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# Chapter 7 - PowerPoint PPT Presentation

Chapter 7. Linear Programming Models. Part One. Basis of Linear Programming Linear Program formulati on. Linear Programming (LP). Linear programming is a optimization model with an objective (in a linear function) and a set of limitations (in linear constraints). A Linear Program.

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### Chapter 7

Linear Programming Models

• Basis of Linear Programming

• Linear Program formulation

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

Max X1 + 2X2

S.T. 3X1 + X2 <= 200

X2 <= 100

X1, X2 >= 0

• 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.

• 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

• X2.5 Log X

• 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.

• 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).

• 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.

• Formulate the problem into a linear program (LP).

• Enter the LP into QM.

• QM solves LP and provide the optimal solution.

• 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.

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

• Definitions of variables:

• LP formulation:

• Solution from QM

• 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’.

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.

• Definitions of variables:

• LP formulation:

• Solution from QM:

• 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’.

• 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.

• 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.

• Sensitivity Analysis

• 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.

• 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.

• 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.

• 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.

• 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.

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

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

• To see sensitivities on following changes, one must solve the changed LP again:

• Changing technological (constraint) coefficients

• 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.

• Do we want a model more sensitive or less sensitive to the inaccuracies (changes) of parameters in it ?