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Quantitative Analysis for Management

Thirteenth Edition

Chapter 9

Transportation, Assignment, and Network Models

Learning Objectives

After completing this chapter, students will be able to:

9.1 Construct LP problems for the transportation, assignment, and transshipment models.

9.2 Solve facility location and other application problems with transportation models.

9.3 Use LP to model and solve maximal-flow problems.

9.4 Use LP to model and solve shortest route problems.

9.5 Solve minimal-spanning tree problems.

Chapter Outline

9.1 The Transportation Problem

9.2 The Assignment Problem

9.3 The Transshipment Problem

9.4 Maximal-Flow Problem

9.5 Shortest-Route Problem

9.6 Minimal-Spanning Tree Problem

Introduction (1 of 2)

LP problems modeled as networks

Helps visualize and understand problems

Transportation problem

Transshipment problem

Assignment problem

Maximal-flow problem

Shortest-route problem

Minimal-spanning tree problem

Specialized algorithms available

Introduction (2 of 2)

Common terminology for network models

Points on the network are referred to as nodes

Typically circles

Lines on the network that connect nodes are called arcs

The Transportation Problem

Deals with the distribution of goods from several points of supply (sources) to a number of points of demand (destinations)

Usually given the capacity of goods at each source and the requirements at each destination

Typically objective is to minimize total transportation and production costs

Linear Program for Transportation (1 of 5)

Executive Furniture Corporation transportation problem

Minimize transportation cost

Meet demand

Not exceed supply

Linear Program for Transportation (2 of 5)

Let Xij = number of units shipped from source i to destination j

Where

i = 1, 2, 3, with 1 = Des Moines, 2 = Evansville, and 3 = Fort Lauderdale

j = 1, 2, 3, with 1 = Albuquerque, 2 = Boston, and 3 = Cleveland

Linear Program for Transportation (3 of 5)

Minimize total cost = 5X11 + 4X12 + 3X13 + 8X21 + 4X22

+ 3X23 + 9X31 +7X32 + 5X33

Subject to:

X11 + X12 + X13 ≤ 100 (Des Moines supply)

X21 + X22 + X23 ≤ 300 (Evansville supply)

X31 + X32 + X33 ≤ 300 (Fort Lauderdale supply)

X11 + X21 + X31 = 300 (Albuquerque demand)

X12 + X22 + X32 = 200 (Boston demand)

X13 + X23 + X33 = 200 (Cleveland demand)

Xij ≥ 0 for all i and j

Linear Program for Transportation (4 of 5)

Optimal solution

100 units from Des Moines to Albuquerque

200 units from Evansville to Boston

100 units from Evansville to Cleveland

200 units from Ft. Lauderdale to Albuquerque

100 units from Ft. Lauderdale to Cleveland

Total cost = $3,900

Linear Program for Transportation (5 of 5)

FIGURE 9.1 Network Representation of a Transportation Problem, with Costs, Demands, and Supplies

Using Excel QM

PROGRAM 9.1 Executive Furniture Corporation Solution in Excel 2016 Using Excel QM

A General LP Model for Transportation Problems (1 of 2)

Let

Xij = number of units shipped from source i to destination j

cij = cost of one unit from source i to destination j

si = supply at source I

dj = demand at destination j

A General LP Model for Transportation Problems (2 of 2)

Subject to:

Facility Location Analysis (1 of 7)

Transportation method especially useful

New location is major financial importance

Several alternative locations evaluated

Subjective factors are considered

Final decision also involves minimizing total shipping and production costs

Alternative facility locations analyzed within the framework of one overall distribution system

Facility Location Analysis (2 of 7)

Hardgrave Machine Company produces computer components in Cincinnati, Salt Lake City, and Pittsburgh

Four warehouses in Detroit, Dallas, New York, and Los Angeles

Two new plant sites being considered – Seattle and Birmingham

Which of the new locations will yield the lowest cost for the firm in combination with the existing plants and warehouses?

