kerimcan ozcanmngt 379 operations research1 transportation, assignment, and transshipment problems...

Kerimcan Ozcan MNGT 379 Operations Research 1

Transportation, Assignment,and Transshipment Problems

Chapter 7

MNGT 379 Operations Research 2Kerimcan Ozcan

Transportation, Assignment, and Transshipment Problems A network model is one which can be represented

by a set of nodes, a set of arcs, and functions (e.g. costs, supplies, demands, etc.) associated with the arcs and/or nodes.

Transportation, assignment, and transshipment problems of this chapter as well as the PERT/CPM problems (in another chapter) are all examples of network problems.

Each of the three models of this chapter can be formulated as linear programs and solved by general purpose linear programming codes.

For each of the three models, if the right-hand side of the linear programming formulations are all integers, the optimal solution will be in terms of integer values for the decision variables.

However, there are many computer packages (including The Management Scientist) that contain separate computer codes for these models which take advantage of their network structure.


Transportation Problem

The transportation problem seeks to minimize the total shipping costs of transporting goods from m origins (each with a supply si) to n destinations (each with a demand dj), when the unit shipping cost from an origin, i, to a destination, j, is cij.

The network representation for a transportation problem with two sources and three destinations is given below.

22

cc11

11cc1212

cc1313

cc2121

cc2222cc2323

dd11

dd22

dd33

ss11

s2

SourcesSources DestinationsDestinations

33

22

11

11


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 js.t. xij < si for each origin i

j

xij = dj for each destination j i

xij > 0 for all i and j

Special CasesThe 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.


Example: Acme Block Co.

Acme Block Company has orders for 80 tons of concrete blocks at three suburban locations as follows: Northwood -- 25 tons, Westwood -- 45 tons, and Eastwood -- 10 tons. Acme 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. How should end of week shipments be made to fill the above orders?

Northwood Westwood EastwoodPlant 1 24 30 40Plant 2 30 40 42

MIN 24x11+30x12+40x13+30x21+40x22+42x23

S.T.

1) 1x11+1x12+1x13<50 2) 1x21+1x22+1x23<50 3) 1x11+1x21=25 4) 1x12+1x22=45 5) 1x13+1x23=10


Example: Acme Block Co.Objective Function Value = 2490.000

Variable Value Reduced Costs -------------- --------------- ------------------ x11 5.000 0.000 x12 45.000 0.000 x13 0.000 4.000 x21 20.000 0.000 x22 0.000 4.000 x23 10.000 0.000

Constraint Slack/Surplus Dual Prices -------------- --------------- ------------------ 1 0.000 6.000 2 20.000 0.000 3 0.000 -30.000 4 0.000 -36.000 5 0.000 -42.000

OBJECTIVE COEFFICIENT RANGES

Variable Lower Limit Current Value Upper Limit ------------ --------------- --------------- --------------- x11 20.000 24.000 28.000 x12 No Lower Limit 30.000 34.000 x13 36.000 40.000 No Upper Limit x21 26.000 30.000 34.000 x22 36.000 40.000 No Upper Limit x23 No Lower Limit 42.000 46.000

RIGHT HAND SIDE RANGES

Constraint Lower Limit Current Value Upper Limit ------------ --------------- --------------- --------------- 1 45.000 50.000 70.000 2 30.000 50.000 No Upper Limit 3 5.000 25.000 45.000 4 25.000 45.000 50.000 5 0.000 10.000 30.000


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 below.

2222

3333

1111

2222

3333

1111 cc1111cc1212

cc1313

cc2121 cc2222

cc2323

cc3131 cc3232

cc3333

AgentsAgents

TasksTasks


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

Note: A modification to the right-hand side of the first constraint set can be made if a worker is permitted to work more than 1 job.

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

Special Cases (continued) The assignment alternatives are evaluated in

terms of revenue or profit: Solve as a maximization problem.

An assignment is unacceptable: Remove the corresponding decision

variable. An agent is permitted to work a tasks:

xij < a for each agent i

j


Example: Who Does What?

