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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

Chapter 7

Linear Programming Models

Part one
Part One

  • Basis of Linear Programming

  • Linear Program formulation

Linear programming lp
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
A Linear Program

Max X1 + 2X2

S.T. 3X1 + X2 <= 200

X2 <= 100

X1, X2 >= 0

Lp components
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
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

    • X2.5 Log X

Format of a linear program
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
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
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!)

George Dantzig


Inventor of Simplex Method.

Professor of Operations

Research and

Computer Science

at Stanford University.

To solve a problem by linear programming
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
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
Flair Furniture, Example, p.252

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

Flair furniture example p 2521
Flair Furniture, Example, p.252

  • Definitions of variables:

  • LP formulation:

  • Solution from QM

Tips of formulating lp
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
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 2701
Holiday Meal Turkey Ranch,p.270

  • 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’.

Team work
“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
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
Part Two

  • Shadow Price (Dual Value)

  • Sensitivity Analysis

Dual price
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
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
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
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
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
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
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
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
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
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.