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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 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).
A Linear Program Max X1 + 2X2 S.T. 3X1 + X2 <= 200 X2 <= 100 X1, X2 >= 0
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!)
George Dantzig 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.