transportation application

7/27/2019 Transportation Application

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TRANSPORTATIONAPPLICATION

Shipping ProblemThe transportation or shipping problem involvesdetermining the amount of goods or items to betransported from a number of origins to a number ofdestinationsThe objective usually is to minimize total shipping costsor distancesThis is a specific case of LP and a special algorithm hasbeen developed to solve it


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The Top Speed Bicycle Co. manufactures andmarkets a line of 10-speed bicyclesThe firm has final assembly plants in two citieswhere labor costs are lowIt has three major warehouses near large marketsThe sales requirements for the next year areNew York 10,000 bicyclesChicago 8,000 bicyclesLos Angeles 15,000 bicycles

The factory capacities areNew Orleans 20,000 bicyclesOmaha 15,000 bicycles


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The cost of shipping bicycles from the plants to thewarehouses is different for each plant and warehouseTO

FROM NEW YORK CHICAGO LOS ANGELES

New Orleans $2 $3 $5

Omaha $3 $1 $4

The company wants to develop a shipping schedulethat will minimize its total annual cost


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The double subscript variables will represent the origin factoryand the destination warehouseX ij = bicycles shipped from factory i to warehouse j

SoX11 = number of bicycles shipped from New Orleans to NewYorkX12 = number of bicycles shipped from New Orleans to ChicagoX13 = number of bicycles shipped from New Orleans to LosAngelesX21 = number of bicycles shipped from Omaha to New YorkX22 = number of bicycles shipped from Omaha to ChicagoX23 = number of bicycles shipped from Omaha to Los Angeles


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Objective functionMinimize totalshipping costs

= 2X11 + 3X12 + 5X13 + 3X21 + 1X22 + 4X23

subject to X11 + X21 = 10,000 (New York demand)

X12 + X22 = 8,000 (Chicago demand)

X13 + X23 = 15,000 (Los Angeles demand)

X11 + X12 + X13 20,000 (New Orleans factory supply)

X21 + X22 + X23 15,000 (Omaha factory supply)

All variables 0


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Formulation for Excels Solver


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Solution from Excels Solver


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Top Speed Bicycle solutionTO

FROM NEW YORK CHICAGO LOS ANGELES

New Orleans 10,000 0 8,000

Omaha 0 8,000 7,000

Total shipping cost equals $96,000Transportation problems are a special caseof LP as the coefficients for every variable inthe constraint equations equal 1This situation exists in assignmentproblems as well as they are a special case ofthe transportation problem


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Truck Loading ProblemThe truck loading problem involves deciding which

items to load on a truck so as to maximize the value of a

load shipped

Goodman Shipping has to ship the following six items

ITEM VALUE ($) WEIGHT (POUNDS)

1 22,500 7,5002 24,000 7,500

3 8,000 3,000

4 9,500 3,500

5 11,500 4,000

6 9,750 3,500


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Maximize

load value

$22,500X1 + $24,000X2 + $8,000X3+ $9,500X4 + $11,500X5 + $9,750X6=

Objective functionsubject to

7,500X1 + 7,500X2 + 3,000X3+ 3,500X4 + 4,000X5 + 3,500X6 10,000 lb capacity

X1 1

X2 1

X3 1

X4 1

X5 1

X6 1

X1,X2, X3, X4, X5, X6 0


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Excel Solver formulation for Goodman Shipping


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The Goodman Shipping problem has aninteresting issueThe solution calls for one third of Item 1 to beloaded on the truckWhat if Item 1 can not be divided into smallerpieces?Rounding down leaves unused capacity on thetruck and results in a value of $24,000Rounding up is not possible since this would

exceed the capacity of the truckUsing integer programming, the solution is toload one unit of Items 3, 4, and 6 for a value of$27,250


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The transportation problem is a special case ofthe transshipment problem When the items are being moved from a source

to a destination through an intermediate point (atransshipment point), the problem is called atransshipment problem


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Distribution CentersFrosty Machines manufactures snowblowers inToronto and DetroitThese are shipped to regional distribution centersin Chicago and BuffaloFrom there they are shipped to supply houses inNew York, Philadelphia, and St LouisShipping costs vary by location and destinationSnowblowers can not be shipped directly from thefactories to the supply houses


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Frosty Machines network

Toronto

Detroit

Source

Chicago

Buffalo

Transshipment Point

New York City

Philadelphia

St Louis

Destination


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Frosty Machines dataTO

FROM CHICAGO BUFFALONEW YORKCITY PHILADELPHIA ST LOUIS SUPPLY

Toronto $4 $7 800

Detroit $5 $7 700

Chicago $6 $4 $5

Buffalo $2 $3 $4

Demand 450 350 300

Frosty would like to minimize the transportationcosts associated with shipping snowblowers tomeet the demands at the supply centers given thesupplies available


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A description of the problem would be to minimize costsubject to1. The number of units shipped from Toronto is not more than 800

2. The number of units shipped from Detroit is not more than 700

3. The number of units shipped to New York is 450

4. The number of units shipped to Philadelphia is 350

5. The number of units shipped to St Louis is 300

6. The number of units shipped out of Chicago is equal to the number of

units shipped into Chicago

7. The number of units shipped out of Buffalo is equal to the number of

units shipped into Buffalo


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The decision variables should represent thenumber of units shipped from each source to thetransshipment points and from there to the finaldestinationsT1 = the number of units shipped from Toronto to Chicago

T2 = the number of units shipped from Toronto to Buffalo

D1 = the number of units shipped from Detroit to Chicago

D2 = the number of units shipped from Detroit to Chicago

C1 = the number of units shipped from Chicago to New York

C2 = the number of units shipped from Chicago to Philadelphia

C3 = the number of units shipped from Chicago to St Louis

B1 = the number of units shipped from Buffalo to New York

B2 = the number of units shipped from Buffalo to Philadelphia

B3 = the number of units shipped from Buffalo to St Louis


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The linear program isMinimize cost =

4T1 + 7T2 + 5D1 + 7D2 + 6C1 + 4C2 + 5C3 + 2B1 + 3B2 + 4B3

subject toT1 + T2 800 (supply at Toronto)

D1 + D2 700 (supply at Detroit)

C1 + B1 = 450 (demand at New York)

C2 + B2 = 350 (demand at Philadelphia)

C3 + B3 = 300 (demand at St Louis)

T1 + D1 = C1 + C2 + C3 (shipping through Chicago)

T2 + D2 = B1 + B2 + B3 (shipping through Buffalo)

T1, T2, D1, D2, C1, C2, C3, B1, B2, B3 0 (nonnegativity)


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The solution from QM for Windows is

transportation application

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