B.5 Binary Indicator Variables
Review Questions
Lesson Topics
Machine Use Indicators are binary variables
where 0 indicates no use and 1 indicates positive use.
They are part of a simple linear model of the fixed cost
of machine use.
Resource Allocation Problems with Fixed
Costs trade off the advantage of using a variety of
inputs to conform to fixed resources with the positive
fixed cost of using each input.
Product Mix Problems with Fixed Costs (1)
trade off the advantage of producing a variety of
goods to conform to fixed resources with the positive
fixed cost of producing each good.
Make or Buy Decisions with Fixed Costs (3)
trade off the lower unit cost of producing a good
yourself with the positive fixed cost of production.
Relational Constraints (2) such as “either
project i or project j is completed” or “both project i
and project j are completed” can be written as linear
constraints in binary indicator variables.
Capital Budgeting (4) Problems maximize the net
present value or net return from a selection of projects
that each require a fixed amount of capital. Relational
constraints are often imposed.
1
B.5 Binary Indicator Variables
Review Questions
Product Mix with Fixed Costs
Question. Muir Manufacturing produces three
popular grades of commercial carpeting among its
many other products. In the coming production
period, Muir needs to decide how many rolls of each
grade should be produced in order to maximize profit.
Each roll of Grade X carpet uses 1 unit of synthetic fiber, requires 3 hours
of production time, and needs 2 units of foam backing.
Each roll of Grade Y carpet uses 2 units of synthetic fiber, requires 1 hour
of production time, and needs 3 units of foam backing.
Each roll of Grade Z carpet uses 3 units of synthetic fiber, requires 2 hours
of production time, and needs 1 unit of foam backing.
The selling price per roll of Grade X carpet is $2; the selling price per roll of
Grade Y carpet is $3; and the selling price per roll of Grade Z carpet is $2.
Muir has already paid for a fixed amount of inputs to be used in the coming
production period. Muir has 6 units of synthetic fiber available for use;
workers have been scheduled to provide up to 12 hours of production time;
and the company has 24 units of foam backing available for use.
Finally, Muir incurs a fixed setup cost of $2 if it produces any positive
quantity of Grade X carpet; a fixed setup cost of $3 if it produces any
positive quantity of Grade Y carpet; and a fixed setup cost of $4 if it
produces any positive quantity of Grade Z carpet.
Formulate a linear program for Muir to maximize its profit. But you need
not compute an optimum.
Tip: Your written answer should define the decision variables, carefully
state which variables are continuous and which are binary, and formulate
the objective and constraints.
2
B.5 Binary Indicator Variables
Review Questions
Answer to Question: Continuous variables are used for the number of
units made:
Let X = the number of rolls of Grade X carpet to make
Let Y = the number of rolls of Grade Y carpet to make
Let Z = the number of rolls of Grade Z carpet to make
Binary variables are used to indicate whether production is positive, and
setup costs are incurred:
Let SX = 1 if Grade X is produced; 0 if not
Let SY = 1 if Grade Y is produced; 0 if not
Let SZ = 1 if Grade Z is produced; 0 if not
Objective: Maximize Profit 2X + 3Y + 2Z – 2SX – 3SY – 4SZ
Input Constraints:
X + 2Y + 3Z < 6 (Synthetic fiber)
3X + Y + 2Z < 12 (Production time)
2X + 3Y + Z < 24 (Foam backing)
Setup Constraints (given input constraints imply X < 6, Y < 12, Z < 24):
X < 6SX
Y < 12SY
Z < 24SZ
3
B.5 Binary Indicator Variables
Review Questions
Make or Buy with Fixed Costs
Question. Danchuk Manufacturing produces a
variety of classic automobiles, including a 1955 Chevy
and a 1955 Thunderbird. Each car consists of three
components that can be manufactured by Danchuk: a
body, an interior, and an engine. Both cars use the same interior and
engine, but different bodies.
