Using binary integer linear programming to deal with yes no decisions.

YES NO DECISIONS.
© The McGraw-Hill Companies, Inc., 2014
1
Present By Kattareeya Prompreing
ID: DA61G209
Advance Quantitative Methods
Instructor: Dr. Russell C. Chang

LEARNING OBJECTIVE,
Describe how binary decision variables are used to represent yes-or-no decisions.
Use binary decision variables to formulate constraints for mutually exclusive alternatives
and contingent decisions.
Formulate a binary integer programming model for the selection of projects.
Formulate a binary integer programming model for the selection of sites for facilities.
Formulate a binary integer programming model for crew scheduling in the travel industry.
Formulate other basic binary integer programming models from a description of the problems.
Use mixed binary integer programming to deal with setup costs for initiating the
production of a product.
© The McGraw-Hill Companies, Inc., 2014 2

Yes or No Decisions. A yes or no decision arises when a
particular option is being considered and the only Possible
choices are
Yes , go ahead with this option,
or No, decline this option.
Binary variable are variables whose possible value are 0 and
1. Thus, when representing a yes or no decision, a binary
decision variable is assigned a value of 1 for
choosing Yes and
a value of 0 for choosing No.© The McGraw-Hill Companies, Inc., 2014 3
INTRODUCTION

4
Previously, decisions needed to be made about how much to do of various
activities (level).
•Here, decisions to be made are yes-or-no decisions.
▫A yes-or-no decision arises when a particular option is being considered and the
only possible choices are yes, go ahead with this option, or no, decline this
option.
•Yes-or-No decisions are binary variables.
▫Variables whose only possible values are 0 and 1. [1 = yes ; 0 = no ]
Should we undertake a particular fixed project?
Should we make a particular fixed investment?
•Models that fit linear programming and use binary decision variables in the
same time are called binary integer programming (BIP) models.
•A pure BIP model is one where all variables are binary variables.
•A mixed BIP model is one where only some of the variables are binary
variables.
•Excel Solver can solve BIP problems of modest size.
INTRODUCTION

5
A Case Study: California Manufacturing (Section 7.1)
Using BIP for Project Selection: Tazer Corp. (Section 7.2)
Using BIP for the Selection of Sites: Caliente City (Section 7.3)
Using BIP for Crew Scheduling: Southwestern Airways (Section
7.4)
Using Mixed BIP to Deal with Setup Costs: Revised Wyndor
(Section 7.5)

You should to know
 Background of Case study or Background of Company
 Dealing with interrelationships between the Decisions
 Approach constraint of information Correctly
 The BIP Model
 The objective function
 Changing Variable
 Subject to the Constraints
 Solver Option By Excel (Spreadsheet)

7
7.1 The California Manufacturing Company is a diversified company
with several factories and warehouses throughout California, but
none yet in Los Angeles or San Francisco.
•A basic issue is whether to build a new factory in Los Angeles or San
Francisco, or perhaps even both.
•Management is also considering building at most one new warehouse, but
will restrict the choice to a city where a new factory is being built.
•$10 million of capital is available.
•There are looking at the most profitable combination of investments = They want
to maximize the total net present value of these investments.
Question: Should the California Manufacturing Company expand with
factories and/or warehouses in Los Angeles and/or San
Francisco?

Decision
Number
Yes-or-No
Question
Decision
Variable
Net Present
Value
(Millions)
Capital
Required
(Millions)
1 Build a factory in Los Angeles? x1 $8 $6
2 Build a factory in San Francisco? x2 5 3
3 Build a warehouse in Los Angeles? x3 6 5
4 Build a warehouse in San Francisco? x4 4 2
Capital Available: $10 million
Question: Should the California Manufacturing Company expand
with factories and/or warehouses in Los Angeles and/or San Francisco?
8

Decision
Number
Decision
Variable
Possible
Value
Interpretation
of a Value of 1
Interpretation
of a Value of 0
1 x1
0 or 1
Build a factory in
Los Angeles
Do not build
this factory
2 x2
0 or 1
Build a factory in
San Francisco
Do not build
this factory
3 x3
0 or 1
Build a warehouse in
Los Angeles
Do not build
this warehouse
4 x4
0 or 1
Build a warehouse in
San Francisco
Do not build
this warehouse

10
Let x1 = 1 if build a factory in L.A.; 0 otherwise
x2 = 1 if build a factory in S.F.; 0 otherwise
x3 = 1 if build a warehouse in Los Angeles; 0 otherwise
x4 = 1 if build a warehouse in San Francisco; 0 otherwise
Maximize NPV = 8x1 + 5x2 + 6x3 + 4x4 ($millions)
subject to
Capital Spent: 6x1 + 3x2 + 5x3 + 2x4 ≤ 10 ($millions)
Max 1 Warehouse: x3 + x4 ≤ 1
Warehouse only if Factory: x3 ≤ x1 , x4 ≤ x2
and
x1, x2, x3, x4 are binary variables.

