Transportation and Assignment Problems

Example: BBC

Building Brick Company (BBC) has orders for 80 tons of bricks at three suburban locations as follows: Northwood -- 25 tons, Westwood -- 45 tons, and Eastwood -- 10 tons. BBC has two plants, each of which can produce 50 tons per week. Delivery cost per ton from each plant to each suburban location is shown below:

Delivery Cost Per Ton Northwood Westwood Eastwood

Plant 1 24 30 40 Plant 2 30 40 42

How should end of week shipments be made to fill the above orders?

Transportation Problem

• Network Representation 11

22

33

11

22

c1

1c12

c13

c21 c22

c23

d1

d2

d3

s1

s2

SOURCES DESTINATIONS

Transportation Problem

• LP FormulationThe LP formulation in terms of the amounts

shipped from the origins to the destinations, xij , can be written as:

Min cijxij

i j

s.t. xij < si for each origin i j

xij = dj for each destination j i

xij > 0 for all i and j

Transportation Problem• LP Formulation Special Cases

The following special-case modifications to the linear programming formulation can be made:– Minimum shipping guarantee from i to j:

xij > Lij

– Maximum route capacity from i to j:

xij < Lij

– Unacceptable route:

Remove the corresponding decision variable.

Assignment Problem

• An assignment problem seeks to minimize the total cost assignment of m workers to m jobs, given that the cost of worker i performing job j is cij.

• It assumes all workers are assigned and each job is performed.

• An assignment problem is a special case of a transportation problem in which all supplies and all demands are equal to 1; hence assignment problems may be solved as linear programs.

• The network representation of an assignment problem with three workers and three jobs is shown on the next slide.

Assignment Problem• Network Representation

22

33

11

22

33

11c11

c12

c13

c21 c22

c23

c31 c32

c33

AGENTS TASKS

Assignment Problem

• LP Formulation

Min cijxij i j

s.t. xij = 1 for each agent i j

xij = 1 for each task j i

xij = 0 or 1 for all i and j

LP Formulation Special Cases

• Number of agents exceeds the number of tasks:

xij < 1 for each agent i j

• Number of tasks exceeds the number of agents:

Add enough dummy agents to equalize the

number of agents and the number of tasks.

The objective function coefficients for these

new variable would be zero.

Assignment Problem

A contractor pays his subcontractors a fixed fee plus mileage for work performed. On a given day the contractor is faced with three electrical jobs associated with various projects. Given below are the distances between the subcontractors and the projects.

Projects Subcontractor A B C Westside 50 36 16

Federated 28 30 18 Goliath 35 32 20

Universal 25 25 14

How should the contractors be assigned to minimize total costs?

Example: Contractor

Network Representation

5036

16

2830

18

35 32

2025 25

14

West.West.

CC

BB

AA

Univ.Univ.

Gol.Gol.

Fed. Fed.

ProjectsSubcontractors

Linear Programming Formulation

Min 50x11+36x12+16x13+28x21+30x22+18x23

+35x31+32x32+20x33+25x41+25x42+14x43

s.t. x11+x12+x13 < 1

x21+x22+x23 < 1 x31+x32+x33 < 1 x41+x42+x43 < 1 x11+x21+x31+x41 = 1 x12+x22+x32+x42 = 1 x13+x23+x33+x43 = 1 xij = 0 or 1 for all i and j

Agents

Tasks

Transshipment Problem• Linear Programming Formulation xij represents the shipment from node i to node j

Min cijxij i j s.t. xij < si for each origin i j xik - xkj = 0 for each intermediate i j node k xij = dj for each destination j i xij > 0 for all i and j

Example: Transshipping

Thomas Industries and Washburn Corporation supply three firms (Zrox, Hewes, Rockwright) with customized shelving for its offices. They both order shelving from the same two manufacturers, Arnold Manufacturers and Supershelf, Inc.

Currently weekly demands by the users are 50 for Zrox, 60 for Hewes, and 40 for Rockwright. Both Arnold and Supershelf can supply at most 75 units to its customers.

Additional data is shown on the next slide.

Example: Transshipping

Because of long standing contracts based on past orders, unit costs from the manufacturers to the suppliers are:

Thomas Washburn Arnold 5 8 Supershelf 7 4

The costs to install the shelving at the various locations are:

Zrox Hewes Rockwright Thomas 1 5 8

Washburn 3 4 4

Example: Transshipping

• Network Representation

ARNOLD

WASHBURN

ZROX

HEWES

75

75

50

60

40

5

8

7

4

15

8

3

44

Arnold

SuperShelf

Hewes

Zrox

Thomas

Wash-Burn

Rock-Wright

TransshipmentWidgetco manufactures widgets at two factories, one in Memphis and one in Denver. The Memphis factory can produce as many as 150 widgets per day, and the Denver factory can produce as many as 200 widgets per day. Widgets are shipped by air to customers in Los Angeles and Boston. The customers in each city require 130 widgets per day. Because of the deregulation of airfares, Widgetco believes that it may be cheaper to first fly some widgets to New York or Chicago and then fly them to their final destinations. The costs of flying a widget are shown in Table 58. Widgetco wants to minimize the total cost of shipping the required widgets to its customers.

Transportation Table

Sources

• Wayne Winston• Anderson Sweeney Williams