Linear Programming:
Formulation and Applications
Chapter 3: Hillier and Hillier
Agenda

Discuss Resource Allocation Problems
– Super Grain Corp. Case Study
– Integer Programming Problems
– TBA Airlines Case Study



Discuss Cost-Benefit-Tradeoff-Problems
Discuss Distribution Network and Transportation
Problems
Characteristics of Transportation Problems
– The Big M Company Case Study
Modeling Variants of Transportation
Problems
 Characteristics of Assignment Problems

– Case Study: The Sellmore Company
Modeling Variants of Assignment Problems
 Mixed Problems

Resource Allocation Problems

It is a linear programming problem that
involves the allocation of resources to
activities.
– The identifying feature for this model is that
constraints looks like the following form:
• Amount of resource used  Amount of resource
available
Resource Constraint

A resource constraint is defined as any
functional constraint that has a  sign in a
linear programming model where the
amount used is to the left of the inequality
sign and the amount available is to the right.
The Super Grain Corp. Case
Study

Super Grain is trying to launch a new cereal
campaign using three different medium:
– TV Commercials (TV)
– Magazines (M)
– Sunday Newspapers (SN)

The have an ad budget of $4 million and a
planning budget of $1 million
The Super Grain Corp. Case
Study Cont.
Costs
Cost
TV
Category
Ad Budget $300,000
Magazine
Newspaper
$150,000
$100,000
Planning
Budget
# of
Exposures
$90,000
$30,000
$40,000
1,300,000
600,000
500,000
The Super Grain Corp. Case
Study Cont.
A further constraint to this problem is that
no more than 5 TV spots can be purchased.
 Currently, the measure of performance is
the number of exposures.
 The problem to solve is what is the best
advertising mix given the measure of
performance and the constraints.

Mathematical Model of Super
Grain’s Problem
Max
w.r .t .TV , M , SN
1300TV  150 M  500 SN
subjectto :
300TV  150 M  100 SN  4000
90TV  30 M  40 SN  1000
TV  5
TV  0, M  0, SN  0
Resource-Allocation Problems
Formulation Procedures
Identify the activities/decision variables of
the problem needs to be solved.
 Identify the overall measure of
performance.
 Estimate the contribution per unit of activity
to the overall measure of performance.
 Identify the resources that can be allocated
to the activities.

Resource-Allocation Problems
Formulation Procedures Cont.





Identify the amount available for each resource
and the amount used per unit of each activity.
Enter the data collected into a spreadsheet.
Designate and highlight the changing cells.
Enter model specific information into the
spreadsheet such as  and create a column that
summarizes the amount used of each resource.
Designate a target cell with the overall
performance measure programmed in.
Types of Integer Programming
Problems

Pure Integer Programming (PIP)
– These problems are those where all the decision
variables must be integers.

Mixed Integer Programming (MIP)
– These problems only require some of the
variables to have integer values.
Types of Integer Programming
Problems Cont.

Binary Integer Programming (BIP)
– These problems are those where all the decision
variables restricted to integer values are further
restricted to be binary variables.
– A binary variable are variables whose only possible
values are 0 and 1.
– BIP problems can be separated into either pure BIP
problems or mixed BIP problems.
– These problems will be examined later in the course.
Case Study: TBA Airlines



TBA Airlines is a small regional company that
uses small planes for short flights.
The company is considering expanding its
operations.
TBA has two choices:
– Buy more small planes (SP) and continue with short
flights
– Buy only large planes (LP) and only expand into larger
markets with longer flights
– Expand by purchasing some small and some large
planes
TBA Airlines Cont.

Question: How many large and small planes
should be purchased to maximize total net
annual profit?
Case Study: TBA Airlines
Net Profit Per Plane
Purchase cost
Maximum Quantity
Small
Plane
$1 million
5 mil.
2
Large
Plane
$5 million
50 mil.
N/A
Capital
Available
$100 mil.
Mathematical Model for TBA
Max SP  5 LP
SP , LP
subject to :
5SP  50LP  100
SP  2
SP, LP  0
Graphical Method for Linear
Programming
Number
of large
airplanes
purchased
L
3
2
(2, 1.8) = Optimal solution
Profit = 11 = S + 5 L
1
0
Feasible
region
(2, 1) = Rounded solution
(Profit = 7)
1
2
3
Number of small airplanes purchased
S
Divisibility Assumption of LP

This assumption says that the decision
variables in a LP model are allowed to have
any values that satisfy the functional and
nonnegativity constraints.
– This implies that the decision variables are not
restricted to integer values.