3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
B C D E F G
NPV ($millions) LA SF
Warehouse 6 4
Factory 8 5
Capital Required
($millions) LA SF
Warehouse 5 2 Capital Capital
Spent Available
Factory 6 3 9 <= 10
Total Maximum
Build? LA SF Warehouses Warehouses
Warehouse 0 0 0 <= 1
<= <=
Factory 1 1
Total NPV ($millions) 13
a binary decision variable is assigned
a value of 1 for choosing Yes and a value of 0 for
choosing No.
Answer is
build 2 Factory

Incorporating the constraints developed in the preceding subsection, the complete BIP model The is shown in
Fingure 7.1 The format is basically the same as for linear programming models. The one key different
arises when using Solver. Each of the decision variables is constrained to be Binary. In Excel’s Solver,
this is accomplished in the Add Constraint dialog box by choosing each Rang of changing cells as the left-
hand side and then choosing bin from the pop-up menu.

•Management’s initial tentative decision had been to make $10 million of
capital available.
•With this much capital, the best plan would be to build a factory in both
Los Angeles and San Francisco, but no warehouses.
•An advantage of this plan is that it only uses $9 million of this capital,
which frees up $1 million for other projects.
•A heavy penalty (a reduction of $4 million in total net present value)
would be paid if the capital made available were to be reduced below $9
million.
•Increasing the capital made available by $1 million (to $11 million) would
enable a substantial ($4 million) increase in the total net present
value. Management decides to do this.
•With this much capital available, the best plan is to build a factory in
both cities and a warehouse in San Francisco.

7.2 PROJECT SELECTION AT TAZER CORP.
SELECTION OF RESEARCH AND DEVELOPMENT PROJECTS.
•Five potential research and development projects:
▫Project Up: Develop a more effect antidepressant that doesn’t cause mood
swings
▫Project Stable: Develop a drug that addresses manic depression
▫Project Choice: Develop a less intrusive birth control method for women
▫Project Hope: Develop a vaccine to prevent HIV infection
▫Project Release: Develop a more effective drug to lower blood pressure
•$1.2 billion available for investment (enough for 2 or 3 projects)
Question: Which projects should be selected to research and
develop? Which of these projects can maximize
the expected total profit?

1
Up
2
Stable
3
Choice
4
Hope
5
Release
R&D ($million) 400 300 600 500 200
Success Rate 50% 35% 35% 20% 45%
Revenue if Successful
($million)
1,400 1,200 2,200 3,000 600
Expected Profit
($million)
300 120 170 100 70

Let xi = 1 if approve project i; 0 otherwise (for i = 1, 2, 3, 4, and 5)
The expected total profit is
Maximize P = 300x1 + 120x2 + 170x3 + 100x4 + 70x5 ($million)
subject to
R&D Budget: 400x1 + 300x2 + 600x3 + 500x4 + 200x5 ≤ 1,200 ($million)
and xi are binary (for i = 1, 2, 3, 4, and 5).

1
2
3
4
5
6
7
8
9
10
A B C D E F G H I J
Tazer Corp. Project Selection Problem
Up Stable Choice Hope Release Total Budget
R&DInvestment ($million) 400 300 600 500 200 1200 <= 1200
Success Rate 50% 35% 35% 20% 45%
Revenue if Successful ($million) 1400 1200 2200 3000 600
Expected Profit ($million) 300 120 170 100 70 540
Do Project? 1 0 1 0 1
a value of 1 for choosing Yes and
a value of 0 for choosing No.
Research and Develop investment on Up,
Choice, Release and The objective cell
indicates that the resulting total expect profit
Is $540 million.

