Table of Contents
CD Chapter 15 (Transportation and Assignment Problems)
The P&T Company Distribution Problem (Section 15.1)
Characteristics of Transportation Problems (Section 15.2)
Variants of Transportation Problems: Better Products (Section 15.3)
Variants of Transportation Problems: Nifty (Section 15.3)
Applications of Transportation Problems: Metro Water (Section 15.4)
Applications of Transportation Problems: Northern Airplane (Section 15.4)
Applications of Transportation Problems: Middletown (Section 15.4)
Applications of Transportation Problems: Energetic (Section 15.4)
A Case Study: Texago Corp. Site Selection Problem (Section 15.5)
Characteristics of Assignment Problems: Sellmore (Section 15.6)
Variants of Assignment Problems: Job Shop (Section 15.7)
Variants of Assignment Problems: Better Products (Section 15.7)
Variants of Assignment Problems: Revised Middletown (Section 15.7)
15.2–15.5
15.6–15.14
15.15–15.17
15.18–15.20
15.21–15.22
15.23–15.25
15.26–15.28
15.29–15.31
15.32–15.46
15.47–15.51
15.52-15.54
15.55
15.56
Transportation & Assignment Problems (UW Lecture)
15.57–15.75
These slides are based upon a lecture to second-year MBA students at the University of
Washington that discusses transportation and assignment problems (as taught by one of the
authors).
McGraw-Hill/Irwin
15.1
© The McGraw-Hill Companies, Inc., 2008
P&T Company Distribution Problem
CANNERY 1
Bellingham
WAREHOUSE 3
Rapid City
CANNERY 2
Eugene
CANNERY 3
Albert Lea
WAREHOUSE 2
Salt Lake City
WAREHOUSE 1
Sacramento
WAREHOUSE 4
Albuquerque
McGraw-Hill/Irwin
15.2
© The McGraw-Hill Companies, Inc., 2008
Shipping Data
Cannery
Output
Warehouse
Allocation
Bellingham
75 truckloads
Sacramento
80 truckloads
Eugene
125 truckloads
Salt Lake City
65 truckloads
Albert Lea
100 truckloads
Rapid City
70 truckloads
Total
300 truckloads
Albuquerque
85 truckloads
Total
300 truckloads
McGraw-Hill/Irwin
15.3
© The McGraw-Hill Companies, Inc., 2008
Current Shipping Plan
Warehouse
\ To
Sacramento
Salt Lake City
Rapid City
Albuquerque
Bellingham
75
0
0
0
Eugene
5
65
55
0
Albert Lea
0
0
15
85
From
Cannery
McGraw-Hill/Irwin
15.4
© The McGraw-Hill Companies, Inc., 2008
Shipping Cost per Truckload
Warehouse
\ To
Sacramento
Salt Lake City
Rapid City
Albuquerque
Bellingham
$464
$513
$654
$867
Eugene
352
416
690
791
Albert Lea
995
682
388
685
From
Cannery
Total shipping cost = 75($464) + 5($352) + 65($416) + 55($690) + 15($388) + 85($685)
= $165,595
McGraw-Hill/Irwin
15.5
© The McGraw-Hill Companies, Inc., 2008
Terminology for a Transportation Problem
P&T Company Problem
General Model
Truckloads of canned peas
Units of a commodity
Canneries
Sources
Warehouses
Destinations
Output from a cannery
Supply from a source
Allocation to a warehouse
Demand at a destination
Shipping cost per truckload from a cannery
to a warehouse
Cost per unit distributed from a source to a
destination
McGraw-Hill/Irwin
15.6
© The McGraw-Hill Companies, Inc., 2008
Characteristics of Transportation Problems
•
The Requirements Assumption
– Each source has a fixed supply of units, where this entire supply must be distributed
to the destinations.
– Each destination has a fixed demand for units, where this entire demand must be
received from the sources.
•
The Feasible Solutions Property
– A transportation problem will have feasible solutions if and only if the sum of its
supplies equals the sum of its demands.
•
The Cost Assumption
– The cost of distributing units from any particular source to any particular
destination is directly proportional to the number of units distributed.
– This cost is just the unit cost of distribution times the number of units distributed.
McGraw-Hill/Irwin
15.7
© The McGraw-Hill Companies, Inc., 2008
The Transportation Model
Any problem (whether involving transportation or not) fits the model for a
transportation problem if
1. It can be described completely in terms of a table like Table 15.5 that
identifies all the sources, destinations, supplies, demands, and unit costs, and
2. satisfies both the requirements assumption and the cost assumption.
The objective is to minimize the total cost of distributing the units.
McGraw-Hill/Irwin
15.8
© The McGraw-Hill Companies, Inc., 2008
The P&T Co. Transportation Problem
Unit Cost
Destination
(Warehouse):
Sacramento
Salt Lake City
Rapid City
Albuquerque
Supply
Bellingham
$464
$513
$654
$867
75
Eugene
352
416
690
791
125
Albert Lea
995
682
388
685
100
Demand
80
65
70
85
Source
(Cannery)
McGraw-Hill/Irwin
15.9
© The McGraw-Hill Companies, Inc., 2008
Spreadsheet Formulation
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
B
Unit Cost
Source
(Cannery)
C
D
Bellingham
Eugene
Albert Lea
Shipment Quantity
(Truckloads)
Source
Bellingham
(Cannery)
Eugene
Albert Lea
Total Received
Demand
McGraw-Hill/Irwin
Sacramento
$464
$352
$995
E
F
Destination (Warehouse)
Salt Lake City
Rapid City
$513
$654
$416
$690
$682
$388
Albuquerque
$867
$791
$685
Sacramento
0
80
0
80
=
80
Destination (Warehouse)
Salt Lake City
Rapid City
20
0
45
0
0
70
65
70
=
=
65
70
Albuquerque
55
0
30
85
=
85
15.10
G
H
I
J
Total Shipped
75
125
100
=
=
=
Supply
75
125
100
Total Cost
$152,535
© The McGraw-Hill Companies, Inc., 2008
Network Representation
De ma nds
Supplie s
Destina tions
Sourc es
464
(Be llingham) 75
867
(E ugene) 125
S2
995
(Alber t Le a)100
McGraw-Hill/Irwin
S3
80 (Sa cr amento)
D2
65 (Sa lt La ke City)
D3
70 (Rapid City)
D4
85 (Albuquerque )
513
S1
352
D1
654
416
690
791
682
388
685
15.11
© The McGraw-Hill Companies, Inc., 2008
The Transportation Problem is an LP
Let xij = the number of truckloads to ship from cannery i to warehouse j
(i = 1, 2, 3; j = 1, 2, 3, 4)
Minimize Cost = $464x11 + $513x12 + $654x13 + $867x14 + $352x21 + $416x22
+ $690x23 + $791x24 + $995x31 + $682x32 + $388x33 + $685x34
subject to
Cannery 1:
x11 + x12 + x13 + x14 = 75
Cannery 2:
x21 + x22 + x23 + x24 = 125
Cannery 3:
x31 + x32 + x33 + x34 = 100
Warehouse 1:
x11 + x21 + x31 = 80
Warehouse 2:
x12 + x22 + x32 = 65
Warehouse 3:
x13 + x23 + x33 = 70
Warehouse 4:
x14 + x24 + x34 = 85
and
xij ≥ 0 (i = 1, 2, 3; j = 1, 2, 3, 4)
McGraw-Hill/Irwin
15.12
© The McGraw-Hill Companies, Inc., 2008
Integer Solutions Property
As long as all its supplies and demands have integer values, any transportation
problem with feasible solutions is guaranteed to have an optimal solution with
integer values for all its decision variables. Therefore, it is not necessary to add
constraints to the model that restrict these variables to only have integer values.
