Transportation Problem and Related Topics

Transportation Problem and Related Topics

2Ardavan Asef-Vaziri June-2013Transportation Problem and Related Topics

There are 3 plants, 3 warehouses.Production of Plants 1, 2, and 3 are 100, 150, 200 respectively.Demand of warehouses 1, 2 and 3 are 170, 180, and 100 units respectively.Transportation costs for each unit of product is given below

Transportation problem : Narrative representation

Warehouse1 2 3

1 12 11 13Plant 2 14 12 16

3 15 11 12Formulate this problem as an LP to satisfy demand at minimum transportation costs.


Plant 1

Warehouse 1

Plant 2 Plant 3

Warehouse 2 Warehouse 3

Data for the Transportation Model

• Quantity demanded at each destination

• Quantity supplied from each origin• Cost between origin and

destination


$12$11

$13

$12

Plant 1 Plant 2 Plant 3

Warehouse 1 Warehouse 2 Warehouse 1

$14 $16$12

$11

$15

Supply Locations

Demand Locations

100 150 200

Data for the Transportation Model


Transportation problem I : decision variables

1

2

1

33

100x11

x12

2150

200 100

180

170

x13x21

x31

x22

x32

x23

x33



x11 = Volume of product sent from P1 to W1x12 = Volume of product sent from P1 to W2x13 = Volume of product sent from P1 to W3x21 = Volume of product sent from P2 to W1x22 = Volume of product sent from P2 to W2x23 = Volume of product sent from P2 to W3x31 = Volume of product sent from P3 to W1x32 = Volume of product sent from P3 to W2x33 = Volume of product sent from P3 to W3Minimize Z = 12 x11 + 11 x12 +13 x13 + 14 x21 + 12 x22 +16 x23 +15 x31 + 11 x32 +12 x33


Transportation problem I : supply and demand constraints: equal only of Total S = Total D

x11 + x12 + x13 = 100x21 + x22 + x23 =150x31 + x32 + x33 = 200x11 + x21 + x31 = 170x12 + x22 + x32 = 180x13 + x23 + x33 = 100

x11, x12, x13, x21, x22, x23, x31, x32, x33 0


Transportation problem I : supply and demand constraints: ≤ for S, ≥ for D always correct

x11 + x12 + x13 ≤ 100x21 + x22 + x23 ≤ 150x31 + x32 + x33 ≤ 200x11 + x21 + x31 ≥ 170x12 + x22 + x32 ≥ 180x13 + x23 + x33 ≥ 100

x11, x12, x13, x21, x22, x23, x31, x32, x33 0


Origins

We have a set of ORIGINsOrigin Definition: A source of material- A set of Manufacturing Plants- A set of Suppliers- A set of Warehouses- A set of Distribution Centers (DC)

In general we refer to them as Origins

m

1

2

i

s1

s2

si

sm

There are m origins i=1,2, ………., m

Each origin i has a supply of si


Destinations

We have a set of DESTINATIONsDestination Definition: A location with a demand for material- A set of Markets- A set of Retailers- A set of Warehouses- A set of Manufacturing plantsIn general we refer to them as Destinations

n

1

2

j

d1

d2

di

dn

There are n destinations j=1,2, ………., n

Each origin j has a supply of dj


There is only one route between each pair of origin and destinationItems to be shipped are all the samefor each and all units sent from origin i to destination j there is a shipping cost of Cij per unit

Transportation Model Assumptions


Cij : cost of sending one unit of product from origin i to destination j

m

1

2

i

n

1

2

jC1n

C12

C11

C2n

C22

C21

Use Big M (a large number) to eliminate unacceptable routes and allocations.


Xij : Units of product sent from origin i to destination j

m

1

2

i

n

1

2

jx1n

x12

x11

x2n

x22

x21


The Problem

m

1

2

i

n

1

2

j

The problem is to determine how much material is sent from each origin to each destination, such that all demand is satisfied at the minimum transportation cost


The Objective Function

m

1

2

i

n

1

2

j

If we send Xij units from origin i to destination j, its cost is Cij Xij

We want to minimize

ijijxCZ



1

2

1

33

100x11

x12

2150

200 100

180

170

x13x21

x31

x22

x32

x23

x33


Transportation problem I : supply and demand constraints

x11 + x12 + x13 =100 +x21 + x22 + x23 =150 +x31 + x32 + x33 =200x11 + x21 + x31 =170 x12 + x22 + x32 =180 x13 + x23 + x33 = 100

In transportation problem. each variable Xij appears only in two constraints, constraints i and constraint m+j, where m is the number of supply nodes. The coefficients of all the variables in the constraints are 1.


