j =1

cij x ij

subject ton

j =1

x ij ≤ s i i = 1 , 2, . . . , m Supply

m

i =1

x ij = d j i = 1 , 2, . . . , n Demmand

x ij ≥ 0 for all i and j




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Computer Solution

Computer Solution

There are several known methods to solve transportation.

The rst, and the easiest solution is by using computersoftware. Since we can express transportation problem in linearprogramming model, therefore we can use GLPK to solve it.

Start with extracting the coefficients and write them in amatrix.




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Computer Solution

The Foster Generators ProblemSystem of Equations

Min.

3x 11 +2 x 12 +7 x 13 +6 x 14 +7 x 21 +5 x 22 +2 x 23 +3 x 24 +2 x 31 +5 x 32 +4 x 33 +5 x 34

s.t

x 11 + x 12 + x 13 + x 14 ≤ 5, 000

x 21 + x 22 + x 23 + x 24 ≤ 6, 000

x 31 + x 32 + x 33 + x 34 ≤ 2, 500

x 11 + x 21 + x 31 = 6 , 000

x 12 + x 22 + x 32 = 4 , 000

x 13 + x 23 + x 33 = 2 , 000

x 14 + x 24 + x 34 = 1 , 500

x ij ≥

0 for i = 1 , 2, 3 and j = 1 , 2, 3, 4IFC6503C PSTI UNSRAT



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Computer Solution

Constraints in Tabular Form

Table 4 : Foster Generators transportation problem constraints coefficients.

x 11 x12 x13 x 14 x 21 x22 x23 x24 x 31 x32 x33 x341 1 1 1 0 0 0 0 0 0 0 00 0 0 0 1 1 1 1 0 0 0 00 0 0 0 0 0 0 0 1 1 1 11 0 0 0 1 0 0 0 1 0 0 00 1 0 0 0 1 0 0 0 1 0 00 0 1 0 0 0 1 0 0 0 1 00 0 0 1 0 0 0 1 0 0 0 1




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Computer Solution

Assigning Variables

Assign the coeffients of the constraints to a matrix. mat


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Computer Solution

Assigning Variablesx 11 x 12 x 13 x 14 x 21 x 22 x 23 x 24 x 31 x 32 x 33 x 34

1 1 1 1 0 0 0 0 0 0 0 00 0 0 0 1 1 1 1 0 0 0 00 0 0 0 0 0 0 0 1 1 1 11 0 0 0 1 0 0 0 1 0 0 00 1 0 0 0 1 0 0 0 1 0 00 0 1 0 0 0 1 0 0 0 1 00 0 0 1 0 0 0 1 0 0 0 1

mat

## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12]## [1,] 1 1 1 1 0 0 0 0 0 0 0 0

## [2,] 0 0 0 0 1 1 1 1 0 0 0 0## [3,] 0 0 0 0 0 0 0 0 1 1 1 1## [4,] 1 0 0 0 1 0 0 0 1 0 0 0## [5,] 0 1 0 0 0 1 0 0 0 1 0 0## [6,] 0 0 1 0 0 0 1 0 0 0 1 0## [7,] 0 0 0 1 0 0 0 1 0 0 0 1




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Computer Solution

Assigning Variables

Assign the right side values to a variable.

rhs


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Solving LP using Rglpk

Do not forget to load the Rglpk package before usingRglpk solve LP() .

library (Rglpk)

## Loading required package: slam ## Using the GLPK callable library version 4.52

Rglpk_solve_LP ( obj =obj, mat =mat, dir =dir, rhs =rhs, max = F)

## $optimum## [1] 39500#### $solution## [1] 3500 1500 0 0 0 2500 2000 1500 2500 0 0## [12] 0#### $status## [1] 0



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Computer Solution

ConclusionThe Optimal Solution

Based on the results of the software calculation, the optimalsolution to the Foster Generators transportation problem is as shonin Table 5.




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Computer Solution

ConclusionThe Optimal Solution

Table 5 : Optimal solution of Foster Generators transportation problem.

Variable 1 From To UnitsShipped Costper Unit ($) Total Cost($)x 11 Cleveland Boston 3500 3 10,500x 12 Cleveland Chicago 1500 2 3,000x 22 Bedford Chicago 2500 5 12,500x 23 Bedford St. Louis 2000 2 4,000x 24 Bedford Lexington 1500 3 4,500x 31 York Boston 2500 2 5,000

Total 39,500

1 Actually, the Variable column is not a part of the solution. It is added only

to ease the comparison with the linear model.IFC6503C PSTI UNSRAT



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References I

[1] D. R. Anderson, D. J. Sweeney, T. A. Williams, J. D. Camm, andK. Martin, An Introduction to Management Science: Quantitative Approaches to Decision Making , revised 13th ed. OH:South-Western/Cengage Learning, 2012.


Or2016 s07 Transportation Problem Pt1 Handout

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