McGraw-Hill/Irwin
15.13
© The McGraw-Hill Companies, Inc., 2008
Distribution System at Proctor and Gamble
•
Proctor and Gamble needed to consolidate and re-design their North American
distribution system in the early 1990’s.
–
–
–
–
50 product categories
60 plants
15 distribution centers
1000 customer zones
•
Solved many transportation problems (one for each product category).
•
Goal: find best distribution plan, which plants to keep open, etc.
•
Closed many plants and distribution centers, and optimized their product
sourcing and distribution location.
•
Implemented in 1996. Saved $200 million per year.
For more details, see 1997 Jan-Feb Interfaces article, “Blending OR/MS, Judgement, and GIS: Restructuring P&G’s Supply
Chain”
McGraw-Hill/Irwin
15.14
© The McGraw-Hill Companies, Inc., 2008
Better Products (Assigning Plants to Products)
The Better Products Company has decided to initiate the product of four new
products, using three plants that currently have excess capacity.
Unit Cost
1
2
3
4
Capacity
Available
1
$41
$27
$28
$24
75
2
40
29
—
23
75
3
37
30
27
21
45
Required production
20
30
30
40
Product:
Plant
Question: Which plants should produce which products?
McGraw-Hill/Irwin
15.15
© The McGraw-Hill Companies, Inc., 2008
Transportation Problem Formulation
Unit Cost
Destination
(Product):
1
2
3
4
Supply
1
$41
$27
$28
$24
75
2
40
29
—
23
75
3
37
30
27
21
45
Demand
20
30
30
40
Source(Plant)
McGraw-Hill/Irwin
15.16
© The McGraw-Hill Companies, Inc., 2008
Spreadsheet Formulation
B
3
4
5
6
7
8
9
10
11
12
13
14
15
16
Unit Cost
Plant 1
Plant 2
Plant 3
Daily Production
Plant 1
Plant 2
Plant 3
Products Produced
Required Production
McGraw-Hill/Irwin
C
Product 1
$41
$40
$37
Product 1
0
0
20
20
=
20
D
Product 2
$27
$29
$30
Product 2
30
0
0
30
=
30
E
Product 3
$28
$27
Product 3
30
0
0
30
=
30
15.17
F
Product 4
$24
$23
$21
Product 4
0
15
25
40
=
40
G
H
I
Produced
At Plant
60
15
45
<=
<=
<=
Capacity
75
75
45
Total Cost
$3,260
© The McGraw-Hill Companies, Inc., 2008
Nifty Co. (Choosing Customers)
•
The Nifty Company specializes in the production of a single product, which it
produces in three plants.
•
Four customers would like to make major purchases. There will be enough to
meet their minimum purchase requirements, but not all of their requested
purchases.
•
Due largely to variations in shipping cost, the net profit per unit sold varies
depending on which plant supplies which customer.
Question: How many units should Nifty sell to each customer and how many
units should they ship from each plant to each customer?
McGraw-Hill/Irwin
15.18
© The McGraw-Hill Companies, Inc., 2008
Data for the Nifty Company
Unit Cost
1
2
3
4
Capacity
Available
1
$41
$27
$28
$24
75
2
40
29
—
23
75
3
37
30
27
21
45
Required production
20
30
30
40
Product:
Plant
Question: How many units should Nifty sell to each customer and how many
units should they ship from each plant to each customer?
McGraw-Hill/Irwin
15.19
© The McGraw-Hill Companies, Inc., 2008
Spreadsheet Formulation
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
B
Unit Profit
Plant 1
Plant 2
Plant 3
C
Customer 1
$55
$37
$29
D
Customer 2
$42
$18
$59
E
Customer 3
$46
$32
$51
F
Customer 4
$53
$48
$35
Plant 1
Plant 2
Plant 3
Customer 1
7,000
0
0
Customer 2
0
0
6,000
Customer 3
1,000
0
1,000
Customer 4
0
5,000
0
7,000
<=
7,000
<=
7,000
3,000
<=
6,000
<=
9,000
2,000
<=
2,000
<=
6,000
0
<=
5,000
<=
8,000
Shipment
Min Purchase
Total Shipped
Max Purchase
McGraw-Hill/Irwin
15.20
G
Total
Production
8,000
5,000
7,000
H
I
=
=
=
Production
Quantity
8,000
5,000
7,000
Total Profit
$1,076,000
© The McGraw-Hill Companies, Inc., 2008
Metro Water (Distributing Natural Resources)
Metro Water District is an agency that administers water distribution in a large
goegraphic region. The region is arid, so water must be brought in from outside
the region.
– Sources of imported water: Colombo, Sacron, and Calorie rivers.
– Main customers: Cities of Berdoo, Los Devils, San Go, and Hollyglass.
Cost per Acre Foot
Berdoo
Los Devils
San Go
Hollyglass
Available
Colombo River
$160
$130
$220
$170
5
Sacron River
140
130
190
150
6
Calorie River
190
200
230
—
5
2
5
4
1.5
(million
acre feet)
Needed
Question: How much water should Metro take from each river, and how
much should they send from each river to each city?
