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 Category TVMagazineNewspaper Ad Budget$300,000$150,000$100,000 Planning Budget $90,000$30,000$40,000 # of Exposures 1,300,000600,000500,000

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

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 Small Plane Large Plane Capital Available Net Profit Per Plane$1 million$5 million Purchase cost5 mil.50 mil.$100 mil. Maximum Quantity2N/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 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

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 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 ShiftsMinimum # of Agents Needed Time PeriodS1S2S3S4S5 6 AM to 8 AM  48 8 AM to 10 AM  AM to 12 PM  PM to 2 PM  87 Daily Cost Per Agent$170$160$175$180$195

Union Airways Case Study Cont. Time Periods Covered by ShiftsMinimum # of Agents Needed Time PeriodS1S2S3S4S5 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 ShiftsMinimum # of Agents Needed Time PeriodS1S2S3S4S5 10 PM to 12 AM  AM to 6 AM  15 Daily Cost Per Agent$170$160$175$180$195

Mathematical Model of Union Airway’s 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 Source 1 Source 2 Source 3 Source n Destination 1 Destination 2 Destination 3 Destination m Supply S1 S2 S3 Sn Demand D1 D2 D3 Dm c 11 c 12 c 13 c 1m c 21 c 22 c 23 c 2m c 31 c 32 c 33 c 3m c n1 c n2 c n3 c nm

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 M’s data is on the next slide.

Big M Company Case Study Cont. Shipping Cost for Each Lathe Customer 1Customer 2Customer 3Output Factory 1$700$900$80012 Factory 2$800$900$70015 Order Size1089

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. Factory 1 12 Lathes Customer 1 10 Lathes Factory 2 15 Lathes Customer 3 9 Lathes Customer 2 8 Lathes $700 $900 $800 $900 $700

Mathematical Model for Big M’s 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 EmployeeWord Processing GraphicsPacketsRegistrationWage Ann $14 Ian $12 Joan $13 Sean $15

Assignment of Variables  x ij –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.