7.3 SELECTION OF SITES FOR EMERGENCY SERVICES:
THE CALIENTE CITY PROBLEM
Caliente City is growing rapidly and spreading well beyond its original borders
They still have only one fire station, located in the congested center of town
The result has been long delays in fire trucks reaching the outer part of the city

Goal: Develop a plan for locating multiple fire
stations throughout the city
=> Project selection problem
New Policy: Response Time ≤ 10 minutes

© The McGraw-Hill Companies, Inc.,2014
23
Fire Station in Tract
1 2 3 4 5 6 7 8
Response times
(minutes) for a
fire in tract
1 2 8 18 9 23 22 16 28
2 9 3 10 12 16 14 21 25
3 17 8 4 20 21 8 22 17
4 10 13 19 2 18 21 6 12
5 21 12 16 13 5 11 9 12
6 25 15 7 21 15 3 14 8
7 14 22 18 7 13 15 2 9
8 30 24 15 14 17 9 8 3
Cost of Station
($thousands)
350 250 450 300 50 400 300 200

ALGEBRAIC FORMULATION OF CALIENTE CITY PROBLEM
Let xj = 1 if tract j is selected to receive a fire station; 0 otherwise (j = 1, 2, … , 8)
Minimize C = 350x1 + 250x2 + 450x3 + 300x4 + 50x5 + 400x6 + 300x7 + 200x8
subject to
Tract 1: x1 + x2 + x4 ≥ 1
Tract 2: x1 + x2 + x3 ≥ 1
Tract 3: x2 + x3 + x6 ≥ 1
Tract 4: x1 + x4 + x7 ≥ 1
Tract 5: x5 + x7 ≥ 1
Tract 6: x3 + x6 + x8 ≥ 1
Tract 7: x4 + x7 + x8 ≥ 1
Tract 8: x6 + x7 + x8 ≥ 1 and xj are binary (for j = 1, 2, … , 8).

26
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
A B C D E F G H I J K L M N
Caliente City Fire Station Location Problem
Fire Station in Tract
1 2 3 4 5 6 7 8
1 2 8 18 9 23 22 16 28
Response 2 9 3 10 12 16 14 21 25
Times 3 17 8 4 20 21 8 22 17
(Minutes) 4 10 13 19 2 18 21 6 12
for a Fire 5 21 12 16 13 5 11 9 12
in Tract 6 25 15 7 21 15 3 14 8
7 14 22 18 7 13 15 2 9
8 30 24 15 14 17 9 8 3
Cost of Station 350 250 450 300 50 400 300 200
($thousands) Number
Covering
1 1 1 0 1 0 0 0 0 1 >= 1
Response 2 1 1 1 0 0 0 0 0 1 >= 1
Time 3 0 1 1 0 0 1 0 0 1 >= 1
<= 4 1 0 0 1 0 0 1 0 1 >= 1
10 5 0 0 0 0 1 0 1 0 1 >= 1
Minutes? 6 0 0 1 0 0 1 0 1 1 >= 1
7 0 0 0 1 0 0 1 1 2 >= 1
8 0 0 0 0 0 1 1 1 2 >= 1
Total
Fire Station in Tract Cost
1 2 3 4 5 6 7 8 ($thousands)
Station in Tract? 0 1 0 0 0 0 1 1 750
Select tracts
2,7,8
as the site for
fire stations.
The resulting
total
Cost is $750,000.

7.4 SOUTHWESTERN AIRWAYS CREW SCHEDULING
Southwestern Airways needs to assign crews to cover all its upcoming
flights.
We will focus on assigning 3 crews based in San Francisco (SFO) to 11
flights.
Question: How should the 3 crews be assigned 3 sequences of flights so
that every one of the 11 flights is covered?
=> Minimize the total cost of the crew assignments that cover
all the flights.

Seattle
(SEA)
San Francisco
(SFO)
Los Angeles
(LAX)
Denver
(DEN)
Chicago
ORD)

Feasible Sequence of Flights
Flights 1 2 3 4 5 6 7 8 9 10 11 12
1. SFO–LAX 1 1 1 1
2. SFO–DEN 1 1 1 1
3. SFO–SEA 1 1 1 1
4. LAX–ORD 2 2 3 2 3
5. LAX–SFO 2 3 5 5
6. ORD–DEN 3 3 4
7. ORD–SEA 3 3 3 3 4
8. DEN–SFO 2 4 4 5
9. DEN–ORD 2 2 2
10. SEA–SFO 2 4 4 5
11. SEA–LAX 2 2 4 4 2
Cost, $1,000s 2 3 4 6 7 5 7 8 9 9 8 9
© The McGraw-Hill Companies, Inc., 2014 7.29