McGraw-Hill/Irwin
15.21
© The McGraw-Hill Companies, Inc., 2008
Spreadsheet Formulation
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
B
Unit Cost ($millions)
Colombo River
Sacron River
Calorie River
Water Distribution
(million acre-feet)
Colombo River
Sacron River
Calorie River
Total To City
McGraw-Hill/Irwin
Needed
C
Berdoo
160
140
190
Berdoo
0
2
0
2
=
2
D
Los Devils
130
130
200
Los Devils
5
0
0
5
=
5
E
San Go
220
190
230
San Go
0
2.5
1.5
4
=
4
15.22
F
Hollyglass
170
150
-
Hollyglass
0
1.5
0
1.5
=
1.5
G
H
I
Total
From River
5
6
1.5
<=
<=
<=
Available
5
6
5
Total Cost
($million)
1,975
© The McGraw-Hill Companies, Inc., 2008
Northern Airplane (Production Scheduling)
Northern Airplane Company produces commercial airplanes. The last stage in
production is to produce the jet engines and install them.
– The company must meet the delivery deadline indicated in column 2.
– Production and storage costs vary from month to month.
Unit Cost of
Production ($million)
Maximum Production
Overtime
Regular
Time
Overtime
Unit Cost
of Storage
($thousand)
20
10
1.08
1.10
15
15
30
15
1.11
1.12
15
3
25
25
10
1.10
1.11
15
4
20
5
10
1.13
1.15
Month
Scheduled
Installations
Regular
Time
1
10
2
Question: How many engines should be produced in each of the four months
so that the total of the production and storage costs will be minimized?
McGraw-Hill/Irwin
15.23
© The McGraw-Hill Companies, Inc., 2008
Spreadsheet Formulation
B
C
D
3 Production Cost
Regular
4 ($millions)
Time
5
Month 1
1.08
6
Month 2
1.11
7
Month 3
1.10
8
Month 4
1.13
9
10
11 Unit Cost
12 ($millions)
1
13
1 (RT)
1.08
14
1 (OT)
1.10
15
2 (RT)
16
Month
2 (OT)
17 Produced
3 (RT)
18
3 (OT)
19
4 (RT)
20
4 (OT)
21
22
23
24 Units Produced
1
25
1 (RT)
10
26
1 (OT)
0
27
2 (RT)
0
28
Month
2 (OT)
0
29 Produced
3 (RT)
0
30
3 (OT)
0
31
4 (RT)
0
32
4 (OT)
0
33
Installed
10
34
=
35 Scheduled Installations
10
36
McGraw-Hill/Irwin
E
Overtime
1.10
1.12
1.11
1.15
F
G
H
Storage Cost
($millions per month)
0.015
Month Installed
2
3
1.10
1.11
1.12
1.13
1.11
1.13
1.12
1.14
1.10
1.11
-
4
1.13
1.15
1.14
1.15
1.12
1.13
1.13
1.15
Month Installed
2
3
5
0
0
0
10
0
0
0
0
25
0
0
0
0
0
0
15
25
=
=
15
25
4
5
0
0
0
0
10
5
0
20
=
20
15.24
Produced
20
0
10
0
25
10
5
0
I
J
<=
<=
<=
<=
<=
<=
<=
<=
Maximum
Production
20
10
30
15
25
10
5
10
Total Cost
($millions)
77.4
© The McGraw-Hill Companies, Inc., 2008
Optimal Production at Northern Airplane
Month
Production
Installations
Stored
1 (RT)
20
10
10
2 (RT)
10
15
5
3 (RT)
25
25
5
3 (OT)
10
0
10
4 (RT)
5
20
0
McGraw-Hill/Irwin
15.25
© The McGraw-Hill Companies, Inc., 2008
Middletown School District
•
Middletown School District is opening a third high school and thus needs to
redraw the boundaries for the area of the city that will be assigned to the
respective schools.
•
The city has been divided into 9 tracts with approximately equal populations.
•
Each school has a minimum and maximum number of students that should be
assigned.
•
The school district management has decided that the appropriate objective is to
minimize the average distance that students must travel to school.
Question: How many students from each tract should be assigned to each
school?
McGraw-Hill/Irwin
15.26
© The McGraw-Hill Companies, Inc., 2008
Data for the Middletown School District
Distance (Miles) to School
Tract
1
2
3
Number of High
School Students
1
2.2
1.9
2.5
500
2
1.4
1.3
1.7
400
3
0.5
1.8
1.1
450
4
1.2
0.3
2.0
400
5
0.9
0.7
1.0
500
6
1.1
1.6
0.6
450
7
2.7
0.7
1.5
450
8
1.8
1.2
0.8
400
9
1.5
1.7
0.7
500
Minimum enrollment
1,200
1,100
1,000
Maximum enrollment
1,800
1,700
1,500
McGraw-Hill/Irwin
15.27
© The McGraw-Hill Companies, Inc., 2008
Spreadsheet Formulation
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
B
Distance (Miles)
Tract
Tract
Tract
Tract
Tract
Tract
Tract
Tract
Tract
McGraw-Hill/Irwin
Number of
Students
Tract
Tract
Tract
Tract
Tract
Tract
Tract
Tract
Tract
1
2
3
4
5
6
7
8
9
C
School 1
2.2
1.4
0.5
1.2
0.9
1.1
2.7
1.8
1.5
D
School 2
1.9
1.3
1.8
0.3
0.7
1.6
0.7
1.2
1.7
E
School 3
2.5
1.7
1.1
2
1
0.6
1.5
0.8
0.7
1
2
3
4
5
6
7
8
9
School 1
0
400
450
0
350
0
0
0
0
School 2
500
0
0
400
150
0
450
0
0
School 3
0
0
0
0
0
450
0
400
500
1,200
<=
1,200
<=
1,800
1,500
<=
1,500
<=
1,700
1,350
<=
1,350
<=
1,500
Min Enrollment
Total At School
Max Enrollment
15.28
F
Total
From Tract
500
400
450
400
500
450
450
400
500
G
H
=
=
=
=
=
=
=
=
=
Total
In Tract
500
400
450
400
500
450
450
400
500
Total Distance
(miles)
3,530
© The McGraw-Hill Companies, Inc., 2008
Energetic (Meeting Energy Needs)
•
The Energetic Company needs to make plans for the energy systems for a new
building.
•
The energy needs fall into three categories:
– electricity (20 units)
– heating water (10 units)
– heating space (30 units)
•
The three possible sources of energy are
– electricity
– natural gas
– solar heating unit (limited to 30 units because of roof size)
Question: How should Energetic meet the energy needs for the new building?