Note: Implicitly in TBA’s problem, it
cannot purchase a fraction of a plane which
implies this assumption is not met.
The Challenges of Rounding
It may be tempting to round a solution from
a non-integer problem, rather than modeling
for the integer value.
 There are three main issues that can arise:

– Rounded Solution may not be feasible.
– Rounded solution may not be close to optimal.
– There can be many rounded solutions
New Mathematical Model for
TBA
Max SP  5LP
SP , LP
subject to :
5SP  50LP  100
SP  2
SP, LP  0
SP, LP integer
The Graphical Solution Method
For Integer Programming




Step 1: Graph the feasible region
Step 2: Determine the slope of the objective
function line
Step 3: Moving the objective function line through
this feasible region in the direction of improving
values of the objective function.
Step 4: Stop at the last instant the the objective
function line passes through an integer point that
lies within this feasible region.
– This integer point is the optimal solution.
Graphical Method for Integer
Programming
L
Number
of large
3
airplanes
purchased
2
1
0
(0, 2) = Optimal solution for the integer programming problem (Profit = 10)
(2, 1.8) = Optimal solution for the
LP relaxation (Profit = 11)
Profit = 10 = S + 5 L
(2, 1) = Rounded solution (Profit = 7)
1
2
3
Number of small airplanes purchased
S
Cost-Benefit-Trade-Off Problems

It is a linear programming problem that
involves choosing a mix of level of various
activities that provide acceptable minimum
levels for various benefits at a minimum
cost.
– The identifying feature for this model is that
constraints looks like the following form:
• Level Achieved  Minimum Acceptable Level
Benefit Constraints

A benefit constraint is defined as any
functional constraint that has a  sign in a
linear programming model where the
benefits achieved from the activities are
represented on the left of the inequality sign
and the minimum amount of benefits is to
the right.
Union Airways Case Study



Union Airways is an airline company trying to
schedule employees to cover it shifts by service
agents.
Union Airways would like find a way of
scheduling five shifts of workers at a minimum
cost.
Due to a union contract, Union Airways is limited
to following the shift schedules dictated by the
contract.
Union Airways Case Study

The shifts Union Airways can use:
–
–
–
–
–

Shift 1: 6 A.M. to 2:00 P.M. (S1)
Shift 2: 8 A.M. to 4:00 P.M. (S2)
Shift 3: 12 P.M. to 8:00 P.M. (S3)
Shift 4: 4 P.M. to 12:00 A.M. (S4)
Shift 5: 10 P.M. to 6:00 A.M. (S5)
A summary of the union limitations are on
the next page.
Union Airways Case Study Cont.
Time Periods Covered by Shifts
Time Period
S1
S2
S3
S4
S5
Minimum
# of
Agents
Needed
6 AM to 8 AM

8 AM to 10 AM


10 AM to 12 PM



65
12 PM to 2 PM



87
Daily Cost Per Agent $170
48
79
$160 $175 $180 $195
Union Airways Case Study Cont.
Time Periods Covered by Shifts
Time Period
S1
2 PM to 4 PM
S2

S3
S4
S5

64

73
6 PM to 8 PM

82
8 PM to 10 PM

43
4 PM to 6 PM
Daily Cost Per Agent $170

Minimum
# of
Agents
Needed
$160 $175 $180 $195
Union Airways Case Study Cont.
Time Periods Covered by Shifts
Time Period
S1
10 PM to 12 AM
12 AM to 6 AM
Daily Cost Per Agent $170
S2
S3
S4

S5
Minimum
# of
Agents
Needed

52

15
$160 $175 $180 $195
Mathematical Model of Union
Airway’s Problem
MIN
w.r .t . S 1, S 2 , S 3, S 4 , S 5
170 * S1  160 * S 2  175 * S 3  180 * S 4  195 * S 5
subjectto :
S1  48,
S1  S 2  79, S1  S 2  65,
S1  S 2  S 3  87, S 2  S 3  64, S 3  S 4  73,
S 3  S 4  82,
S 4  43,
S 4  S 5  52
S 5  15
S1, S 2, S 3, S 4, S 5  0
Cost-Benefit-Trade-Off Problems
Formulation Procedures

The procedures for this type of problem is
equivalent with the resource allocation
problem.
Distribution Network Problems

This is a problem that is concerned with the
optimal distribution of goods through a
distribution network.
– The constraints in this model tend to be fixedrequirement constraints, i.e., constraints that are met
with equality.
– The left hand side of the equality represents the amount
provided of some type of quantity, while the right hand
side represents the required amount of that quantity.
Transportation Problems


Transportation problems are characterized by
problems that are trying to distribute commodities
from a any supply center, known as sources, to
any group of receiving centers, known as
destinations.
Two major assumptions are needed in these types
of problems:
– The Requirements Assumption
– The Cost Assumption
Transportation Assumptions

The Requirement Assumption
– Each source has a fixed supply which must be
distributed to destinations, while each
destination has a fixed demand that must be
received from the sources.

