Chapter 6 Optimizing Models with Integer Variables
Binary variable: A binary variable is a 0-1 variable that must equal 0 or 1. Usually, a 0-1
variable corresponds to an activity that either is or is not undertaken. If the 0-1 variable is equal
to 1, then the activity is undertaken; if it is equal to 0, the activity is not undertaken.
Examples discussed in this chapter
 Capital budgeting models
 Fixed-cost models
 Conditional production quantity – Minimum production
 Set-covering model – Hub location problem
 Location-Assignment model
 Manufacturing and distribution
Capital Budgeting Models
Example 6.1 Selecting Investments (CapitalBudgeting1.xls)
From the following seven investments select the best set of investments that maximizes NPV
given only $15,000 of cash is available for investment. Each investment must be taken in full,
no partial investment is allowed.
Investment
1
2
3
4
5
6
7
Changing cells
Set target cell
Constraint
Cash required
5000
2500
3500
6000
7000
4500
3000
NPV
16000
8000
10000
19500
22000
12000
7500
Binary variable for each investment (1 = Invest, 0= do not invest)
Maximize sum of NPV of selected investments
Total cash required <= $15,000
Sensitivity analysis: Solver table for budget from $15,000 to $25,000 in increments of $1000
Extensions:
Condition
At most two investments can be selected
If investment 2 is selected, then
investment 1 must also be selected
Either investment 1, or 2 or both must be
selected
Either investment 1 or 2, but not both
must be selected
Add constraint
Sum of 0-1 variables <= 2
0-1 variable for investment 1 >= 0-1 variable for
investment 2
Sum of the 0-1 variables for investment 1 and 2 >=1
Sum of the 0-1 variables for investments 1 and 2 =1
Two-Period Capital Budgeting Model (CapitalBudgeting2.xls)
The following table shows investments when cash is required over two years. The available
budget for year 1 is $14,000 and for year 2 is $4,500.
Investment
1
2
3
4
5
6
7
Cash required
in year 1
5000
2500
3500
6500
7000
4500
3000
Cash required
in year 2
2000
1500
2000
0
500
1500
0
NPV
16000
8000
10000
20000
22000
12000
8000
The only change to the previous model required is an additional budget constraint for year 2.
Fixed Cost Models
Fixed cost is incurred if an activity is chosen. In addition there will be variable cost that varies
with the level of the activity.
Example 6.2 Textile manufacturing (FixedCostMfg.xls)
The company manufactures five products. Each requires a machine to be rented if it is chosen
for production. In addition, there is labor cost, cloth cost, and other variable cost for each unit
produced. The data is given in the table below.
Shirts
Shorts
Pants
Skirts
Jackets
Resource available:
Changing cells
Set target cell
Constraints
Rental
cost
1500
1200
1600
1500
1600
Labor
Hours
2
1
6
4
8
Cloth
(Sq. yd)
3
2.5
4
4.5
5.5
Sales
price
35
40
65
70
110
Unit Variable
cost
20
10
25
30
35
4000 hours of labor and 4500 square yards of cloth
 Binary variable for each product (1 = Produce, 0= do not produce)
 Production quantity for each product
Maximize total profit
 Labor hours used < Labor hours available
 Square yards of cloth used <= Square yards of cloth available
 Unit produced <= Logical upper limit (0 if the product is not
selected, otherwise limited by the smaller of labor hours and
cloth availability.
Extension:
At least three types of clothing is
produced at positive level of at least 100
Sum of 0-1 variables>= 3
A logical minimum production level of 100 for each
type of clothing
Example 6.3 Manufacturing at Dorian Auto (EitherOrManufacturing.xls)
The company produces three types of cars and two types of minivans. Each automobile has a
minimum production level if it is chosen for production.
