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### Chapter 7

Linear Programming Models

Part One

- Basis of Linear Programming
- Linear Program formulation

Linear Programming (LP)

Linear programming is a optimization model with an objective (in a linear function) and a set of limitations (in linear constraints).

LP Components

- Decision variables - their values are to be found in the solution.
- One objective function – tells our goal.
- Constraints - reflect limitations.
- Only linear terms are allowed.

Linear Terms

- A term is linear if it contains one variable with exponent one, or if it is a constant.
- Examples of linear terms:
- 3.5X 68.83 (3.78)6X1

- Examples of non-linear terms:
- 5X2 X1X2 sin X X3
- X2.5 Log X

Format of a Linear Program

- Align columns of inequality signs, variable terms, and constants.
- Variable terms are at left, constant terms are at right (called right-hand-side, RHS).
- Non-negative constraints must be there.

LP Solution

- A solution is a set of values each for a variable.
- A feasible solution satisfies all constraints.
- An infeasible solution violates at least one constraint.
- The optimal solution is a feasible solution that makes the objective function value maximized (or minimized).

LP Solution Methods

- Trial-and-Error
(brute force)

- Graphic Method
(Won’t work if more than 2 variables)

- Simplex Method (by George Dentzig)
(Elegant, but time-taking if by hand)

- Computerized simplex method
(We’ll use it!)

1914-2005

Inventor of Simplex Method.

Professor of Operations

Research and

Computer Science

at Stanford University.

To Solve a Problem by Linear Programming

- Formulate the problem into a linear program (LP).
- Enter the LP into QM.
- QM solves LP and provide the optimal solution.

Formulate a Problem into LP

- To formulate a decision making problem into a linear program:
- Understand the problem thoroughly;
- Define decision variables in unambiguous terms;
- Describe the problem with one objective function and a few constraints, in terms of the variables.

Flair Furniture, Example, p.252

Find how many tables and chairs should be produced to maximize the total profit.

Flair Furniture, Example, p.252

- Definitions of variables:
- LP formulation:
- Solution from QM

Tips of Formulating LP

- What are variables?
- Those amounts you want to decide.

- What is the ‘objective’?
- Profit (or cost) you do not know but you want to maximize (or minimize).

- What are ‘constraints’?
- Restrictions of reaching your ‘objective’.

Holiday Meal Turkey Ranch,p.270

Find how many pounds of brand 1feed and brand 2 feed should be purchased with lowest cost, which meet the minimum requirements of a turkey for each ingredient.

Holiday Meal Turkey Ranch,p.270

- Definitions of variables:
- LP formulation:
- Solution from QM:

Formulating

- To formulate a business problem into a linear program is to re-describe the problem with a ‘language’ that a computer understands.
- The key concern of formulation is:
- whether the LP tells the story exactly the same as the original one.

- Formulating is synonymous with ‘describing’ and ‘translating’. It is NOT ‘solving’.

“Team Work”

- The process of solving a business problem by using linear programming is a team work between us and computers:
- We formulate the problem in LP so that computers can understand;
- Computers solve the LP, providing us with the solution to the problem.

Irregular LP Problems

- A regular LP has one optimal solution.
- An irregular LP has no or many optimal solutions:
- Infeasible problem
- Unbounded problem
- Multiple optimal solutions

- Redundancy refers to having extra and un-useful constraints.

Part Two

- Shadow Price (Dual Value)
- Sensitivity Analysis

Dual Price

- Each dual price is associated with a constraint. It is the amount of improvement in the objective function value that is caused by a one-unit increase in the RHS of the constraint.
- It is also called Shadow Price.

In a product-mix problem

- As in the Flair Furniture example, a dual price is:
- the contribution of an additional unit of a resource to the objective function value (total profit), i.e.,
- the marginal value of a resource, i.e.,
- The highest “price” the company would be willing to pay for one additional unit of a resource.

Primal and Dual in LP

- Each linear program has another associated with it. They are called a pair of primal and dual.
- The dual LP is the “transposition” of the primal LP.
- Primal and dual have equal optimal objective function values.
- The solution of the dual is the dual prices of the primal, and vice versa.

More on Dual Price:

- A dual price can be negative, which shows a negative ( or worse off) contribution to the objective function value by an additional unit of RHS increase of the constraint.

Sensitivity Analysis (S.A.)

- S.A. is the analysis of the effect of parameter changes on the optimal solution.
- S. A. is conducted after the optimal solution is obtained.

S.A. on Objective Coefficients

- Sensitivity range for an objective coefficient is the range of values over which the coefficient can change without changing the current optimal solution.

S.A. on RHS

- Sensitivity range for a RHS value is the range of values over which the RHS value can change without changing the dual prices.

S.A. on other changes

- To see sensitivities on following changes, one must solve the changed LP again:
- Changing technological (constraint) coefficients
- Adding a new constraint
- Adding a new variable

Why doing S.A.?

- LP is used for decision making on something in the future.
- Rarely does a manager know all of the parameters exactly. Many parameters are inaccurate “estimates” when a model is formed and solved.
- We want to see to what extent the optimal solution is stable to the inaccurate parameters.

Sensitive or In-sensitive?

- Do we want a model more sensitive or less sensitive to the inaccuracies (changes) of parameters in it ?
- Answer:
Less sensitive.

- Why?
- An optimal solution that is insensitive to inaccuracies of parameters is more likely valid in the real world situation.

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