The Cost Assumption
– The cost of distributing commodities from the
source to the destination is directly proportional
to the number of units distributed.
The General Model of a
Transportation Problem

Any problem that attempts to minimize the
total cost of distributing units of
commodities while meeting the requirement
assumption and the cost assumption and has
information pertaining to sources,
destinations, supplies, demands, and unit
costs can be formulated into a transportation
model.
Feasible Solution Property

A transportation problem will have a
feasible solution if and only if the sum of
the supplies is equal to the sum of the
demands.
– Hence the constraints in the transportation
problem must be fixed requirement constraints.
Visualizing the Transportation
Model

When trying to model a transportation
model, it is usually useful to draw a network
diagram of the problem you are examining.
– A network diagram shows all the sources,
destinations, and unit cost for each source to
each destination in a simple visual format like
the example on the next slide.
Network Diagram
Supply
c11
S1 Source 1
c12
c13
c1m
S2 Source 2
c21
c22
c23
c2m
S3 Source 3
.
.
.
Sn Source n
Demand
Destination 1
D1
Destination 2
D2
Destination 3
D3
c31
c32
c33
c3m
cn1
cn2
cn3
cnm
.
.
.
Destination m
Dm
General Mathematical Model of
Transportation Problems
c11x11  c12 x12    c1m x1m 
n
m
Min i 1  j 1 cij xij  c21x21  c22 x22    c2 m x2 m   
x11 , x12 ,..., x1 m
x21 , x22 ,..., x2 m
cn1 xn1  cn 2 xn 2    cnm xnm

xn1 , xn 2 ,..., xnm
General Mathematical Model of
Transportation Problems Cont.
Subject to :
x11  x12    x1m  S1
x 21  x 22    x 2m  S 2

x n1  x n2    x nm  Sn
x11  x 21    x m1  D1
x12  x 22    x m2  D 2

x1m  x 2m    x nm  Dm
x ij  0 (i  1,2,..., n; j  1,2,..., m)
Solving a Transportation
Problem
When Excel solves a transportation
problem, it uses the regular simplex
method.
 Due to the characteristics of the
transportation problem, a faster solution can
be found using the transportation simplex
method.

– Unfortunately, the transportation simplex
model is not programmed in Solver.
Integer Solutions Property

If all the supplies and demands have integer
values, then the transportation problem with
feasible solutions is guaranteed to have an
optimal solution with integer values for all
its decision variables.
– This implies that there is no need to add
restrictions on the model to force integer
solutions.
Big M Company Case Study
Big M Company is a company that has two
lathe factories that it can use to ship lathes
to its three customers.
 The goal for Big M is to minimize the cost
of sending the lathes to its customer while
meeting the demand requirements of the
customers.

Big M Company Case Study
Cont.

Big M has two sets of requirements.
– The first set of requirements dictates how many
lathes can be shipped from factories 1 and 2.
– The second set of requirements dictates how
much each customer needs to get.

A summary of Big M’s data is on the next
slide.
Big M Company Case Study
Cont.
Shipping Cost for Each Lathe
Customer 1 Customer 2 Customer 3
Output
Factory 1
$700
$900
$800
12
Factory 2
$800
$900
$700
15
8
9
Order Size 10
Big M Company Case Study
Cont.

