Chapter 7 Linear Programming Models

Part One n Basis of Linear Programming n 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 Max X 1 + 2X 2 S.T.3X 1 + X 2 <= 200 X 2 <= 100 X 1, X 2 >= 0

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

Linear Terms n A term is linear if it contains one variable with exponent one, or if it is a constant. n Examples of linear terms: –  3.5X 68.83(3.78) 6 X 1 n Examples of non-linear terms: – 5X 2 X 1 X 2 sin XX 3 – X  2.5 Log X

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

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

LP Solution Methods n Trial-and-Error (brute force) n Graphic Method (Won’t work if more than 2 variables) n Simplex Method (by George Dentzig) (Elegant, but time-taking if by hand) n 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 n Formulate the problem into a linear program (LP). n Enter the LP into QM. n QM solves LP and provide the optimal solution.

Formulate a Problem into LP n 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 n Definitions of variables: n LP formulation: n Solution from QM

Tips of Formulating LP n What are variables? – Those amounts you want to decide. n What is the ‘objective’? – Profit (or cost) you do not know but you want to maximize (or minimize). n 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 n Definitions of variables: n LP formulation: n Solution from QM:

Formulating n To formulate a business problem into a linear program is to re-describe the problem with a ‘language’ that a computer understands. n The key concern of formulation is: – whether the LP tells the story exactly the same as the original one. n Formulating is synonymous with ‘describing’ and ‘translating’. It is NOT ‘solving’.

“Team Work” n 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 n A regular LP has one optimal solution. n An irregular LP has no or many optimal solutions: – Infeasible problem – Unbounded problem – Multiple optimal solutions n Redundancy refers to having extra and un-useful constraints.

Part Two n Shadow Price (Dual Value) n Sensitivity Analysis

Dual Price n 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. n It is also called Shadow Price.

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

More on Dual Price: n 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.) n S.A. is the analysis of the effect of parameter changes on the optimal solution. n S. A. is conducted after the optimal solution is obtained.

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

Sensitive or In-sensitive? n Do we want a model more sensitive or less sensitive to the inaccuracies (changes) of parameters in it ? n Answer: Less sensitive. n Why? – An optimal solution that is insensitive to inaccuracies of parameters is more likely valid in the real world situation.