An electrical 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.

ProjectsSubcontractor 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 mileage costs?



Network Representation

5050

3636

1616

28283030

1818

3535 3232

20202525 2525

1414

WestWest..WestWest..

CCCC

BBBB

AAAA

Univ.Univ.Univ.Univ.

Gol.Gol.Gol.Gol.

Fed.Fed. 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

The optimal assignment is: Subcontractor Project Distance Westside C 16 Federated A 28

Goliath (unassigned) Universal B 25 Total Distance = 69 miles


Transshipment Problem

Transshipment problems are transportation problems in which a shipment may move through intermediate nodes (transshipment nodes) before reaching a particular destination node.

Transshipment problems can be converted to larger transportation problems and solved by a special transportation program.

Transshipment problems can also be solved by general purpose linear programming codes.

The network representation for a transshipment problem with two sources, three intermediate nodes, and two destinations is shown below.

22 22

3333

4444

5555

6666

77 77

11 11cc1313

cc1414

cc2323

cc2424cc2525

cc1515

ss11

cc3636

cc3737

cc4646

cc4747

cc5656

cc5757

dd11

dd22

SourcesSources DestinationsDestinationsss22

DemanDemandd

SupplSupplyy

Intermediate NodesIntermediate Nodes


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: Zeron Shelving

The Northside and Southside facilities of Zeron Industries supply three firms (Zrox, Hewes, Rockrite) 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 Rockrite. Both Arnold and Supershelf can supply at most 75 units to its customers.

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

Zeron N Zeron S Arnold 5 8 Supershelf 7 4

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

Zrox Hewes Rockrite Thomas 1 5 8

Washburn 3 4 4



Network Representation

7575

7575

5050

6060

4040

55

88

77

44

1155

88

33

4444

ArnoldArnold

SuperSuperShelfShelf

HewesHewes

ZroxZrox

ZeronZeronNN

ZeronZeronSS

Rock-Rock-RiteRite



Linear Programming Formulation Decision Variables Defined

xij = amount shipped from manufacturer i to supplier j xjk = amount shipped from supplier j to customer k where i = 1 (Arnold), 2 (Supershelf) j = 3 (Zeron N), 4 (Zeron S) k = 5 (Zrox), 6 (Hewes), 7 (Rockrite)

Objective Function DefinedMinimize Overall Shipping Costs:

Min 5x13 + 8x14 + 7x23 + 4x24 + 1x35 + 5x36 + 8x37 + 3x45 + 4x46 + 4x47

Constraints DefinedAmount Out of Arnold: x13 + x14 < 75Amount Out of Supershelf: x23 + x24 < 75Amount Through Zeron N: x13 + x23 - x35 - x36 - x37 = 0Amount Through Zeron S: x14 + x24 - x45 - x46 - x47 = 0Amount Into Zrox: x35 + x45 = 50Amount Into Hewes: x36 + x46 = 60Amount Into Rockrite: x37 + x47 = 40

Non-negativity of Variables: xij > 0, for all i and j.



Objective Function Value = 1150.000

Variable Value Reduced Costs X13 75.000 0.000X14 0.000 2.000X23 0.000 4.000X24 75.000 0.000X35 50.000 0.000X36 25.000 0.000X37 0.000 3.000X45 0.000 3.000X46 35.000 0.000X47 40.000 0.000

Constraint Slack/Surplus Dual Prices

1 0.000 0.0002 0.000 2.0003 0.000 -5.0004 0.000 -6.0005 0.000 -6.0006 0.000 -10.0007 0.000 -10.000



Optimal SolutionOptimal Solution

7575

7575

5050

6060

4040

55

88

77

44

1155

88

33 44

44

ArnoldArnold

SuperSuperShelfShelf

HewesHewes

ZroxZrox

ZeronZeronNN

ZeronZeronSS

Rock-Rock-RiteRite

7575

7575

5050

2525

3535

4040

kerimcan ozcanmngt 379 operations research1 transportation, assignment, and transshipment problems...

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