Facility Location Analysis (3 of 7)

TABLE 9.1 Hardgrave’s Demand and Supply Data

WAREHOUSE	MONTHLY DEMAND (UNITS)	PRODUCTION PLANT	MONTHLY SUPPLY	COST TO PRODUCE ONE UNIT ($)
Detroit	10,000	Cincinnati	15,000	48
Dallas	12,000	Salt Lake City	6,000	50
New York	15,000	Pittsburgh	14,000	52
Los Angeles	9,000	Blank	35,000	Blank
Blank	46,000	Blank	Blank	Blank

Supply needed from a new plant = 46,000 − 35,000 = 11,000 units per month

ESTIMATED PRODUCTION COST PER UNIT AT PROPOSED PLANTS

Seattle	$53
Birmingham	$49

Facility Location Analysis (4 of 7)

TABLE 9.2 Hardgrave’s Shipping Costs

FROM	TO	DETROIT	DALLAS	NEW YORK	LOS ANGELES
CINCINNATI		$25	$55	$40	$60
SALT LAKE CITY		35	30	50	40
PITTSBURGH		36	45	26	66
SEATTLE		60	38	65	27
BIRMINGHAM		35	30	41	50

Solve two transportation problems – one for each combination

Facility Location Analysis (5 of 7)

Xij = number of units shipped from source i to destination j

Where

i = 1, 2, 3, 4 with 1 = Cincinnati, 2 = Salt Lake City, 3 = Pittsburgh, and 4 = Seattle

j = 1, 2, 3, 4 with 1 = Detroit, 2 = Dallas, 3 = New York, and 4 = Los Angeles

Facility Location Analysis (6 of 7)

Minimize total cost = 73X11 + 103X12 + 88X13 + 108X14 + 85X21 + 80X22 + 100X23 + 90X24 + 88X31 + 97X32 + 78X33

+ 118X34 + 113X41 + 91X42 + 118X43 + 80X44

Subject to:

X11 + X21 + X31 + X41 = 10,000 Detroit demand

X12 + X22 + X32 + X42 = 12,000 Dallas demand

X13 + X23 + X33 + X43 = 15,000 New York demand

X14 + X24 + X34 + X44 = 9,000 Los Angeles demand

X11 + X12 + X13 + X14 ≤ 15,000 Cincinnati supply

X21 + X22 + X23 + X24 ≤ 6,000 Salt Lake City supply

X31 + X32 + X33 + X34 ≤ 14,000 Pittsburgh supply

X41 + X42 + X43 + X44 ≤ 11,000 Seattle supply

All variables Xij ≥ 0

Facility Location Analysis (7 of 7)

The total cost for the Seattle alternative = $3,704,000

Reformulating the problem for the Birmingham alternative and solving, the total cost = $3,741,000

Using Excel QM (1 of 2)

PROGRAM 9.2 Facility Location (Seattle) Solution in Excel 2016 Using Excel QM

Using Excel QM (2 of 2)

PROGRAM 9.3 Facility Location (Birmingham) Solution in Excel 2016 Using Excel QM

The Assignment Problem

This class of problem determines the most efficient assignment of people or equipment to particular tasks

Objective is typically to minimize total cost or total task time

Linear Program for Assignment Example (1 of 5)

The Fix-it Shop has just received three new repair projects that must be repaired quickly

A radio

A toaster oven

A coffee table

Three workers with different talents are able to do the jobs

Owner estimates wage cost for workers on projects

Objective – minimize total cost

Linear Program for Assignment Example (2 of 5)

FIGURE 9.2 Example of an Assignment Problem in a Transportation Network Format

Linear Program for Assignment Example (3 of 5)

Let

where

i = 1, 2, 3, with 1 = Adams, 2 = Brown, and 3 = Cooper

j = 1, 2, 3, with 1 = Project 1, 2 = Project 2, and 3 = Project 3

Linear Program for Assignment Example (4 of 5)

Minimize total cost = 11X11 + 14X12 + 6X13 + 8X21

+ 10X22 + 11X23 + 9X31

+ 12X32 + 7X33

subject to

X11 + X12 + X13 = 1

X21 + X22 + X23 = 1

X31 + X32 + X33 = 1

X11 + X21 + X31 = 1

X12 + X22 + X32 = 1

X13 + X23 + X33 = 1

Xij = 0 or 1 for all i and j

Linear Program for Assignment Example (5 of 5)

Solution

X13 = 1, Adams assigned to Project 3

X22 = 1, Brown assigned to Project 2

X31 = 1, Cooper is assigned to Project 1

Total cost = $25

Using Excel QM

PROGRAM 9.4 Mr. Fix-It Shop Assignment Solution in Excel 2016 Using Excel QM

The Transshipment Problem (1 of 8)

Items are being moved from a source to a destination through an intermediate point (a transshipment point)

Transshipment problem

The Transshipment Problem (2 of 8)

Frosty Machines manufactures snow blowers in Toronto and Detroit

Shipped to regional distribution centers in Chicago and Buffalo

Then shipped to supply houses in New York, Philadelphia, and St. Louis

Shipping costs vary by location and destination

Snow blowers cannot be shipped directly from the factories to the supply houses

The Transshipment Problem (3 of 8)

FIGURE 9.3 Network Representation of a Transshipment Example

The Transshipment Problem (4 of 8)