ALGEBRAIC FORMULATION
Letxj = 1 if flight sequence j is assigned to a crew; 0 otherwise. (j = 1, 2, … , 12).
Minimize Cost = 2x1 + 3x2 + 4x3 + 6x4 + 7x5 + 5x6 + 7x7 + 8x8 + 9x9 + 9x10 + 8x11 + 9x12
(in $thousands)
subject to
Flight 1 covered:x1 + x4 + x7 + x10 ≥ 1
Flight 2 covered:x2 + x5 + x8 + x11 ≥ 1
Flight 11 covered: x6 + x9 + x10 + x11 + x12 ≥ 1
Three Crews: x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 + x10 + x11 + x12 ≤ 3
and xj are binary (j = 1, 2, … , 12).

32
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
B C D E F G H I J K L M N O P Q
Flight Sequence
1 2 3 4 5 6 7 8 9 10 11 12
Cost ($thousands) 2 3 4 6 7 5 7 8 9 9 8 9 At
Least
Includes Segment? Total One
SFO-LAX 1 0 0 1 0 0 1 0 0 1 0 0 1 >= 1
SFO-DEN 0 1 0 0 1 0 0 1 0 0 1 0 1 >= 1
SFO-SEA 0 0 1 0 0 1 0 0 1 0 0 1 1 >= 1
LAX-ORD 0 0 0 1 0 0 1 0 1 1 0 1 1 >= 1
LAX-SFO 1 0 0 0 0 1 0 0 0 1 1 0 1 >= 1
ORD-DEN 0 0 0 1 1 0 0 0 1 0 0 0 1 >= 1
ORD-SEA 0 0 0 0 0 0 1 1 0 1 1 1 1 >= 1
DEN-SFO 0 1 0 1 1 0 0 0 1 0 0 0 1 >= 1
DEN-ORD 0 0 0 0 1 0 0 1 0 0 1 0 1 >= 1
SEA-SFO 0 0 1 0 0 0 1 1 0 0 0 1 1 >= 1
SEA-LAX 0 0 0 0 0 1 0 0 1 1 1 1 1 >= 1
Total Number
1 2 3 4 5 6 7 8 9 10 11 12 Sequences of Crew s
Fly Sequence? 0 0 1 1 0 0 0 0 0 0 1 0 3 <= 3
Total Cost ($thousands) 18
X3=1 assign sequence 3 to a crew
Total cost of $18,000 .

7.5 WYNDOR WITH SETUP COSTS
Suppose that two changes are made to the original
Wyndor problem:
1.For each product, producing any units requires a substantial
one-time setup cost for setting up the production facilities.
2.The production runs for these products will be ended after one
week, so D and W in the original model now represent the total
number of doors and windows produced, respectively,
rather than production rates. Therefore, these two variables need
to be restricted to integer values.

Net Profit ($)
Number of
Units Produced Doors Windows
0 0(300) – 0 = 0 0 (500) – 0 = 0
1 1(300) – 700 = –400 1(500) – 1,300 = –800
2 2(300) – 700 = –100 2(500) – 1,300 = –300
3 3(300) – 700 = 200 3(500) – 1,300 = 200
4 4(300) – 700 = 500 4(500) – 1,300 = 700
5 Not feasible 5(500) – 1,300 = 1,200
6 Not feasible 6(500) – 1,300 = 1,700
© The McGraw-Hill Companies, Inc., 2014 7.35

36

3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
B C D E F G H
Doors Window s
Unit Profit $300 $500
Setup Cost $700 $1,300
Hours Hours
Used Available
Plant 1 1 0 0 <= 4
Plant 2 0 2 12 <= 12
Plant 3 3 2 12 <= 18
Doors Window s
Units Produced 0 6
<= <= Production Profit $3,000
Only If Setup 0 99 - Total Setup Cost $1,300
Setup? 0 1 Total Profit $1,700
Hours Used Per Unit Produced

41
MAKING “YES-OR-NO” TYPE DECISIONS
SET-COVERING PROBLEMS
•FIXED COSTS
CONCLUSION APPLICATIONS OF BINARY VARIABLES

42
CONCLUSION
APPLICATIONS
Investment Analysis
▫Should we make a certain fixed investment?
•Site Selection
▫Should a certain site be selected for the location of a new facility?
•Designing a Production and Distribution Network
▫Should a certain plant remain open?
▫Should a certain site be selected for a new plant? Should a distribution center remain
open?
▫Should a certain site be selected for a new distribution center?
▫Should a certain distribution center be assigned to serve a certain market area?