Danchuk’s sales forecast indicates that 400 Chevys and 600 Thunderbirds
will be needed to satisfy demand during the next year. Because only 2000
hours of in-house manufacturing time is available, Danchuk is considering
purchasing some, or all, of the components from outside suppliers. If
Denchuk manufactures a component in-house, it incurs a fixed setup cost
as well as a variable manufacturing cost. The following table shows the
setup cost, the manufacturing time per component, the manufacturing cost
per component, and the cost to purchase each of the components from an
outside supplier
Manufacturing
Purchase
Setup Cost Manufacturing Cost per Unit Cost per Unit
(thousands Time per Unit (thousands of (thousands
Component
of dollars)
(hours)
dollars)
of dollars)
Chevy Body
100
3
4
5
Thunderbird
90
2
2
3
Body
Interior
30
1
1
2
Engine
20
2
4
5
Formulate a linear program for Denchuk to minimize total cost to meet the
sales forecasts. Formulate the program, but you need not solve for the
optimum.
Tip: Your written answer should define the decision variables, carefully
state which variables are continuous and which are binary, and formulate
the objective and constraints.
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B.5 Binary Indicator Variables
Review Questions
Answer to Question:
Continuous (or integer) variables are used for the number of units made:
Let CB = the number of Chevy Bodies to make
Let TB = the number of Thunderbird Bodies to make
Let IN = the number of Interiors to make
Let E = the number of Engines to make
Continuous (or integer) variables are also used for the number of units
purchased:
Let CBP = the number of Chevy Bodies to purchase
Let TBP = the number of Thunderbird Bodies to purchase
Let INP = the number of Interiors to purchase
Let EP = the number of Engines to purchase
Binary variables are used to indicate whether production is positive, and
setup costs are incurred:
Let CBS = 1 if Chevy Bodies are produced; 0, if not.
Let TBS = if Thunderbird Bodies are produced; 0, if not.
Let INS = if Interiors are produced; 0, if not.
Let ES = if Engines are produced; 0, if not.
Objective: Minimize Total Cost
4CB + 2TB + 1IN + 4E (production costs)
+5CBP + 3TBP + 2INP + 5EP (purchase costs)
+100CBS + 90TBS + 30INS + 20ES (setup costs)
Input Constraints:
3CB + 2TB + 1IN + 2E < 2000 (In-house hours)
Sales Constraints:
CB + CBP = 400 (manufactured or purchased bodies used for Chevys)
TB + TBP = 600 (manufactured or purchased bodies used for
Thunderbirds)
IN + INP = 1000 (manufactured or purchased interiors used for both
Chevys and Thunderbirds)
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B.5 Binary Indicator Variables
Review Questions
E + EP = 1000 (manufactured or purchased engines used for both Chevys
and Thunderbirds)
Setup Constraints (given sales constraints imply CB < 400, TB < 600, IN <
1000, and E < 1000):
CB < 400CBS
TB < 600TBS
IN < 1000INS
E < 1000ES
6
B.5 Binary Indicator Variables
Review Questions
Make or Buy with Fixed Costs
Question. Danchuk Manufacturing produces a
variety of classic automobiles, including a 1955 Chevy
and a 1955 Thunderbird. Each car consists of three
components that can be manufactured by Danchuk: a
body, an interior, and an engine. Both cars use the same engine, but
different bodies and different interiors.
Danchuk’s sales forecast indicates that 300 Chevys and 500 Thunderbirds
will be needed to satisfy demand during the next year. Because only 2000
hours of in-house manufacturing time is available, Danchuk is considering
purchasing some, or all, of the components from outside suppliers. If
Denchuk manufactures a component in-house, it incurs a fixed setup cost
as well as a variable manufacturing cost. The following table shows the
setup cost, the manufacturing time per component, the manufacturing cost
per component, and the cost to purchase each of the components from an
outside supplier
Manufacturing
Purchase
Setup Cost Manufacturing Cost per Unit Cost per Unit
(thousands Time per Unit (thousands of (thousands
Component
of dollars)
(hours)
dollars)
of dollars)
Chevy Body
100
3
4
5
Thunderbird
90
2
2
3
Body
Chevy
10
1
2
3
Interior
Thunderbird
20
2
3
4
Interior
Engine
20
2
4
5
Formulate a linear program for Denchuk to minimize total cost to meet the
sales forecasts. But you need not compute an optimum.