Vehicle type
Compact car
Midsize car
Large car
Midsize minivan
Large minivan
Steel (tons)/unit
1.5
3
5
6
8
Labor
hours/unit
30
25
40
45
55
Minimum
production (if any)
1000
1000
1000
200
200
Profit
contribution/unit
2000
2500
3000
5500
7000
Resources available: 6500 tons of steel and 65,000 hours of labor.
Changing cells
Set target cell
Constraints
 Binary variable for each vehicle (1 = Produce, 0= do not produce)
 Production quantity for each vehicle
Maximize total profit
 Labor hours used < Labor hours available
 Tons of steel used <= Tons of steel available
 Unit produced <= Logical upper limit (0 if the product is not selected,
otherwise limited by the smaller of labor hours and steel availability.
 Units produced >= Logical lower limit (0 if the vehicle is not chosen,
given minimum value if chosen
Set-Covering and Location-Assignment Models
Determine the minimum number of offices or service centers required and their respective
locations such that all the service areas are covered. Fixed costs, operating costs, and
transportation costs are considered.
Example 6.4 Hub location at Western Airlines (LocatingHubs1.xls)
The airlines wishes to locate minimum number of hubs to serve cities within 1000 miles.
City
AT
BO
CH
DE
HO
LA
NO
NY
PI
SL
SF
SE
Cities within 1000 miles
AT, CH, HO, NO, NY, PI
BO, NY, PI
AT, CH, NY, NO, PI
DE, SL
AT, HO, NO
LA, SL, SF
AT, CH, HO, NO
AT, BO, CH, NY PI
AT, BO, CH, NY PI
DE, LA SL, SF, SE
LA, SL, SF, SE
SL, SF, SE
Decision variable
Set target cell
Constraints
Binary variable for each city (1 =
Locate a hub, 0= do not locate)
Minimize number of hubs
required to cover all cities
Each city must be covered by at
least one hub within 1000 miles
Extension:
What if the distance limit from hub to cities is reduced
to 800 miles? We need the actual distances between
cities to model this. See LocatingHubs2.xls.
Example 6.5 Locating and Assigning Service Centers at United Copiers
(LocatingServiceCenters1.xls)
United Copiers sells copy machines in 11 cities. Service center must be located in 3 of these 11
cities and each city must be assigned to one of the three service centers. Set target cell is to
minimize total distance traveled from service centers to cities annually.
Given data:
Table of distances and number of service trips required at each city
Changing cells
Set target cell
Constraints
 Binary variable for each city (1 = Selected for service center, 0= not
selected)
 Binary variable for assigning each city to one of the potential service
centers i.e. all 11 cities (1 = assigned, 0= not assigned)
Minimize annual distance traveled = distance between a city and the
service center to which the city is assigned times number of trips per year
 Total cities selected for service center <= 3
 Number of service centers to which a given city is assigned = 1
 A city can be assigned to a service center in a city only if a service
center is located in that city
Sensitivity: How will the annual distance traveled vary if the number of service centers is
anywhere between 1 and 11?
Example 6.6 Manufacturing and Distributing Fertilizer at Green Grass
(FixedCostTransportation.xls)
Given data: Customer order bids (quantity and price), and distances between each pair of cities.
Assume plant capacity of 2500 pounds per month. Fixed cost operating a plant is $60,000.
Production cost is $10.25 per pound. Shipping cost is $0.02 per pound per mile.
Changing cells
Set target cell
Constraints
 Binary variable for each city (1 = Selected for production, 0= not
selected)
 Binary variable for assigning each customer order to one of the
potential plants i.e. all 8 cities (1 = assigned, 0= not assigned)
Maximize monthly profit = Sum of bid prices accepted – Production cost
x Quantity produced – Shipping cost x quantity shipped x distance – Fixed
operating cost of number of plants selected for production
 Number of plants to which a given customer order is assigned <= 1
(Note: “<=” allows for an order to be not fulfilled)
 A customer order can be assigned to a plant in a city only if a plant is
selected for production
Sensitivity: How much larger Miami’s bid would need to be before it is accepted?