TABLE 9.3 Frosty Machines Transshipment Data

Blank	Blank	Blank	TO	Blank	Blank	Blank
FROM	CHICAGO	BUFFALO	NEW YORK CITY	PHILADELPHIA	ST. LOUIS	SUPPLY
Toronto	$4	$7	—	—	—	800
Detroit	$5	$7	—	—	—	700
Chicago	—	—	$6	$4	$5	—
Buffalo	—	—	$2	$3	$4	—
Demand	—	—	450	350	300	Blank

Minimize transportation costs associated with shipping snow blowers subject to demands and supplies

The Transshipment Problem (5 of 8)

Minimize cost subject to

The number of units shipped from Toronto is not more than 800

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

The number of units shipped to New York is 450

The number of units shipped to Philadelphia is 350

The number of units shipped to St. Louis is 300

The number of units shipped out of Chicago is equal to the number of units shipped into Chicago

The number of units shipped out of Buffalo is equal to the number of units shipped into Buffalo

The Transshipment Problem (6 of 8)

Decision variables

Xij = number of units shipped from location (node) i to location (node) j

where

i = 1, 2, 3, 4

j = 3, 4, 5, 6, 7

The Transshipment Problem (7 of 8)

Minimize cost = 4X13 + 7X14 + 5X23 + 7X24 + 6X35 + 4X36 + 5X37

+ 2X45 + 3X46 + 4X47

subject to

X13 + X14 ≤ 800 (Supply at Toronto [node 1])

X23 + X24 ≤ 700 (Supply at Detroit [node 2])

X35 + X45 = 450 (Demand at New York [node 5])

X36 + X46 = 350 (Demand at Philadelphia [node 6])

X37 + X47 = 300 (Demand at St. Louis [node 7])

X13 + X23 = X35 + X36 + X37 (Shipping through Chicago [node 3])

X14 + X24 = X45 + X46 + X47 (Shipping through Buffalo [node 4])

Xij ≥ 0 for all i and j (nonnegativity)

The Transshipment Problem (8 of 8)

Ship 650 units from Toronto to Chicago

Ship 150 units from Toronto to Buffalo

Ship 300 units from Detroit to Buffalo

Ship 350 units from Chicago to Philadelphia

Ship 300 units form Chicago to St. Louis

Ship 450 units from Buffalo to New York

Total cost = $9,550

Using Excel QM

PROGRAM 9.5 Excel QM Solution to Frosty Machines Transshipment Problem in Excel 2016

Maximal-Flow Problem (1 of 4)

Determining the maximum amount of material that can flow from one point (the source) to another (the sink) in a network

Two common methods

Linear programming

Maximal-flow technique

Maximal-Flow Problem (2 of 4)

Determine maximum number of cars from east to west for Waukesha WI road system

FIGURE 9.4 Road Network for Waukesha Maximal-Flow Example

Maximal-Flow Problem (3 of 4)

Variables

Xij = flow from node i to node j

where

i = 1, 2, 3, 4, 5, 6

j = 1, 2, 3, 4, 5, 6

Constraints necessary for

Capacity of each arc

Equal flows into and out of each arc

Maximal-Flow Problem (4 of 4)

Maximize flow = X61

subject to

X12	≤	3	X13	≤	10	X14	≤	2	Capacities for arcs from node 1
X21	≤	1	X24	≤	1	X26	≤	2	Capacities for arcs from node 2
X34	≤	3	X35	≤	2	Blank	Blank	Blank	Capacities for arcs from node 3
X42	≤	1	X43	≤	1	X46	≤	1	Capacities for arcs from node 4
X53	≤	1	X56	≤	1	Blank	Blank	Blank	Capacities for arcs from node 5
X62	≤	2	X64	≤	1	Blank	Blank	Blank	Capacities for arcs from node 6

(X21 + X61) − (X12 + X13 + X14)	=	Flows into = flows out of node 1
(X12 + X42 + X62) − (X21 + X24 + X26)	=	Flows into = flows out of node 2
(X13 + X43 + X53) − (X34 + X35)	=	Flows into = flows out of node 3
(X14 + X24 + X34 + X64) − (X42 + X43 + X46)	=	Flows into = flows out of node 4
(X35) − (X56 + X53)	=	Flows into = flows out of node 5
(X26 + X46 + X56) − (X61 + X62 + X64)	=	Flows into = flows out of node 6
Xij	≥	Blank

Using Excel QM

PROGRAM 9.6 Waukesha Maximal-Flow Solution in Excel 2016 Using Excel QM

Shortest-Route Problem (1 of 5)