McGraw-Hill/Irwin
15.29
© The McGraw-Hill Companies, Inc., 2008
Cost Data for Energetic
Unit Cost
Energy Need:
Electricity
Water Heating
Space Heating
Electricity
$400
$500
$600
Natural gas
—
600
500
Solar heater
—
300
400
Source of Energy
McGraw-Hill/Irwin
15.30
© The McGraw-Hill Companies, Inc., 2008
Spreadsheet Formulation
B
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
C
Unit Cost ($/day)
Source
Electricity
of
Natural Gas
Energy
Solar Heater
Daily Energy Use
Source
Electricity
of
Natural Gas
Energy
Solar Heater
Total Supplied
McGraw-Hill/Irwin
Demand
D
Electricity
400
-
E
Energy Need
Water Heating
500
600
300
Electricity
20
0
0
20
=
20
Energy Need
Water Heating
0
0
10
10
=
10
15.31
F
G
H
I
<=
Max Solar
30
Space Heating
600
500
400
Space Heating
0
10
20
30
=
30
Total
Used
20
10
30
Total Cost
($/day)
24,000
© The McGraw-Hill Companies, Inc., 2008
Location of Texago’s Facilities
Type of Facility
Locations
Oil fields
1. Several in Texas
2. Several in California
3. Several in Alaska
Refineries
1. Near New Orleans, Lousiana
2. Near Charleston, South Carolina
3. Near Seattle, Washington
Distribution Centers
1. Pittsburgh, Pennsylvania
2. Atlanta, Georgia
3. Kansas City, Missouri
4. San Francisco, California
McGraw-Hill/Irwin
15.32
© The McGraw-Hill Companies, Inc., 2008
Potential Sites for Texago’s New Refinery
Potential Site
Main Advantages
Near Los Angeles, California
1. Near California oil fields.
2. Ready access from Alaska oil fields.
3. Fairly near San Francisco distribution center.
Near Galveston, Texas
1. Near Texas oil fields.
2. Ready access from Middle East imports.
3. Near corporate headquarters.
Near St. Louis, Missouri
1. Low operating costs.
2. Centrally located for distribution centers.
3. Ready access to crude oil via the Mississippi River.
McGraw-Hill/Irwin
15.33
© The McGraw-Hill Companies, Inc., 2008
Production Data for Texago
Refinery
Crude Oil
Needed Annually
(Million Barrels)
Oil Fields
Crude Oil
Produced Annually
(Million Barrels)
New Orleans
100
Texas
80
Charleston
60
California
60
Seattle
80
Alaska
100
New site
120
Total
240
Total
360
Needed imports = 360 – 240 = 120
McGraw-Hill/Irwin
15.34
© The McGraw-Hill Companies, Inc., 2008
Cost Data for Shipping to Refineries
Cost per Unit Shipped to Refinery or Potential Refinery
(Millions of Dollars per Million Barrels)
New
Orleans
Seattle
Los
Angeles
Charleston
Galveston
St. Louis
Texas
2
4
5
3
1
1
California
5
5
3
1
3
4
Alaska
5
7
3
4
5
7
Middle East
2
3
5
4
3
4
Source
McGraw-Hill/Irwin
15.35
© The McGraw-Hill Companies, Inc., 2008
Cost Data for Shipping to Distribution Centers
Cost per Unit Shipped to Distribution Center
(Millions of Dollars)
Pittsburgh
Atlanta
Kansas City
San Francisco
6.5
5.5
6
8
Charleston
7
5
4
7
Seattle
7
8
4
3
Los Angeles
8
6
3
2
Galveston
5
4
3
6
St. Louis
4
3
1
5
100
80
80
100
Refinery
New Orleans
Potential Refinery
Number of units needed
McGraw-Hill/Irwin
15.36
© The McGraw-Hill Companies, Inc., 2008
Estimated Operating Costs for Refineries
Site
McGraw-Hill/Irwin
Annual Operating Cost
(Millions of Dollars)
Los Angeles
620
Galveston
570
St. Louis
530
15.37
© The McGraw-Hill Companies, Inc., 2008
Basic Spreadsheet for Shipping to Refineries
B
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
C
Unit Cost ($millions)
Texas
Oil
California
Fields
Alaska
Middle East
Shipment Quantity
(millions of barrels)
Texas
Oil
California
Fields
Alaska
Middle East
Total Received
Demand
McGraw-Hill/Irwin
D
E
New Orleans
2
5
5
2
F
Refineries
Charleston
Seattle
4
5
5
3
7
3
3
5
New Orleans
0
0
0
0
0
=
100
Refineries
Charleston
Seattle
0
0
0
0
0
0
0
0
0
0
=
=
60
80
15.38
G
H
I
J
Total Shipped
0
0
0
0
=
=
=
=
Supply
80
60
100
120
New Site
New Site
0
0
0
0
0
=
120
Total Cost
($millions)
0
© The McGraw-Hill Companies, Inc., 2008
Shipping to Refineries, Including Los Angeles
B
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
C
Unit Cost ($millions)
Texas
Oil
California
Fields
Alaska
Middle East
Shipment Quantity
(millions of barrels)
Texas
Oil
California
Fields
Alaska
Middle East
Total Received
Demand
McGraw-Hill/Irwin
D
E
New Orleans
2
5
5
2
F
Refineries
Charleston
Seattle
4
5
5
3
7
3
3
5
Los Angeles
3
1
4
4
New Orleans
40
0
0
60
100
=
100
Refineries
Charleston
Seattle
0
0
0
0
0
80
60
0
60
80
=
=
60
80
Los Angeles
40
60
20
0
120
=
120
15.39
G
H
I
J
Total Shipped
80
60
100
120
=
=
=
=
Supply
80
60
100
120
Total Cost
($millions)
880
© The McGraw-Hill Companies, Inc., 2008
Shipping to Refineries, Including Galveston
B
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
C
Unit Cost ($millions)
Texas
Oil
California
Fields
Alaska
Middle East
Shipment Quantity
(millions of barrels)
Texas
Oil
California
Fields
Alaska
Middle East
Total Received
Demand
McGraw-Hill/Irwin
D
E
New Orleans
2
5
5
2
F
Refineries
Charleston
Seattle
4
5
5
3
7
3
3
5
Galveston
1
3
5
3
New Orleans
20
0
20
60
100
=
100
Refineries
Charleston
Seattle
0
0
0
0
0
80
60
0
60
80
=
=
60
80
Galveston
60
60
0
0
120
=
120
15.40
G
H
I
J
Total Shipped
80
60
100
120
=
=
=
=
Supply
80
60
100
120
Total Cost
($millions)
920
© The McGraw-Hill Companies, Inc., 2008
Shipping to Refineries, Including St. Louis
B
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
C
Unit Cost ($millions)
Texas
Oil
California
Fields
Alaska
Middle East
Shipment Quantity
(millions of barrels)
Texas
Oil
California
Fields
Alaska
Middle East
Total Received
Demand
