Intro to Management science

Slides by

John

Loucks

St. Edward’s

University

‹#›

‹#›

Chapter 6, Part A: Distribution and Network Models

Supply Chain Models

Transportation Problem

Transshipment Problem

Assignment Problem

‹#›

‹#›

Transportation, Transshipment, and Assignment 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, transshipment, assignment, shortest-route, and maximal flow problems of this chapter as well as the PERT/CPM problems (in Chapter 13) are all examples of network problems.

‹#›

‹#›

Transportation, Transshipment, and Assignment Problems

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

For each of the problems, 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 that contain separate computer codes for these problems which take advantage of their network structure.

‹#›

‹#›

Supply Chain Models

A supply chain describes the set of all interconnected resources involved in producing and distributing a product.

In general, supply chains are designed to satisfy customer demand for a product at minimum cost.

Those that control the supply chain must make decisions such as where to produce a product, how much should be produced, and where it should be sent.

‹#›

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 on the next slide.

‹#›

‹#›

Transportation Problem

Network Representation

2

c11

c12

c13

c21

c22

c23

d1

d2

d3

s1

s2

Sources

Destinations

3

2

1

1

‹#›

‹#›

Linear Programming Formulation

Using the notation:

xij = number of units shipped from

origin i to destination j

cij = cost per unit of shipping from

origin i to destination j

si = supply or capacity in units at origin i

dj = demand in units at destination j

continued

Transportation Problem

‹#›

‹#›

Transportation Problem

Linear Programming Formulation (continued)

xij > 0 for all i and j

‹#›

LP Formulation Special Cases

Total supply exceeds total demand:

Total demand exceeds total supply:

Add a dummy origin with supply equal to the

shortage amount. Assign a zero shipping cost

per unit. The amount “shipped” from the

dummy origin (in the solution) will not actually

be shipped.

Transportation Problem

No modification of LP formulation is necessary.

‹#›

Transportation Problem

LP Formulation Special Cases (continued)

The objective is maximizing profit or revenue:

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.

Solve as a maximization problem.

‹#›

‹#›

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

on the next slide.

How should end of week shipments be made to fill

the above orders?

Transportation Problem: Example #1

‹#›

‹#›

Delivery Cost Per Ton

Northwood Westwood Eastwood

Plant 1 24 30 40

Plant 2 30 40 42

Transportation Problem: Example #1

‹#›

‹#›

Optimal Solution

Variable From To Amount Cost

x11 Plant 1 Northwood 5 120

x12 Plant 1 Westwood 45 1,350

x13 Plant 1 Eastwood 0 0

x21 Plant 2 Northwood 20 600

x22 Plant 2 Westwood 0 0

x23 Plant 2 Eastwood 10 420

Total Cost = $2,490

Transportation Problem: Example #1

‹#›

‹#›

The Navy has 9,000 pounds of material in Albany,

Georgia that it wishes to ship to three installations:

San Diego, Norfolk, and Pensacola. They require 4,000,

2,500, and 2,500 pounds, respectively. Government

regulations require equal distribution of shipping

among the three carriers.

Transportation Problem: Example #2

‹#›

‹#›

The shipping costs per pound for truck, railroad,

and airplane transit are shown on the next slide.

Formulate and solve a linear program to determine the

shipping arrangements (mode, destination, and

quantity) that will minimize the total shipping cost.

Transportation Problem: Example #2

‹#›

‹#›

Destination

Mode San Diego Norfolk Pensacola

Truck $12 $ 6 $ 5

Railroad 20 11 9

Airplane 30 26 28

Transportation Problem: Example #2

‹#›

‹#›

Define the Decision Variables

We want to determine the pounds of material, xij , to be shipped by mode i to destination j. The following table summarizes the decision variables:

San Diego Norfolk Pensacola

Truck x11 x12 x13

Railroad x21 x22 x23

Airplane x31 x32 x33

Transportation Problem: Example #2

‹#›

‹#›

Define the Objective Function

Minimize the total shipping cost.

Min: (shipping cost per pound for each mode per

destination pairing) x (number of pounds shipped

by mode per destination pairing).

Min: 12x11 + 6x12 + 5x13 + 20x21 + 11x22 + 9x23

+ 30x31 + 26x32 + 28x33

Transportation Problem: Example #2

‹#›

‹#›

Define the Constraints

Equal use of transportation modes:

(1) x11 + x12 + x13 = 3000

(2) x21 + x22 + x23 = 3000

(3) x31 + x32 + x33 = 3000

Destination material requirements:

(4) x11 + x21 + x31 = 4000

(5) x12 + x22 + x32 = 2500

(6) x13 + x23 + x33 = 2500

Non-negativity of variables:

xij > 0, i = 1, 2, 3 and j = 1, 2, 3

Transportation Problem: Example #2

‹#›

‹#›

Computer Output

Objective Function Value = 142000.000

Variable Value Reduced Cost

x11 1000.000 0.000

x12 2000.000 0.000

x13 0.000 1.000

x21 0.000 3.000

x22 500.000 0.000

x23 2500.000 0.000

x31 3000.000 0.000

x32 0.000 2.000

x33 0.000 6.000

Transportation Problem: Example #2

‹#›

‹#›

Solution Summary

San Diego will receive 1000 lbs. by truck

and 3000 lbs. by airplane.