Find the shortest distance from one location to another

Can be modeled as

A linear programming problem with 0-1 variables

A special type of transshipment problem

Using specialized algorithm

Shortest-Route Problem (2 of 5)

Ray Design transports beds, chairs, and other furniture items from the factory to the warehouse

Travel through several cities

No direct interstate highways

Find the route with the shortest distance

FIGURE 9.5 Roads from Ray’s Plant to Warehouse

Shortest-Route Problem (3 of 5)

Variables

Xij = 1 if arc from node i to node j is selected and Xij = 0 otherwise

where

i = 1, 2, 3, 4, 5

j = 2, 3, 4, 5, 6

Constraints specify the number of units going into a node must equal the number of units going out of the node

Shortest-Route Problem (4 of 5)

Origin point must ship one unit

X12 + X13 = 1

Final destination must have one unit shipped into it

X46 + X56 = 1

Intermediate nodes must have same amounts entering and leaving

X12 + X32 = X23 + X24 + X25

X12 + X32 − X23 − X24 − X25 = 0

Shortest-Route Problem (5 of 5)

Minimize distance = 100X12 + 200X13 + 50X23 + 50X32

+ 200X24 + 200X42 + 100X25

+ 100X52 + 40X35 + 40X53 + 150X45

+ 150X54 + 100X46 + 100X56

subject to

X12 + X13	=	1	Node 1
X12 + X32 − X23 − X24 − X25	=	0	Node 2
X13 + X23 − X32 − X35	=	0	Node 3
X24 + X54 − X42 − X45 − X46	=	0	Node 4
X25 + X35 + X45 − X52 − X53 − X54 − X56	=	0	Node 5
X46 + X56	=	1	Node 6
All variables	=	0	or 1

Using Excel QM (1 of 2)

PROGRAM 9.7 Ray Designs, Inc., Solution in Excel 2016 Using Excel QM

Using Excel QM (2 of 2)

Solution

X12 = X23 = X35 = X56 = 1

Route is City 1 to City 2 to City 3 to City 5 to City 6

Total distance traveled = 290 miles

Minimal-Spanning Tree Problem (1 of 8)

Connecting all points of a network together while minimizing the total distance of the connections

Linear programming can be used but is complex

Minimal-spanning tree technique is quite easy

Minimal-Spanning Tree Problem (2 of 8)

Steps for the Minimal-Spanning Tree Technique

Select any node in the network.

Connect this node to the nearest node that minimizes the total distance.

Considering all of the nodes that are now connected, find and connect the nearest node that is not connected. If there is a tie for the nearest node, select one arbitrarily. A tie suggests there may be more than one optimal solution.

Repeat the third step until all nodes are connected.

Minimal-Spanning Tree Problem (3 of 8)

Lauderdale Construction

Housing project in Panama City Beach

Determine the

least expensive

way to provide

water and power

to each house

FIGURE 9.6 Network for Lauderdale Construction

Minimal-Spanning Tree Problem (4 of 8)

Step 1 – Arbitrarily select node 1

Step 2 – Connect node 1 to node 3 (nearest)

FIGURE 9.7 First Iteration for Lauderdale Construction

Minimal-Spanning Tree Problem (5 of 8)

Step 3 – Connect next nearest unconnected node, node 4

Continue for other unconnected nodes

FIGURE 9.8 Second and Third Iterations for Lauderdale Construction

Minimal-Spanning Tree Problem (6 of 8)

Step 4 – Repeat the process

FIGURE 9.9 Last Four Iterations for Lauderdale Construction

Minimal-Spanning Tree Problem (7 of 8)

Step 4 – Repeat the process

FIGURE 9.9 Last Four Iterations for Lauderdale Construction

Minimal-Spanning Tree Problem (8 of 8)

TABLE 9.4 Summary of Steps in Lauderdale Construction Minimal-Spanning Tree Problem

STEP	CONNECTED NODES	UNCONNECTED NODES	CLOSE UNCONNECTED NODES	ARC SELECTED	ARC LENGTH	TOTAL DISTANCE
1	1	2, 3, 4, 5, 6, 7, 8	3	1–3	2	2
2	1, 3	2, 4, 5, 6, 7, 8	4	3–4	2	4
3	1, 3, 4	2, 5, 6, 7, 8	2 or 6	3–2	3	7
4	1, 2, 3, 4	5, 6, 7, 8	5 or 6	2–5	3	10
5	1, 2, 3, 4, 5	6, 7, 8	6	3–6	3	13
6	1, 2, 3, 4, 5, 6	7, 8	8	6–8	1	14
7	1, 2, 3, 4, 5, 6, 8	7	7	8–7	2	16

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