McGraw-Hill/Irwin
D
E
New Orleans
2
5
5
2
F
Refineries
Charleston
Seattle
4
5
5
3
7
3
3
5
St. Louis
1
4
7
4
New Orleans
0
0
20
80
100
=
100
Refineries
Charleston
Seattle
0
0
20
0
0
80
40
0
60
80
=
=
60
80
St. Louis
80
40
0
0
120
=
120
15.41
G
H
I
J
Total Shipped
80
60
100
120
=
=
=
=
Supply
80
60
100
120
Total Cost
($millions)
960
© The McGraw-Hill Companies, Inc., 2008
Basic Spreadsheet for Shipping to D.C.’s
B
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
C
Unit Cost ($millions)
New Orleans
Refineries
Charleston
Seattle
New Site
Shipment Quantity
(millions of barrels)
New Orleans
Refineries
Charleston
Seattle
New Site
Total Received
Demand
McGraw-Hill/Irwin
D
Pittsburgh
6.5
7
7
E
F
Distribution Center
Atlanta
Kansas City
5.5
6
5
4
8
4
San Francisco
8
7
3
Pittsburgh
0
0
0
0
0
=
100
Distribution Center
Atlanta
Kansas City
0
0
0
0
0
0
0
0
0
0
=
=
80
80
San Francisco
0
0
0
0
0
=
100
15.42
G
H
I
J
Total Shipped
0
0
0
0
=
=
=
=
Supply
100
60
80
120
Total Cost
($millions)
0
© The McGraw-Hill Companies, Inc., 2008
Shipping to D.C.’s When Choose Los Angeles
B
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
C
Unit Cost ($millions)
New Orleans
Refineries
Charleston
Seattle
Los Angeles
Shipment Quantity
(millions of barrels)
New Orleans
Refineries
Charleston
Seattle
Los Angeles
Total Received
Demand
McGraw-Hill/Irwin
D
Pittsburgh
6.5
7
7
8
E
F
Distribution Center
Atlanta
Kansas City
5.5
6
5
4
8
4
6
3
San Francisco
8
7
3
2
Pittsburgh
80
0
20
0
100
=
100
Distribution Center
Atlanta
Kansas City
20
0
60
0
0
0
0
80
80
80
=
=
80
80
San Francisco
0
0
60
40
100
=
100
15.43
G
H
I
J
Total Shipped
100
60
80
120
=
=
=
=
Supply
100
60
80
120
Total Cost
($millions)
1,570
© The McGraw-Hill Companies, Inc., 2008
Shipping to D.C.’s When Choose Galveston
B
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
C
Unit Cost ($millions)
New Orleans
Refineries
Charleston
Seattle
Galveston
Shipment Quantity
(millions of barrels)
New Orleans
Refineries
Charleston
Seattle
Galveston
Total Received
Demand
McGraw-Hill/Irwin
D
Pittsburgh
6.5
7
7
5
E
F
Distribution Center
Atlanta
Kansas City
5.5
6
5
4
8
4
4
3
San Francisco
8
7
3
6
Pittsburgh
100
0
0
0
100
=
100
Distribution Center
Atlanta
Kansas City
0
0
60
0
0
0
20
80
80
80
=
=
80
80
San Francisco
0
0
80
20
100
=
100
15.44
G
H
I
J
Total Shipped
100
60
80
120
=
=
=
=
Supply
100
60
80
120
Total Cost
($millions)
1,630
© The McGraw-Hill Companies, Inc., 2008
Shipping to D.C.’s When Choose St. Louis
B
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
C
Unit Cost ($millions)
New Orleans
Refineries
Charleston
Seattle
St. Louis
Shipment Quantity
(millions of barrels)
New Orleans
Refineries
Charleston
Seattle
St. Louis
Total Received
Demand
McGraw-Hill/Irwin
D
Pittsburgh
6.5
7
7
4
E
F
Distribution Center
Atlanta
Kansas City
5.5
6
5
4
8
4
3
1
San Francisco
8
7
3
5
Pittsburgh
100
0
0
0
100
=
100
Distribution Center
Atlanta
Kansas City
0
0
60
0
0
0
20
80
80
80
=
=
80
80
San Francisco
0
0
80
20
100
=
100
15.45
G
H
I
J
Total Shipped
100
60
80
120
=
=
=
=
Supply
100
60
80
120
Total Cost
($millions)
1,430
© The McGraw-Hill Companies, Inc., 2008
Annual Variable Costs
Total Cost
of Shipping
Crude Oil
Total Cost
of Shipping
Finished Product
Operating Cost
for New
Refinery
Total
Variable
Cost
Los Angeles
$880 million
$1.57 billion
$620 million
$3.07 billion
Galveston
920 million
1.63 billion
570 million
3.12 billion
St. Louis
960 million
1.43 billion
530 million
2.92 billion
Site
McGraw-Hill/Irwin
15.46
© The McGraw-Hill Companies, Inc., 2008
Sellmore Company Assignment Problem
•
The marketing manager of Sellmore Company will be holding the company’s
annual sales conference soon.
•
He is hiring four temporary employees:
–
–
–
–
•
Ann
Ian
Joan
Sean
Each will handle one of the following four tasks:
–
–
–
–
Word processing of written presentations
Computer graphics for both oral and written presentations
Preparation of conference packets, including copying and organizing materials
Handling of advance and on-site registration for the conference
Question: Which person should be assigned to which task?
McGraw-Hill/Irwin
15.47
© The McGraw-Hill Companies, Inc., 2008
Data for the Sellmore Problem
Required Time per Task (Hours)
Temporary
Employee
Word
Processing
Graphics
Packets
Registrations
Hourly
Wage
Ann
35
41
27
40
$14
Ian
47
45
32
51
12
Joan
39
56
36
43
13
Sean
32
51
25
46
15
McGraw-Hill/Irwin
15.48
© The McGraw-Hill Companies, Inc., 2008
Spreadsheet Formulation
B
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
C
D
E
F
G
H
I
J
Task
Required Time
(Hours)
Assignee
Ann
Ian
Joan
Sean
Word
Processing
35
47
39
32
Ann
Ian
Joan
Sean
Word
Processing
$490
$564
$507
$480
Graphics
41
45
56
51
Packets
27
32
36
25
Registrations
40
51
43
46
Packets
$378
$384
$468
$375
Registrations
$560
$612
$559
$690
Packets
1
0
0
0
1
=
1
Registrations
0
0
1
0
1
=
1
Hourly
Wage
$14
$12
$13
$15
Task
Cost
Assignee
Graphics
$574
$540
$728
$765
Task
Assignment
Ann
Ian
Joan
Sean
Total Assigned
Assignee
Demand
McGraw-Hill/Irwin
Word
Processing
0
0
0
1
1
=
1
Graphics
0
1
0
0
1
=
1
15.49
Total
Assignments
1
1
1
1
=
=
=
=
Supply
1
1
1
1
Total Cost
$1,957
© The McGraw-Hill Companies, Inc., 2008
The Model for Assignment Problems
Given a set of tasks to be performed and a set of assignees who are available to
perform these tasks, the problem is to determine which assignee should be
assigned to each task.
