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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
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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. • 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.
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?
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
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.
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.
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 c1m c13 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
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. • 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
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.