Our Task

Our main task is to formulate the problem.

By problem formulation we mean to prepare a tabular representation for this problem.

Then we can simply pass our formulation ( tabular representation) to EXCEL.

EXCEL will return the optimal solution.

What do we mean by formulation?


Cost Table

Cost TableWarhouse1 Warhouse2 Warhouse3

Plant 1 12 11 13Plant 2 14 12 16Plant 3 15 11 12

`Decision Variable Table

Warhouse1 Warhouse2 Warhouse3Plant 1Plant 2Plant 3


Right Hand Side (RHS)

San Siego Norfolk Pensacola LHS RHSTruck 1000 2000 0 3000 ≤ 3000Railroad 0 500 2500 3000 ≤ 3000Airplane 3000 0 0 3000 ≤ 3000LHS 4000 2500 2500 142000

≥ ≥ ≥RHS 4000 2500 2500


Decision Variable TableWarhouse1 Warhouse2 Warhouse3 RHS

Plant 1 170 180 0 =SUM(B11:D11) 100Plant 2 0 0 0 =SUM(B12:D12) 150Plant 3 0 0 100 =SUM(B13:D13) 200

=SUM(B11:B13) =SUM(C11:C13) =SUM(D11:D13) =SUMPRODUCT(B5:D7,B11:D13)

RHS 170 180 100

Left Hand Side (RHS), and Objective Function


≤ for Supply, ≥ for Demand unless Some Equality Requirement is Enforced


Plant 1 0 ≤ 100Plant 2 0 ≤ 150Plant 3 0 ≤ 200

0 0 0 0≥ ≥ ≥

RHS 170 180 100


≤ for Supply, ≥ for Demand unless Some Equality Requirement is Enforced


Plant 1 0 ≤ 100Plant 2 0 ≤ 150Plant 3 0 ≤ 200

0 0 0 0≥ ≥ ≥

RHS 170 180 100



Plant 1 100 0 0 100 ≤ 100Plant 2 70 80 0 150 ≤ 150Plant 3 0 100 100 200 ≤ 200

170 180 100 5440≥ ≥ ≥

RHS 170 180 100

Optimal Solution

Extra Credit. How the colors were generated and what they mea?Using Conditional formatting.Green if the decision variable is >0Red if the constraint is binding LHS = RHS


Example: Narrative Representation

We have 3 factories and 4 warehouses.Production of factories are 100, 200, 150 respectively.Demand of warehouses are 80, 90, 120, 160 respectively.Transportation cost for each unit of material from each origin to each destination is given below.

Destination1 2 3 4

1 4 7 7 1Origin 2 12 3 8 8

3 8 10 16 5

Formulate this problem as a transportation problem


Excel : Data


11 repairmen and 10 tasks. The time (in minutes) to complete each job by each repairman is given below.

Assign each task to one repairman in order to minimize to total repair time by all the repairmen.In the assignment problem, all RHSs are 1. That is the only difference with the transportation problem,.

The Assignment Problem : Example

Cost Table TaskTime of task j if done by repairman i1 2 3 4 5 6 7 8 9 10

1 40 40 45 30 45 35 50 20 45 302 30 50 30 30 35 30 55 30 55 403 50 20 30 55 30 40 55 25 30 204 35 40 35 55 35 20 45 55 45 455 45 35 50 30 35 20 55 35 40 20

Repairman 6 30 35 50 35 45 35 50 30 55 407 50 55 35 40 45 25 55 35 45 358 20 40 40 25 45 55 35 30 40 409 20 20 45 50 20 50 50 30 25 50

10 20 40 40 35 20 40 40 30 50 3511 45 50 55 30 50 35 55 50 45 40


The Assignment Problem : Example

Decision Variables Task1 2 3 4 5 6 7 8 9 10

1 0 0 0 0 0 0 0 1 0 0 1 ≤ 12 0 0 1 0 0 0 0 0 0 0 1 ≤ 13 0 1 0 0 0 0 0 0 0 0 1 ≤ 14 0 0 0 0 0 1 0 0 0 0 1 ≤ 1