To fit the model for an assignment problem, the following assumptions need to be
satisfied:
1.
2.
3.
4.
5.
The number of assignees and the number of tasks are the same.
Each assignee is to be assigned to exactly one task.
Each task is to be performed by exactly one assignee.
There is a cost associated with each combination of an assignee performing a task.
The objective is to determine how all the assignments should be made to minimize
the total cost.
McGraw-Hill/Irwin
15.50
© The McGraw-Hill Companies, Inc., 2008
The Network Representation
Assigne es
(Ann)
Tasks
490
A1
574
T1
(Word pr oc essing)
T2
(Gr aphics)
T3
(Pa ckets)
T4
(Registrations)
378
560
564
(I an)
540
A2
612
728
507
(Joa n)
384
468
A3
559
765
375
480
(Sea n)
McGraw-Hill/Irwin
A4
690
15.51
© The McGraw-Hill Companies, Inc., 2008
Job Shop (Assigning Machines to Locations)
•
The Job Shop Company has purchased three new machines of different types.
•
There are five available locations where the machine could be installed.
•
Some of these locations are more desirable for particular machines because of
their proximity to work centers that will have a heavy work flow to these
machines.
Question: How should the machines be assigned to locations?
McGraw-Hill/Irwin
15.52
© The McGraw-Hill Companies, Inc., 2008
Materials-Handling Cost Data
Cost per Hour
Location:
1
2
3
4
5
1
$13
$16
$12
$14
$15
2
15
—
13
20
16
3
4
7
10
6
7
Machine
McGraw-Hill/Irwin
15.53
© The McGraw-Hill Companies, Inc., 2008
Spreadsheet Formulation
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
B
Cost ($/hour)
Machine 1
Machine 2
Machine 3
C
Location 1
13
15
4
D
Location 2
16
7
E
Location 3
12
13
10
F
Location 4
14
20
6
G
Location 5
15
16
7
Assignment
Machine 1
Machine 2
Machine 3
Total Assigned
Location 1
0
0
1
1
<=
1
Location 2
0
0
0
0
<=
1
Location 3
0
1
0
1
<=
1
Location 4
1
0
0
1
<=
1
Location 5
0
0
0
0
<=
1
Demand
McGraw-Hill/Irwin
15.54
H
Total
Assignments
1
1
1
I
J
=
=
=
Supply
1
1
1
Total Cost
($/hour)
31
© The McGraw-Hill Companies, Inc., 2008
Better Products (No Product Splitting)
B
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
Unit Cost
Plant 1
Plant 2
Plant 3
Required Production
Cost ($/day)
Plant 1
Plant 2
Plant 3
Assignment
Plant 1
Plant 2
Plant 3
Total Assigned
Demand
McGraw-Hill/Irwin
C
Product 1
$41
$40
$37
D
Product 2
$27
$29
$30
E
Product 3
$28
$27
F
Product 4
$24
$23
$21
20
30
30
40
Product 1
$820
$800
$740
Product 2
$810
$870
$900
Product 3
$840
$810
Product 4
$960
$920
$840
Product 1
0
1
0
1
=
1
Product 2
1
0
0
1
=
1
Product 3
1
0
0
1
=
1
15.55
Product 4
0
0
1
1
=
1
G
H
I
Total
Assignments
2
1
1
<=
<=
=
Supply
2
2
1
Total Cost
$3,290
© The McGraw-Hill Companies, Inc., 2008
Middletown School District (No Tract Splitting)
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
B
Distance
(Miles)
Tract
Tract
Tract
Tract
Tract
Tract
Tract
Tract
Tract
1
2
3
4
5
6
7
8
9
Assignment
Tract 1
Tract 2
Tract 3
Tract 4
Tract 5
Tract 6
Tract 7
Tract 8
Tract 9
Total Assigned
Demand
McGraw-Hill/Irwin
C
D
E
School 1
2.2
1.4
0.5
1.2
0.9
1.1
2.7
1.8
1.5
School 2
1.9
1.3
1.8
0.3
0.7
1.6
0.7
1.2
1.7
School 3
2.5
1.7
1.1
2
1
0.6
1.5
0.8
0.7
School 1
0
1
1
0
1
0
0
0
0
3
=
3
School 2
1
0
0
1
0
0
1
0
0
3
=
3
School 3
0
0
0
0
0
1
0
1
1
3
=
3
F
Number of
Students
500
400
450
400
500
450
450
400
500
Total
Assignments
1
1
1
1
1
1
1
1
1
G
=
=
=
=
=
=
=
=
=
H
Cost
(Miles)
Tract
Tract
Tract
Tract
Tract
Tract
Tract
Tract
Tract
1
2
3
4
5
6
7
8
9
I
J
K
School 1
1100
560
225
480
450
495
1215
720
750
School 2
950
520
810
120
350
720
315
480
850
School 3
1250
680
495
800
500
270
675
320
350
Supply
1
1
1
1
1
1
1
1
1
Total Distance
(Miles)
3560
15.56
© The McGraw-Hill Companies, Inc., 2008
The Transportation Problem
•
A common problem in logistics is how to transport goods from a set of sources
(e.g., plants, warehouses, etc.) to a set of destinations (e.g., warehouses,
customers, etc.) at the minimum possible cost.
•
Given
– a set of sources, each with a given supply,
– a set of destinations, each with a given demand,
– a cost table (cost/unit to ship from each source to each destination)
•
Goal
– Choose shipping quantities from each source to each destination so as to minimize
total shipping cost.
McGraw-Hill/Irwin
15.57
© The McGraw-Hill Companies, Inc., 2008
The Network Representation
Sources
Destinations
Demand1
Supply1
Demand2
Supply2
Demand3
Supply3
Costij
Shipment Quantityij
McGraw-Hill/Irwin
15.58
Demand4
© The McGraw-Hill Companies, Inc., 2008
Transportation Problem Example
A company has two plants (in Seattle and Atlanta) producing a certain product that
is to be shipped to three distribution centers (in Sacramento, St. Louis, and
Pittsburgh).
– The unit production costs are the same at the two plants, and the shipping costs per
unit are shown in the table below.
– Shipments are made once per week.
– During each week, each plant produces at most 60 units and each distribution center
needs at least 40 units.