Tip: Your written answer should define the decision variables, carefully
state which variables are continuous and which are binary, and formulate
the objective and constraints.
7
B.5 Binary Indicator Variables
Review Questions
Answer to Question:
Continuous (or integer) variables are used for the number of units made:
Let CB = the number of Chevy Bodies to make
Let TB = the number of Thunderbird Bodies to make
Let CI = the number of Chevy Interiors to make
Let TI = the number of Thunderbird Interiors to make
Let E = the number of Engines to make
Continuous (or integer) variables are also used for the number of units
purchased:
Let CBP = the number of Chevy Bodies to purchase
Let TBP = the number of Thunderbird Bodies to purchase
Let CIP = the number of Chevy Interiors to purchase
Let TIP = the number of Thunderbird Interiors to purchase
Let EP = the number of Engines to purchase
Binary variables are used to indicate whether production is positive, and
setup costs are incurred:
Let CBS = 1 if Chevy Bodies are produced; 0, if not.
Let TBS = if Thunderbird Bodies are produced; 0, if not.
Let CIS = 1 if Chevy Interiors are produced; 0, if not.
Let TIS = if Thunderbird Interiors are produced; 0, if not.
Let ES = if Engines are produced; 0, if not.
Objective: Minimize Total Cost
4CB + 2TB + 2CI + 3TI + 4E (production costs)
+5CBP + 3TBP + 3CIP + 4TIP + 5EP (purchase costs)
+100CBS + 90TBS + 10CIS + 20TIS + 20ES (setup costs)
Input Constraints:
3CB + 2TB + 1CI + 2TI + 2E < 2000 (In-house hours)
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B.5 Binary Indicator Variables
Review Questions
Sales Constraints:
CB + CBP = 300 (manufactured or purchased bodies used for Chevys)
TB + TBP = 500 (manufactured or purchased bodies used for
Thunderbirds)
CI + CIP = 300 (manufactured or purchased interiors used for Chevys)
TI + TIP = 500 (manufactured or purchased interiors used for
Thunderbirds)
E + EP = 800 (manufactured or purchased engines used for both Chevys
and Thunderbirds)
Setup Constraints (given sales constraints imply CB < 300, TB < 500, CI <
300, TI < 500, and E < 800):
CB < 300CBS
TB < 500TBS
CI < 300CIS
TI < 500TIS
E < 800ES
9
B.5 Binary Indicator Variables
Review Questions
Make or Buy with Fixed Costs
Question. Dell Inc. produces a variety of personal
computers, including the OptiPlex brand and the
Precision brand. Each brand consists of four
components that can be manufactured by Dell: a
motherboard, a monitor, a hard drive, and a case. Both brands use the
same monitor and hard drive and case, but different motherboards.
Dell’s sales forecast indicates that 500 OptiPlexs and 300 Precisions will be
needed to satisfy demand during the next month. Because only 2000
hours of in-house manufacturing time is available each month, Dell is
considering purchasing some, or all, of the components from outside
suppliers. If Dell manufactures a component in-house during a month, it
incurs a fixed setup cost as well as a variable manufacturing cost. The
following table shows the setup cost, the manufacturing time per
component, the manufacturing cost per component, and the cost to
purchase each of the components from an outside supplier
Manufacturing
Purchase
Setup Cost Manufacturing Cost per Unit Cost per Unit
(thousands Time per Unit (thousands of (thousands
Component
of dollars)
(hours)
dollars)
of dollars)
OptiPlex
40
3
4
5
motherboard
Precision
50
2
2
3
motherboard
Monitor
20
1
1
2
Hard Drive
10
4
5
1
Case
30
5
3
4
Formulate a linear program for Dell to minimize total cost to meet the sales
forecasts. But you need not compute an optimum.