Repairman 5 0 0 0 0 0 0 0 0 0 1 1 ≤ 16 1 0 0 0 0 0 0 0 0 0 1 ≤ 17 0 0 0 0 0 0 0 0 0 0 0 ≤ 18 0 0 0 0 0 0 1 0 0 0 1 ≤ 19 0 0 0 0 0 0 0 0 1 0 1 ≤ 1

10 0 0 0 0 1 0 0 0 0 0 1 ≤ 111 0 0 0 1 0 0 0 0 0 0 1 ≤ 1

1 1 1 1 1 1 1 1 1 1 250≥ ≥ ≥ ≥ ≥ ≥ ≥ ≥ ≥ ≥1 1 1 1 1 1 1 1 1 1


Texaco Corp. Site Selection Problem: Sec. 5.5 Hillier and Hillier Texaco produces most of its oil in its own fields and import

the rest from middle east.

Texaco has several oil fields, several refineries, and several distribution centers.

Texaco is planning to add a new refinery. There are several candidate locations for this new refinery.

There are 3 important factors1. Distance from row material locations (Transportation

cost).2. Distance from distribution locations (Transportation

cost). .3. Production cost


From Plants to Existing and Future WarehousesThree manufacturing PlantsThree WarehouseCapacity, and Distribution costs are given below

Warehouse 1 Warehouse 2 Warehouse 3Plant CapacityPlant 1 2 4 5 80Plant 2 4 5 2 60Plant 3 5 7 3 100Plant 4 2 2 5 120Warehouse Capacity 100 60 80Capacity of the warehouses are nor enogh. Three candidate locations for one additional warehouse.Capacity, and Distribution costs are given below

Candidate 1 Candidate 2 Candidate 3Plant 1 3 1 3Plant 2 1 3 4Plant 3 4 5 6Plant 4 4 3 4Candidate Capacity 120 120 120


From Warehouses to Distribution CentersFour distribution centersThe distance from warehouse to distribution center given below

DC1 DC2 DC3 DC4Warehouse 1 5 2 6 8 100Warehouse 2 6 4 3 5 60Warehouse 3 7 8 4 3 80Candidate 1 8 6 3 2 120Candidate 2 5 4 3 6 120Candidate 3 4 3 1 5 120

100 80 80 100Operations cost at each plant is given belowOperations Cost 620 570 530Which candidate warehouse leads to the minimal system costs


From Plants To Candidate Warehouse 1Warehouse 1 Warehouse 2 Warehouse 3 Candidate 1

Plant 1 2 4 5 3Plant 2 4 5 2 1Plant 3 5 7 3 4Plant 4 2 2 5 4

Warehouse 1 Warehouse 2 Warehouse 3 Candidate 1 SupplyPlant 1 40 0 0 40 80 ≤ 80Plant 2 0 0 0 60 60 ≤ 60Plant 3 0 0 80 20 100 ≤ 100Plant 4 60 60 0 0 120 ≤ 120

100 60 80 120 820≥ ≥ ≥ ≥

Capacity 100 60 80 120





100 60 80 120 860≥ ≥ ≥ ≥

Capacity 100 60 80 120





100 60 80 120 1040≥ ≥ ≥ ≥

Capacity 100 60 80 120


From Existing Warehouses and Candidate 1 to DCs

DC1 DC2 DC3 DC4Warehouse 1 5 2 6 8Warehouse 2 6 4 3 5Warehouse 3 7 8 4 3Candidate 1 8 6 3 2

DC1 DC2 DC3 DC4 SupplyWarehouse 1 20 80 0 0 100 ≤ 100Warehouse 2 60 0 0 0 60 ≤ 60Warehouse 3 20 0 0 60 80 ≤ 80Candidate 1 0 0 80 40 120 ≤ 120

100 80 80 100 1260≥ ≥ ≥ ≥

Demand 100 80 80 100





100 80 80 100 1240≥ ≥ ≥ ≥

Demand 100 80 80 100





100 80 80 100 1080≥ ≥ ≥ ≥

Demand 100 80 80 100


Total System Costs

Candidate 1 Candidate 2 Candidate 3From Plant to Warehouse costs 820 860 1040

From Warehouse to DC costs 1260 1240 1080Productin Costs 620 570 530

Total 2700 2670 2650

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Transportation Problem and Related Topics