The decision variables for Big M are the
following:
–
–
–
–
–
–
How much factory 1 ships to customer 1 (F1C1)
How much factory 1 ships to customer 2 (F1C2)
How much factory 1 ships to customer 3 (F1C3)
How much factory 2 ships to customer 1 (F2C1)
How much factory 2 ships to customer 2 (F2C2)
How much factory 2 ships to customer 3 (F2C3)
Big M Company Case Study
Cont.
Customer 1
10 Lathes
$700
Factory 1
12 Lathes
$900
$800
Customer 2
8 Lathes
$800
Factory 2
15 Lathes
$900
Customer 3
9 Lathes
$700
Mathematical Model for Big M’s
Problem
MIN
w.r .t . F 1C1, F 1C 2 , F 1C 3,
F 2 C1, F 2 C 2 , F 2 C 3
7 * F1C1  9 * F1C 2  8 * F1C 3  8 * F 2C1  9 * F 2C 2  7 * F 2C 3
subjectto :
F1C1  F1C 2  F1C 3  12
F 2C1  F 2C 2  F 2C 3  15
F1C1  F 2C1  10
F1C 2  F 2C 2  8
F1C 3  F 2C 3  9
F1C1, F1C 2, F1C 3, F 2C1, F 2C 2, F 2C 3  0
Modeling Variants of
Transportation Problems
In many transportation models, you are not
going to always see supply equals demand.
 With small problems, this is not an issue
because the simplex method can solve the
problem relatively efficiently.
 With large transportation problems it may
be helpful to transform the model to fit the
transportation simplex model.

Issues That Arise with
Transportation Models

Some of the issues that may arise are:
– The sum of supply exceeds the sums of demand.
– The sum of the supplies is less than the sum of
demands.
– A destination has both a minimum demand and
maximum demand.
– Certain sources may not be able to distribute
commodities to certain destinations.
– The objective is to maximize profits rather than
minimize costs.
Method for Handling Supply Not
Equal to Demand


When supply does not equal demand, you can use
the idea of a slack variable to handle the excess.
A slack variable is a variable that can be
incorporated into the model to allow inequality
constraints to become equality constraints.
– If supply is greater than demand, then you need a slack
variable known as a dummy destination.
– If demand is greater than supply, then you need a slack
variable known as a dummy source.
Handling Destinations that
Cannot Be Delivered To

There are two ways to handle the issue
when a source cannot supply a particular
destination.
– The first way is to put a constraint that does not
allow the value to be anything but zero.
– The second way of handling this issue is to put
an extremely large number into the cost of
shipping that will force the value to equal zero.
Assignment Problems
Assignment problems are problems that
require tasks to be handed out to assignees
in the cheapest method possible.
 The assignment problem is a special case of
the transportation problem.

Characteristics of Assignment
Problems





The number of assignees and the number of task
are the same.
Each assignee is to be assigned exactly one task.
Each task is to be assigned 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 of the
assignments should be made to minimize the total
cost.
Case Study: Sellmore Company



Sellmore is a marketing company that needs to
prepare for an upcoming conference.
Instead of handling all the preparation work inhouse with current employees, they decide to hire
temporary employees.
The tasks that need to be accomplished are:
–
–
–
–
Word Processing
Computer Graphics
Preparation of Conference Packets
Handling Registration
Case Study: Sellmore Company
Cont.

The assignees for the task are:
–
–
–
–

Ann
Ian
Joan
Sean
A summary of each assignees productivity
and costs are given on the next slide.
Case Study: Sellmore Company
Cont.
Required Time Per Task
Employee Word
Processing
Graphics
Packets Registration 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
Assignment of Variables

xij
– i = 1 for Ann, 2 for Ian, 3 for Joan, 4 for Sean
– j = 1 for Processing, 2 for Graphics, 3 for
Packets, 4 for Registration
Mathematical Model for Sellmore
Company
490 x11  574 x12  378 x13  560 x14 
Minimize 564 x21  540 x22  384 x23  612 x24 
x11 , x12 , x13 , x14
x21 , x22 , x23 , x24 507 x31  728 x32  468 x33  559 x34 
x31 , x32 , x33 , x34
480 x31  765 x32  375 x33  690 x34
Mathematical Model for Sellmore
Company Cont.
Subject to :
x11  x12  x13  x14  1
1  x11, x12 , x13 , x14  0
x21  x22  x23  x24  1 1  x21 , x22 , x23 , x24  0
x31  x32  x33  x34  1 1  x31 , x32 , x33 , x34  0
x41  x42  x43  x44  1 1  x31 , x32 , x33 , x34  0
x11  x21  x31  x41  1
x12  x22  x32  x42  1
x13  x23  x33  x43  1
x14  x24  x34  x44  1
Modeling Variants of the
Assignment Problem

Issues that arise:
– Certain assignees are unable to perform certain tasks.
– There are more task than there are assignees, implying
some tasks will not be completed.
– There are more assignees than there are tasks, implying
some assignees will not be given a task.
– Each assignee can be given multiple tasks
simultaneously.
– Each task can be performed jointly by more than one
assignee.
Mixed Problems
A mixed linear problem is one that has
some combination of resource constraints,
benefit constraints, and fixed requirement
constraints.
 Mixed problems tend to be the type of
linear programming problem seen most.
