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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 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

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 Grains Problem

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

SmallPlane LargePlane CapitalAvailable

Net Profit Per Plane 1 million 5 million

Purchase cost 5 mil. 50 mil. 100 mil.

Maximum Quantity 2 N/A

Mathematical Model for TBA

Graphical Method for Linear Programming

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.

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

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

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 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.

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

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

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

Mathematical Model of Union Airways Problem

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

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.

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

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

General Mathematical Model of Transportation

Problems

General Mathematical Model of Transportation

Problems Cont.

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 Ms data is on the next slide.

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

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 Ms Problem

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

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

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 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

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

Mathematical Model for Sellmore Company Cont.

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.