Norfolk will receive 2000 lbs. by truck

and 500 lbs. by railroad.

Pensacola will receive 2500 lbs. by railroad.

The total shipping cost will be $142,000.

Transportation Problem: Example #2

‹#›

‹#›

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 on the next slide.

‹#›

‹#›

Transshipment Problem

Network Representation

c13

c14

c23

c24

c25

c15

s1

c36

c37

c46

c47

c56

c57

d1

d2

Intermediate Nodes

Sources

Destinations

s2

Demand

Supply

2

3

4

5

6

7

1

‹#›

‹#›

Linear Programming Formulation

Using the notation:

xij = number of units shipped from node i to node j

cij = cost per unit of shipping from node i to node j

si = supply at origin node i

dj = demand at destination node j

continued

Transshipment Problem

‹#›

‹#›

xij > 0 for all i and j

Linear Programming Formulation (continued)

continued

Transshipment Problem

Origin nodes i

Transshipment nodes

Destination nodes j

s.t.

‹#›

‹#›

LP Formulation Special Cases

Total supply not equal to total demand

Maximization objective function

Route capacities or route minimums

Unacceptable routes

The LP model modifications required here are

identical to those required for the special cases in

the transportation problem.

Transshipment Problem

‹#›

‹#›

Transshipment Problem: Example

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.

Additional data is shown on the next slide.

‹#›

‹#›

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

Transshipment Problem: Example

‹#›

‹#›

Transshipment Problem: Example

Network Representation

75

75

50

60

40

5

8

7

4

1

5

8

3

4

4

Arnold

Super

Shelf

Hewes

Zrox

Zeron

N

Zeron

S

Rock-

Rite

‹#›

‹#›

Transshipment Problem: Example

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 Defined

Minimize Overall Shipping Costs:

Min 5x13 + 8x14 + 7x23 + 4x24 + 1x35 + 5x36 + 8x37

+ 3x45 + 4x46 + 4x47

‹#›

‹#›

Transshipment Problem: Example

Constraints Defined

Amount Out of Arnold: x13 + x14 < 75

Amount Out of Supershelf: x23 + x24 < 75

Amount Through Zeron N: x13 + x23 - x35 - x36 - x37 = 0

Amount Through Zeron S: x14 + x24 - x45 - x46 - x47 = 0

Amount Into Zrox: x35 + x45 = 50

Amount Into Hewes: x36 + x46 = 60

Amount Into Rockrite: x37 + x47 = 40

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

‹#›

‹#›

Computer Output

Objective Function Value = 1150.000

Variable Value Reduced Cost

X13 75.000 0.000

X14 0.000 2.000

X23 0.000 4.000

X24 75.000 0.000

X35 50.000 0.000

X36 25.000 0.000

X37 0.000 3.000

X45 0.000 3.000

X46 35.000 0.000

X47 40.000 0.000

Transshipment Problem: Example

‹#›

‹#›

Solution

75

75

50

60

40

5

8

7

4

1

5

8

3

4

4

75

75

50

25

35

40

Arnold

Super

Shelf

Hewes

Zrox

Zeron

N

Zeron

S

Rock-

Rite

Transshipment Problem: Example

‹#›

‹#›

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

2

3

1

2

3

1

c11

c12

c13

c21

c22

c23

c31

c32

c33

Agents

Tasks

‹#›

‹#›

Linear Programming Formulation

Using the notation:

xij = 1 if agent i is assigned to task j

0 otherwise

cij = cost of assigning agent i to task j

continued

Assignment Problem

‹#›

‹#›

Assignment Problem

Linear Programming Formulation (continued)

xij > 0 for all i and j

‹#›

‹#›

LP Formulation Special Cases

Number of agents exceeds the number of tasks:

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.

Extra agents simply remain unassigned.

Assignment Problem

‹#›

‹#›

LP Formulation 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 t tasks:

Assignment Problem

‹#›

‹#›

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.

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 so that total

mileage is minimized?

Assignment Problem: Example

‹#›

‹#›

Network

Representation

50

36

16

28

30

18

35

32

20

25

25

14

West.

C

B

A

Univ.

Gol.

Fed.

Projects

Subcontractors

Assignment Problem: Example

‹#›

‹#›

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

Assignment Problem: Example

‹#›

‹#›

The optimal assignment is:

Subcontractor Project Distance

Westside C 16

Federated A 28

Goliath (unassigned)

Universal B 25

Total Distance = 69 miles

Assignment Problem: Example

‹#›

‹#›

End of Chapter 6, Part A

‹#›

‹#›