Tip: Your written answer should define the decision variables, carefully
state which variables are continuous and which are binary, and formulate
the objective and constraints.
10
B.5 Binary Indicator Variables
Review Questions
Answer to Question:
Continuous (or integer) variables are used for the number of units made:
Let OM = the number of OptiPlex motherboards to make each month
Let PM = the number of Precision motherboards to make each month
Let M = the number of Monitors to make each month
Let H = the number of Hard Drives to make each month
Let C = the number of Cases to make each month
Continuous (or integer) variables are also used for the number of units
purchased:
Let OMP = the number of OptiPlex motherboards to purchase each month
Let PMP = the number of Precision motherboards to purchase each month
Let MP = the number of Monitors to purchase each month
Let HP = the number of Hard Drives to purchase each month
Let CP = the number of Cases to purchase each month
Binary variables are used to indicate whether production is positive, and
setup costs are incurred:
Let OMS = 1 if OptiPlex motherboards are produced; 0, if not
Let PMS = 1 if Precision motherboards are produced; 0, if not
Let MS = 1 if Monitors are produced; 0, if not
Let HS = 1 if Hard Drives are produced; 0, if not
Let CS = 1 if Cases are produced; 0, if not
Objective: Minimize Total Cost
4OM + 2PM+ 1M + 5H + 3C (production costs)
+5OMP + 3PMP+ 2MP + 1HP + 4CP (purchase costs)
+40OMS + 50PMS+ 20MS + 10HS + 30CS (setup costs)
Input Constraints:
3OM + 2PM+ 1M + 4H + 5C < 2000 (In-house hours)
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B.5 Binary Indicator Variables
Review Questions
Sales Constraints:
OM + OMP = 500 (manufactured or purchased OptiPlex motherboards)
PM + PMP = 300 (manufactured or purchased Precision motherboards)
M +MP = 800 (manufactured or purchased Monitors used for both
OptiPlexs and Precisions)
H +HP = 800 (manufactured or purchased Hard Drives used for both
OptiPlexs and Precisions)
C +CP = 800 (manufactured or purchased Cases used for both OptiPlexs
and Precisions)
Setup Constraints (given sales constraints imply OM < 500, PM < 300, M <
800, H < 800, and C < 800):
OM < 500OMS
PM < 300PMS
M < 800MS
H < 800HS
C < 800CS
12
B.5 Binary Indicator Variables
Review Questions
Relational Constraints
Question. Tower Engineering Corporation is
considering undertaking several proposed projects for
the next fiscal year. The following table summarizes
the number of engineers and the number of support
personnel required for each project, and the expected profits for each
project:
Project
1
2
3
4
5
6
Engineers Required
20
55
47
38
90
63
Support Personnel Required
15
45
50
40
70
70
Profit ($1,000,000s)
1.0
1.8
2.0
1.5
3.6
2.2
Formulate an integer program that maximizes Tower's profit subject to the
following management constraints:
1) Use no more than 175 engineers
2) Use no more than 150 support personnel
3) If either project 4 or project 6 is done, both must be done
4) Project 2 can be done only if project 1 is done
5) If project 5 is done, project 3 must not be done. And if project 3
is done, project 5 must not be done.
6) No more than three projects are to be done.
Formulate the program, but you need not compute an optimum.
Tip: Your written answer should define the decision variables, and
formulate the objective and constraints.
13
B.5 Binary Indicator Variables
Review Questions
Answer to Question:
Define binary variables, where Pi = 1 if project i is done, and = 0 if not.
Max P1 + 1.8P2 + 2P3 + 1.5P4 + 3.6P5 + 2.2P6
s.t.
20P1 + 55P2 + 47P3 + 38P4 + 90P5 + 63P6 < 175
15P1 + 45P2 + 50P3 + 40P4 + 70P5 + 70P6 < 150
P4 - P6 = 0
P1 - P2 > 0
P3 + P5 < 1
P1 + P2 + P3 + P4 + P5 + P6 < 3
Pi = 0 or 1
(Note: “P3 + P5 < 1” says the same thing as “If project 5 is done, project 3
must not be done”. And “if project 3 is done, project 5 must not be done”.)