43
CONCLUSION
APPLICATIONSDispatching Shipments
▫Should a certain route be selected for a truck? Should a certain size
truck be used? Should a certain time period for departure be used?
•Scheduling Interrelated Activities
▫Should a certain activity begin in a certain time period?
•Scheduling Asset Divestitures
▫Should a certain asset be sold in a certain time period?
•Airline Applications:
▫Should a certain type of airplane be assigned to a certain flight leg?
Should a certain sequence of flight legs be assigned to a crew?

PRACTICE.
7.s1 Capital and Budgeting with Contingency Constraints
A company is planning its capital budget over the next several years. There are eight
potential projects under consideration. A calculation has been made of the expected net
Present value of each project , along with the cash outflow that would be required over
That next four years. These data, along with the cash that is available each year, are
Shown in the next table. There also are the following contingency constraints:
(a) At least one of project 1,2 or 3 must be done
(b) Project 6 and 7 cannot both be done, and
(c) Project 5 can only be done if project 6 is done.
Formulate and solve a BIP model in a spreadsheet to determine which projects should be
Pursued to maximize the total expected net present value.

7.S1 Capital Budgeting with Contingency Constraints
This is a resource a location problem.
The activities are the various projects and the limited resources are the
available cash in each year. We will therefore build a spreadsheet following
the template for resource-allocation problems.

Entering the data. The data for this problem are the NPV for each project, the cash outflow for the projects ,
and the Cash available each year. Following the template, the data in the spreadsheet would be enter as displayed
below, Where rang name of NPV(C5:J5) and TotalAvailable (M8:M11) are assigned to the corresponding data
cells.
The decisions to be made in this problem are whether or not to do each project. Thus is a binary variable
Is defined for each project in the changing cells Undertake? (C15:J15). The values in Undertake (C15:J15)
Will eventually are determined by Solver. For Now, arbitrary values are entered.

The goal is to maximize the total expected NPV. Thus, the objective cell should calculate the total NPV.
In this case, the total NPV will be
Total NPV = SUMPRODUCT(NPV,Undertake?)
This formula is entered into TotalNPV(M15)
M
13 Total NPV
14 ($million)
15 =SUMPRODUCT(NPV,Under
take?)

The functional constraints in this problem take into account the limited resources of the case available each year.
Given Undertake? (the changing cell in C15:J15), We calculate the total outflow in(K8:K11). For year 1,this will be =
SUMPRODUCT(C8:J8,Undertake?) Using a range name or and absolute reference for Undertake? , this formula can be copied into
Cells K9:K11 to calculate the total outflow in year2,3, and 4. The TotalOutflow(K8:K11) Must be <= TotalAvailable(M8:M11), as
indicated by the <=in L8:L11 Further more, there are contingency constraints.(a) At Least One Of Projects1, 2, Or 3 Must Be done.
Therefore, we Add The Changing Cells For Projects1,2,and 3 in C18 And constrainC18>=1. (b) Project 6 and 7 cannot both be don.
Therefore,we add the changing cellsfor projects 6 and 7 in C19 and constrain C19<=1. (c)Project5 can only be done if project 6 is done.
Therefore we constrain the changing cell for Project 5 to be<= the changing cell for Project6 in C20:E20

The Solver information and solved spreadsheet are shown below.
Thus They should undertake projects 1,3,4,5,6 and 8 to obtain a total expected NPV of $95 million.

Thus, when representing a yes or no decision, a binary decision
variable is assigned
a value of 1 for choosing Yes and a value of 0 for
choosing No.
Thus They should undertake projects 1,3,4,5,6 and 8 to obtain a total expected NPV of $95 million.
Answer 6 Project.

Thank you very much for
your time
52
If you have any questions about this
document please don’t hesitate to contact
me at:
▪ da61g209@stust.edu.tw
▪ Kattareeya Prompreing

Using binary integer linear programming to deal with yes no decisions.

More Related Content

What's hot (20)

Similar to Using binary integer linear programming to deal with yes no decisions. (20)

Recently uploaded (20)

Using binary integer linear programming to deal with yes no decisions.