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## Linear Programming: Formulation and Applications

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### Linear Programming: ... (2, 1.8) as the optimal solution. Rounding L = 1.8 down then gives ... Introduction to Management Science Author: shurley – PowerPoint PPT presentation

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Title: Linear Programming: Formulation and Applications

1
Linear Programming Formulation and Applications
• Chapter 3 Hillier and Hillier

2
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

3
• Modeling Variants of Transportation Problems
• Characteristics of Assignment Problems
• Case Study The Sellmore Company
• Modeling Variants of Assignment Problems
• Mixed Problems

4
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

5
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.

6
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

7
The Super Grain Corp. Case Study Cont.
Costs Costs Costs
Cost Category TV Magazine Newspaper
Ad Budget 300,000 150,000 100,000
Planning Budget 90,000 30,000 40,000
of Exposures 1,300,000 600,000 500,000
8
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.

9
Mathematical Model of Super Grains Problem
10
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.

11
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.

12
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.

13
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.

14
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

15
TBA Airlines Cont.
• Question How many large and small planes should
be purchased to maximize total net annual profit?

16
Case Study TBA Airlines
SmallPlane LargePlane CapitalAvailable
Net Profit Per Plane 1 million 5 million
Purchase cost 5 mil. 50 mil. 100 mil.
Maximum Quantity 2 N/A
17
Mathematical Model for TBA
18
Graphical Method for Linear Programming
19
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 TBAs problem, it cannot
purchase a fraction of a plane which implies this
assumption is not met.

20
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

21
New Mathematical Model for TBA
22
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.

23
Graphical Method for Integer Programming
24
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

25
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.

26
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.

27
Union Airways Case Study
• The shifts Union Airways can use
• Shift 1 6 A.M. to 200 P.M. (S1)
• Shift 2 8 A.M. to 400 P.M. (S2)
• Shift 3 12 P.M. to 800 P.M. (S3)
• Shift 4 4 P.M. to 1200 A.M. (S4)
• Shift 5 10 P.M. to 600 A.M. (S5)
• A summary of the union limitations are on the
next page.

28
Union Airways Case Study Cont.
Time Periods Covered by Shifts Time Periods Covered by Shifts Time Periods Covered by Shifts Time Periods Covered by Shifts Time Periods Covered by Shifts Time Periods Covered by Shifts Minimum of Agents Needed
Time Period S1 S2 S3 S4 S5 Minimum of Agents Needed
6 AM to 8 AM ? 48
8 AM to 10 AM ? ? 79
10 AM to 12 PM ? ? ? 65
12 PM to 2 PM ? ? ? 87
Daily Cost Per Agent 170 160 175 180 195
29
Union Airways Case Study Cont.
Time Periods Covered by Shifts Time Periods Covered by Shifts Time Periods Covered by Shifts Time Periods Covered by Shifts Time Periods Covered by Shifts Time Periods Covered by Shifts Minimum of Agents Needed
Time Period S1 S2 S3 S4 S5 Minimum of Agents Needed
2 PM to 4 PM ? ? 64
4 PM to 6 PM ? ? 73
6 PM to 8 PM ? 82
8 PM to 10 PM ? 43
Daily Cost Per Agent 170 160 175 180 195
30
Union Airways Case Study Cont.
Time Periods Covered by Shifts Time Periods Covered by Shifts Time Periods Covered by Shifts Time Periods Covered by Shifts Time Periods Covered by Shifts Time Periods Covered by Shifts Minimum of Agents Needed
Time Period S1 S2 S3 S4 S5 Minimum of Agents Needed
10 PM to 12 AM ? ? 52
12 AM to 6 AM ? 15
Daily Cost Per Agent 170 160 175 180 195
31
Mathematical Model of Union Airways Problem
32
Cost-Benefit-Trade-Off Problems Formulation
Procedures
• The procedures for this type of problem is
equivalent with the resource allocation problem.

33
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
fixed-requirement 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.

34
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

35
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.

36
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.

37
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.

38
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.

39
Network Diagram
Demand
Supply
Source 1
Destination 1
c11
D1
S1
c12
c13
c1m
c21
Source 2
Destination 2
D2
S2
c22
c23
c2m
c31
Source 3
Destination 3
D3
S3
c32
c33
c3m
. . .
. . .
cn1
cn2
Source n
Destination m
Dm
Sn
cn3
cnm
40
General Mathematical Model of Transportation
Problems
41
General Mathematical Model of Transportation
Problems Cont.
42
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.

43
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.

44
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.

45
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 Ms data is on the next slide.

46
Big M Company Case Study Cont.
Shipping Cost for Each Lathe Shipping Cost for Each Lathe Shipping Cost for Each Lathe Shipping Cost for Each Lathe Shipping Cost for Each Lathe
Customer 1 Customer 2 Customer 3 Output
Factory 1 700 900 800 12
Factory 2 800 900 700 15
Order Size 10 8 9
47
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)

48
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
49
Mathematical Model for Big Ms Problem
50
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.

51
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.

52
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.

53
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.

54
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.

55
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.

56
Case Study Sellmore Company
• Sellmore is a marketing company that needs to
prepare for an upcoming conference.
• Instead of handling all the preparation work
in-house 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

57
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.

58
Case Study Sellmore Company Cont.
Required Time Per Task Required Time Per Task Required Time Per Task 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
59
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

60
Mathematical Model for Sellmore Company
61
Mathematical Model for Sellmore Company Cont.
62
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.

63
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.
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