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B.5 Binary Indicator Variables
Review Questions
Relational Constraints
Question. Grush Consulting has five projects to
consider. Each will require time in the next two
quarters according to the table below.
Project
Time in first quarter
Time in second quarter
Revenue
A
5
8
12000
B
3
12
10000
C
7
5
15000
D
2
3
5000
E
15
1
20000
Revenue from each project is also shown. Formulate and solve a model
whose solution would maximize revenue; meet the time budget of 25 in the
first quarter and 20 in the second quarter; not do both projects C and D;
and do project D only if project E is done.
Tip: Your written answer should define the decision variables, formulate the
objective and constraints, and solve for the optimum. --- You will not earn
full credit if you just solve for the optimum; you must also define the
decision variables, and formulate the objective and constraints.
15
B.5 Binary Indicator Variables
Review Questions
Answer to Question:
Let A = 1 if project A is done, 0 otherwise; same for B, C, D, and E
Max 12000A + 10000B + 15000C + 5000D + 20000E
s.t. 5A + 3B + 7C + 2D + 15E ≤ 25
8A + 12B + 5C + 3D + 1E ≤ 20
C+D≤1
D-E≤0
16
B.5 Binary Indicator Variables
Review Questions
Capital Budgeting
Question. Frys Electronics is planning to expand its
sales operation by offering new electronic appliances.
The company has identified seven new product lines it
can carry.
Product Line
1. ProjectionTVs
2. Cell Phones
3. Plasma TVs
4. IPODs
5. DVD Players
6. PDAs
7. Computers
Initial
Floor Space
Invest.
(Sq.Ft.)
$ 6,000
125
$ 12,000
150
$ 20,000
200
$14,000
40
$15,000
40
$ 2,000
20
$32,000
100
Exp. Rate
of Return
18.1%
9.5
11.0
10.2
10.5
14.1
13.2
Frys will not stock Projection TVs unless they stock both Plasma TVs and
IPODs. They will not stock both Cell Phones and PDAs. And they will
stock DVD Players only if they stock Computers. Finally, the company
wishes to introduce at most four new product lines.
If the company has $45,000 to invest and 420 sq. ft. of floor space
available, formulate an integer linear program for Metropolitan to maximize
its overall expected return. But you need not compute an optimum.
Tip: Your written answer should define the decision variables, and
formulate the objective and constraints.
17
B.5 Binary Indicator Variables
Review Questions
Answer to Question:
Define the Decision Variables:
x1 = 1 if product line 1 is introduced, and = 0 otherwise. And so on.
Product line 1. Projection TVs
Product line 2. Cell Phones
Product line 3. Plasma TVs
Product line 4. IPODs
Product line 5. DVD Players
Product line 6. PDAs
Product line 7. Computers
Define the Objective: Maximize total expected return.
Max .181(6000)x1 + .095(12000)x2 + .11(20000)x3 + .102(14000)x4
+ .105(15000)x5 + .141(2000)x6
+ .132(32000)x7
1. Constrain total investment by $45,000:
6000x1 + 12000x2
+ 20000x3 + 14000x4 + 15000x5 + 2000x6 + 32000x7 < 45,000
2. Constrain space by 420 square feet: 125x1 +150x2 +200x3 +40x4 +
40x5 + 20x6 + 100x7 < 420
3. Frys will not stock Projection TVs unless they stock both Plasma TVs
and IPODs: x3 > x1 and x4 > x1. Another linear way to write that
constraint is x3 + x4 > 2x1, and a nonlinear way is x3x4 > x1.
4. Frys will not stock both Cell Phones and PDAs:
x2 + x6 < 1
5. Frys will stock DVD Players only if they stock Computers: x5 < x7
6. Frys will introduce at most four new product lines:
x1 + x2 + x3 + x4 + x5 + x6 + x7 < 4
18
B.5 Binary Indicator Variables
Review Questions
Capital Budgeting
Question. Frys Electronics is planning to expand its
sales operation by offering new electronic appliances.