Unit Shipping Cost
Plant
Distribution Center
Sacramento
St. Louis
Pittsburgh
Seattle
$2
$6
$8
Atlanta
$7
$5
$3
Question: How many units should be shipped from each plant to each
distribution center?
McGraw-Hill/Irwin
15.59
© The McGraw-Hill Companies, Inc., 2008
Spreadsheet Solution
B
3
4
5
6
7
8
9
10
11
12
13
14
15
Cost
Seattle Plant
Atlanta Plant
Shipment
Quantities
Seattle Plant
Atlanta Plant
Shipped
Needed
McGraw-Hill/Irwin
C
D
E
Sacramento
Dist. Center
$2
$7
St. Louis
Dist. Center
$6
$5
Pittsburgh
Dist. Center
$8
$3
Sacramento
Dist. Center
40
0
40
>=
40
St. Louis
Dist. Center
20
20
40
>=
40
Pittsburgh
Dist. Center
0
40
40
>=
40
15.60
F
Shipped
60
60
Cost
G
H
<=
<=
=
Available
60
60
$420
© The McGraw-Hill Companies, Inc., 2008
Shipping from D.C.’s to Customers
The same company ships one of its products from its three distribution centers to
four different customers
–
–
–
–
The shipping costs per unit are shown in the table below.
Shipments are made once per week.
During each week, each distribution center has received 40 units.
Customer demand is also shown in the table below.
Unit Shipping Cost
Customer
1
2
3
4
Sacramento
$8
$10
$7
$11
St. Louis
$12
$11
$9
$6
Pittsburgh
$10
$9
$15
$10
Customer Demand
40
30
25
25
Distribution
Center
Question: How many units should be shipped from each distribution center to
each customer?
McGraw-Hill/Irwin
15.61
© The McGraw-Hill Companies, Inc., 2008
Spreadsheet Solution
3
4
5
6
7
8
9
10
11
12
13
14
15
16
B
C
D
E
F
Cost
Customer 1
$8
$12
$10
Customer 2
$10
$11
$9
Customer 3
$7
$9
$15
Customer 4
$11
$6
$10
Customer 1
30
0
10
40
>=
40
Customer 2
0
0
30
30
>=
30
Customer 3
10
15
0
25
>=
25
Customer 4
0
25
0
25
>=
25
Sacramento DC
St. Louis DC
Pittsburgh DC
Shipment
Quantities
Sacramento DC
St. Louis DC
Pittsburgh DC
Shipped
Needed
McGraw-Hill/Irwin
15.62
G
H
I
Shipped
40
40
40
Cost
<=
<=
<=
=
Available
40
40
40
$965
© The McGraw-Hill Companies, Inc., 2008
Managing the Whole Supply Chain
(Plant to D.C. to Customer)
B
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
Cost
Seattle Plant
Atlanta Plant
Shipment
Quantities
Seattle Plant
Atlanta Plant
Shipped
Distribution
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
Cost
Shipment
Quantities
Sacramento DC
St. Louis DC
Pittsburgh DC
Shipped
McGraw-Hill/Irwin
G
H
Pittsburgh
Dist. Center
0
35
35
Shipped
60
60
Cost
<=
<=
=
Available
60
60
$350
Shipped
60
25
35
Cost
<=
<=
<=
=
D
E
St. Louis
Dist. Center
$6
$5
Pittsburgh
Dist. Center
$8
$3
Sacramento
Dist. Center
60
0
60
St. Louis
Dist. Center
0
25
25
I
from DC's to Customers
Sacramento DC
St. Louis DC
Pittsburgh DC
Needed
F
C
Sacramento
Dist. Center
$2
$7
Customer 1
$8
$12
$10
Customer 2
$10
$11
$9
Customer 3
$7
$9
$15
Customer 4
$11
$6
$10
Customer 1
35
0
5
40
>=
40
Customer 2
0
0
30
30
>=
30
Customer 3
25
0
0
25
>=
25
Customer 4
0
25
0
25
>=
25
Total Cost =
15.63
Available
60
25
35
$925
$1,275
© The McGraw-Hill Companies, Inc., 2008
Site Selection
•
The lease is up on their distribution center in St. Louis. They now must decide
whether to sign a new lease in St. Louis, or move the distribution center to a
new location.
•
One possible new location is Omaha, Nebraska, which is offering a better deal
on the lease.
Question: Should they move their distribution center to Omaha?
McGraw-Hill/Irwin
15.64
© The McGraw-Hill Companies, Inc., 2008
Spreadsheet Solution to Site Selection
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
B
C
D
E
F
G
H
I
<=
<=
=
Available
60
60
$300
Shipped
60
0
60
Cost
<=
<=
<=
=
Available
60
0
60
$1,025
Total Cost =
$1,325
Distribution from Plants to DC's
Cost
Seattle Plant
Atlanta Plant
Shipment
Quantities
Seattle Plant
Atlanta Plant
Shipped
Sacramento
Dist. Center
$2
$7
Omaha
Dist. Center
$5
$6
Pittsburgh
Dist. Center
$8
$3
Sacramento
Dist. Center
60
0
60
Omaha
Dist. Center
0
0
0
Pittsburgh
Dist. Center
0
60
60
Shipped
60
60
Cost
Distribution from DC's to Customers
Cost
Sacramento DC
Omaha DC
Pittsburgh DC
Shipment
Quantities
Sacramento DC
Omaha DC
Pittsburgh DC
Shipped
Needed
McGraw-Hill/Irwin
Customer 1
$8
$13
$10
Customer 2
$10
$10
$9
Customer 3
$7
$8
$15
Customer 4
$11
$8
$10
Customer 1
35
0
5
40
>=
40
Customer 2
0
0
30
30
>=
30
Customer 3
25
0
0
25
>=
25
Customer 4
0
0
25
25
>=
25
15.65
© The McGraw-Hill Companies, Inc., 2008
Distribution System at Proctor and Gamble
•
Proctor and Gamble needed to consolidate and re-design their North American
distribution system in the early 1990’s.
–
–
–
–
50 product categories
60 plants
15 distribution centers
1000 customer zones
•
Solved many transportation problems (one for each product category).
•
Goal: find best distribution plan, which plants to keep open, etc.
•
Closed many plants and distribution centers, and optimized their product
sourcing and distribution location.
•
Implemented in 1996. Saved $200 million per year.
For more details, see 1997 Jan-Feb Interfaces article, “Blending OR/MS, Judgement, and GIS: Restructuring P&G’s Supply
Chain”
McGraw-Hill/Irwin
15.66
© The McGraw-Hill Companies, Inc., 2008
The Assignment Problem
•
The job of assigning people (or machines or whatever) to a set of tasks is
called an assignment problem.
