Linear Programming: Formulation and Applications

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Linear Programming Formulation and Applications

• Chapter 3 Hillier and Hillier

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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

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

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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

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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.

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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

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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
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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.

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Mathematical Model of Super Grains Problem
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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.

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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.

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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.

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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.

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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

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TBA Airlines Cont.

• Question How many large and small planes should
  be purchased to maximize total net annual profit?

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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
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Mathematical Model for TBA
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Graphical Method for Linear Programming
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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.

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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

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New Mathematical Model for TBA
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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.

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Graphical Method for Integer Programming
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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

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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.

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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.

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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.

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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
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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
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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
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Mathematical Model of Union Airways Problem
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Cost-Benefit-Trade-Off Problems Formulation
Procedures

• The procedures for this type of problem is
  equivalent with the resource allocation problem.

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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.

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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

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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.

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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.

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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.

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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.

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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
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General Mathematical Model of Transportation
Problems
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General Mathematical Model of Transportation
Problems Cont.
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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.

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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.

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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.

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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.

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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
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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)

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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
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Mathematical Model for Big Ms Problem
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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.

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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.

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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.

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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.

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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.

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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.

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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

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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.

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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
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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

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Mathematical Model for Sellmore Company
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Mathematical Model for Sellmore Company Cont.
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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.

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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.