The company has identified seven new product lines it
can carry.
Product Line
Investment
Floor Space
($)
(Square Feet)
Projection TVs
6,000
125
Cell Phones
12,000
150
Plasma TVs
20,000
200
IPODs
14,000
40
DVD Players
15,000
40
PDAs
2,000
20
Computers
32,000
100
Expected Return
(Percent)
16
9
11
10
8
14
13
Frys will not stock Projection TVs unless they stock both Plasma TVs and
IPODs. They will not stock both Cell Phones and PDAs. And they will
stock DVD Players only if they stock Computers. Finally, the company
wishes to introduce at most four new product lines.
If the company has $45,000 to invest and 420 square feet of floor space
available, formulate an integer linear program for Frys to maximize its
overall expected return. But you need not compute an optimum.
Tip: Your written answer should define the decision variables, and
formulate the objective and constraints.
19
B.5 Binary Indicator Variables
Review Questions
Answer to Question: Define the Decision Variables:
x1 = 1 if product line 1 is introduced, and = 0 otherwise. And so on.
Product line 1. Projection TVs
Product line 2. Cell Phones
Product line 3. Plasma TVs
Product line 4. IPODs
Product line 5. DVD Players
Product line 6. PDAs
Product line 7. Computers
Define the Objective: Maximize total expected return.
Max .16(6000)x1 + .09(12000)x2 + .11(20000)x3 + .10(14000)x4
.08(15000)x5 + .141(2000)x6
+ .132(32000)x7
+
1. Constrain total investment by $45,000:
6000x1 + 12000x2
+ 20000x3 + 14000x4 + 15000x5 + 2000x6 + 32000x7 < 45,000
2. Constrain space by 420 square feet: 125x1 +150x2 +200x3 +40x4 +
40x5 + 20x6 + 100x7 < 420
3. Frys will not stock Projection TVs unless they stock both Plasma TVs
and IPODs: x3 > x1 and x4 > x1. Another linear way to write that
constraint is x3 + x4 > 2x1, and a nonlinear way is x3x4 > x1.
4. Frys will not stock both Cell Phones and PDAs:
x2 + x6 < 1
5. Frys will stock DVD Players only if they stock Computers: x5 < x7
6. Frys will introduce at most four new product lines:
x1 + x2 + x3 + x4 + x5 + x6 + x7 < 4
20
B.5 Binary Indicator Variables
Review Questions
Capital Budgeting
Question. Frys Electronics is planning to expand its
sales operation by offering new electronic appliances.
The company has identified seven new product lines it
can carry.
Product Line
1. ProjectionTVs
2. Cell Phones
3. Plasma TVs
4. IPODs
5. DVD Players
6. PDAs
7. Computers
Initial
Floor Space
Invest.
(Sq.Ft.)
$ 6,000
125
$ 12,000
150
$ 20,000
200
$14,000
40
$15,000
40
$ 2,000
20
$32,000
100
Exp. Rate
of Return
8.1%
9.5
11.0
10.2
10.5
14.1
13.2
If Frys stocks Projection TVs, then they stock both Plasma TVs and IPODs.
They will not stock both Cell Phones and PDAs. And they will stock DVD
Players if, and only if, they stock Computers. Finally, the company wishes
to introduce at most four new product lines.
If the company has $45,000 to invest and 420 sq. ft. of floor space
available, formulate an integer linear program for Metropolitan to maximize
its overall expected return. But you need not compute an optimum.
Tip: Your written answer should define the decision variables, and
formulate the objective and constraints.
21
B.5 Binary Indicator Variables
Review Questions
Answer to Question: Define the Decision Variables:
x1 = 1 if product line 1 is introduced, and = 0 otherwise. And so on.
Product line 1.
Product line 2.
Product line 3.
Product line 4.