•
Given
– a set of assignees
– a set of tasks
– a cost table (cost associated with each assignee performing each task)
•
Goal
– Match assignees to tasks so as to perform all of the tasks at the minimum possible
cost.
McGraw-Hill/Irwin
15.67
© The McGraw-Hill Companies, Inc., 2008
Network Representation
Assignees
Tasks
Costij
McGraw-Hill/Irwin
15.68
© The McGraw-Hill Companies, Inc., 2008
Assignment Problem Example
The coach of a swim team needs to assign swimmers to a 200-yard medley relay
team (four swimmers, each swims 50 yards of one of the four strokes). Since most
of the best swimmers are very fast in more than one stroke, it is not clear which
swimmer should be assigned to each of the four strokes. The five fastest
swimmers and their best times (in seconds) they have achieved in each of the
strokes (for 50 yards) are shown below.
Backstroke
Breaststroke
Butterfly
Freestyle
Carl
37.7
43.4
33.3
29.2
Chris
32.9
33.1
28.5
26.4
David
33.8
42.2
38.9
29.6
Tony
37.0
34.7
30.4
28.5
Ken
35.4
41.8
33.6
31.1
Question: How should the swimmers be assigned to make the fastest relay team?
McGraw-Hill/Irwin
15.69
© The McGraw-Hill Companies, Inc., 2008
Algebraic Formulation
Let xij = 1 if swimmer i swims stroke j; 0 otherwise
tij = best time of swimmer i in stroke j
Minimize Time = ∑ i ∑ j tij xij
subject to
each stroke swum:
∑ i xij = 1 for each stroke j
each swimmer swims 1: ∑ j xij ≤ 1 for each swimmer i
and
xij ≥ 0 for all i and j.
McGraw-Hill/Irwin
15.70
© The McGraw-Hill Companies, Inc., 2008
Spreadsheet Formulation
3
4
5
6
7
8
9
B
C
D
E
F
Best Times
Backstroke
37.7
32.9
33.8
37.0
35.4
Breastroke
43.4
33.1
42.2
34.7
41.8
Butterfly
33.3
28.5
38.9
30.4
33.6
Freestyle
29.2
26.4
29.6
28.5
31.1
Backstroke
0
0
1
0
0
1
=
1
Breastroke
0
0
0
1
0
1
=
1
Butterfly
0
1
0
0
0
1
=
1
Freestyle
1
0
0
0
0
1
=
1
Carl
Chris
David
Tony
Ken
G
H
I
10
11
12
13
14
15
16
17
18
19
Assignment
Carl
Chris
David
Tony
Ken
McGraw-Hill/Irwin
15.71
1
1
1
1
0
Time
<=
1
<=
1
<=
1
<=
1
<=
1
= 126.2
© The McGraw-Hill Companies, Inc., 2008
Bidding for Classes
•
In the MBA program at a prestigious university in the Pacific Northwest,
students bid for electives in the second year of their program.
•
Each of the 10 students has 100 points to bid (total) and must take two
electives.
•
There are four electives available:
–
–
–
–
•
Quantitative Methods
Finance
Operations Management
Accounting
Each class is limited to 5 students.
Question: How should students be assigned to the classes?
McGraw-Hill/Irwin
15.72
© The McGraw-Hill Companies, Inc., 2008
Points Bid for Electives
Electives
Student
Quantitative
Methods
Finance
Operations
Management
Accounting
George
60
10
10
20
Fred
20
20
40
20
Ann
45
45
5
5
Eric
50
20
5
25
Susan
30
30
30
10
Liz
50
50
0
0
Ed
70
20
10
0
David
25
25
35
15
Tony
35
15
35
15
Jennifer
60
10
10
20
McGraw-Hill/Irwin
15.73
© The McGraw-Hill Companies, Inc., 2008
Spreadsheet Solution
(Maximizing Total Points)
3
4
5
6
7
8
9
10
11
12
13
14
B
C
D
E
F
Points
QMETH
60
20
45
50
30
50
70
25
35
60
Finance
10
20
45
20
30
50
20
25
15
10
Op Mgt.
10
40
5
5
30
0
10
35
35
10
Accounting
20
20
5
25
10
0
0
15
15
20
George
Fred
Ann
Eric
Susan
Liz
Ed
David
Tony
Jennifer
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
Assignment
George
Fred
Ann
Eric
Susan
Liz
Ed
David
Tony
Jennifer
Capacity
McGraw-Hill/Irwin
QMETH
1
0
1
0
0
1
1
0
0
1
5
<=
5
Finance
0
0
1
1
1
1
0
1
0
0
5
<=
5
Op Mgt.
0
1
0
0
1
0
1
1
1
0
5
<=
5
15.74
Accounting
1
1
0
1
0
0
0
0
1
1
5
<=
5
G
Total
Classes
2
2
2
2
2
2
2
2
2
2
H
I
=
=
=
=
=
=
=
=
=
=
Classes
to Take
2
2
2
2
2
2
2
2
2
2
Total Points
J
K
Student
Points
80
60
90
45
60
100
80
60
50
80
=
705
© The McGraw-Hill Companies, Inc., 2008
Spreadsheet Solution
(Maximizing the Minimum Student Point Total)
3
4
5
6
7
8
9
10
11
12
13
14
B
C
D
E
F
Points
QMETH
60
20
45
50
30
50
70
25
35
60
Finance
10
20
45
20
30
50
20
25
15
10
Op Mgt.
10
40
5
5
30
0
10
35
35
10
Accounting
20
20
5
25
10
0
0
15
15
20
QMETH
1
0
0
1
0
0
1
0
1
1
5
<=
5
Finance
0
1
1
0
1
1
0
1
0
0
5
<=
5
Op Mgt.
0
1
0
1
1
0
0
1
1
0
5
<=
5
Accounting
1
0
1
0
0
1
1
0
0
1
5
<=
5
George
Fred
Ann
Eric
Susan
Liz
Ed
David
Tony
Jennifer
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
Assignment
George
Fred
Ann
Eric
Susan
Liz
Ed
David
Tony
Jennifer
Capacity
McGraw-Hill/Irwin
15.75
G
Total
Classes
2
2
2
2
2
2
2
2
2
2
H
I
=
=
=
=
=
=
=
=
=
=
Classes
to Take
2
2
2
2
2
2
2
2
2
2
J
K
Points
80
60
50
55
60
50
70
60
70
80
Total Points
=
635
Min Points
=
50
L
M
>=
>=
>=
>=
>=
>=
>=
>=
>=
>=
Min
Points
50
50
50
50
50
50
50
50
50
50
© The McGraw-Hill Companies, Inc., 2008