Product line 5.
Product line 6.
Product line 7.
Projection TVs
Cell Phones
Plasma TVs
IPODs
DVD Players
PDAs
Computers
Define the Objective: Maximize total expected return.
Max .081(6000)x1 + .095(12000)x2 + .11(20000)x3 + .102(14000)x4
+ .105(15000)x5 + .141(2000)x6
+ .132(32000)x7
1. Constrain total investment by $45,000:
6000x1 + 12000x2
+ 20000x3 + 14000x4 + 15000x5 + 2000x6 + 32000x7 < 45,000
2. Constrain space by 420 square feet: 125x1 +150x2 +200x3 +40x4 +
40x5 + 20x6 + 100x7 < 420
3. If Frys stocks Projection TVs, then they stock both Plasma TVs and
IPODs: x3 > x1 and x4 > x1. Another linear way to write that
constraint is x3 + x4 > 2x1, and a nonlinear way is x3x4 > x1.
4. Frys will not stock both Cell Phones and PDAs:
x2 + x6 < 1
5. Frys will stock DVD Players if, and only if, they stock Computers: x5 =
x7
6. Frys will introduce at most four new product lines:
x1 + x2 + x3 + x4 + x5 + x6 + x7 < 4
22
B.5 Binary Indicator Variables
Review Questions
Capital Budgeting
Question. Newegg.com offers the best prices on
computer parts, laptop computers, digital cameras,
electronics and more with fast shipping. The company
has identified seven new product lines it can carry.
Each requires a specific investment and floor space at the warehouse.
Investment
Floor Space
Expected Return
Product Line
($)
(Square Feet)
(Percent)
Internal Hard
8,000
120
14
Drive
LCD Monitor
7,000
100
9
Laptop Computer
20,000
200
11
Portable DVD
14,000
40
10
Player
Universal
15,000
40
8
Remote Control
Flash Card
2,000
20
14
Computer Server
32,000
100
13
Newegg.com will not stock Portable DVD Players unless they stock both
LCD Monitors and Flash Cards. They will not stock both Internal Hard
Drives and Computer Servers. And they will stock LCD Monitors only if
they stock Universal Remote Controls. Finally, the company will introduce
at least three new product lines.
If the company has $45,000 to invest and 420 square feet of floor space
available, formulate an integer linear program for Newegg.com to maximize
its overall expected return. But you need not compute an optimum.
Tip: Your written answer should define the decision variables, and
formulate the objective and constraints.
23
B.5 Binary Indicator Variables
Review Questions
Answer to Question: Define the Decision Variables:
x1 = 1 if product line 1 is introduced, and = 0 otherwise. And so on.
Product line 1. Internal Hard Drive
Product line 2. LCD Monitor
Product line 3. Laptop Computer
Product line 4. Portable DVD Player
Product line 5. Universal Remote Control
Product line 6. Flash Card
Product line 7. Computer Server
Define the Objective: Maximize total expected return.
Max .14(8000)x1 + .09(7000)x2 + .11(20000)x3 + .10(14000)x4
.08(15000)x5 + .141(2000)x6
+ .132(32000)x7
+
1. Constrain total investment by $45,000:
8000x1 + 7000x2 +
20000x3 + 14000x4 + 15000x5 + 2000x6 + 32000x7 < 45,000
2. Constrain space by 420 square feet: 120x1 +100x2 +200x3 +40x4 +
40x5 + 20x6 + 100x7 < 420
3. … will not stock Portable DVD Players unless they stock both LCD
Monitors and Flash Cards: x2 > x4 and x6 > x4. Another linear way
to write that constraint is x2 + x6 > 2x4. (And if we allowed nonlinear
constraints x2x6 > x4.)
4. … will not stock both Internal Hard Drives and Computer Servers:
x1 + x7 < 1
5. will stock LCD Monitors only if they stock Universal Remote Controls:
x2 < x5
6. … will introduce at least three new product lines:
x1 + x2 + x3 + x4 + x5 + x6 + x7 > 3
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