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**Linear Programming: Fundamentals**

Chapter 2 Linear Programming: Fundamentals

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**Linear Programming (LP)**

Linear programming is a optimization model with an objective and a set of constraints. LP is a model for restricted decision making.

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**An Application Example**

Find how many bowls and mugs should be produced to maximize the profit.

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LP Components Decision variables - their values are to be found in the solution. One linear objective function. Linear constraints - reflect limitations.

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**A Linear Program Max 7X1 + 4X2 S.T. 3X1 + X2 <= 580**

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**Format of a Linear Program**

No variable is in denominator. At most one term for each variable. Variable terms are at left, constant terms are at right (called right-hand-side, RHS). Align columns of inequality signs, variable terms, and constants. Put non-negative constraints in at last.

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**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 meets the objective.

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**LP Solution Methods Trial-and-Error (brute force) Graphic Method**

(Won’t work if more than 2 variables) Simplex Method (Elegant, but time-taking if by hand) Computerized simplex method (We’ll use it programmed in QM)

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**Process of Solving a Problem By Using LP**

Step 1. Formulate the problem into a linear program (by us) Step 2. Solving the linear program (by computer) Step 3. Understanding the result and sensitivity analysis (by us / computer)

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LP Formulation Before using QM to solve a problem, we must first formulate the problem into a linear program, which is a description of the problem in terms of LP. Therefore, the process of formulating a problem in LP is a process of describing the problem by using an objective function and a couple of constraints.

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LP Example 1, p.32-33 Find how many bowls and mugs should be produced to maximize the profit.

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**Steps for LP Formulating**

Define variables unambiguously. Describe the objective function by using the variables. Describing restrictions one at a time by using the variables, which form constraints.

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LP Example 2 p.47-49 How many bags of each brand should be purchased in order to minimize the total cost?

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**Irregular LP Problems A regular LP has one optimal solution.**

Irregular cases: Multiple optimal solutions Infeasible problem Unbounded problem

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**Linear Programming: Sensitivity Analysis**

Chapter 3 Linear Programming: Sensitivity Analysis

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**Sensitivity Analysis (SA)**

SA is the analysis of the effect of parameter changes on the optimal solution. SA is conducted after the optimal solution is obtained.

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**Shadow Price (Dual Value)**

A shadow price is associated with a constraint in the solution.

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**In a product-mix problem**

as in example of ‘bowls and mugs’, a shadow price means: the marginal value of a resource, i.e., the contribution of an additional unit of a resource to the objective function value, i.e., The highest “price” the company would be willing to pay for one additional unit of a resource.

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What Is “Dual”? Each linear program has another LP 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 shadow prices of the primal, and vice versa.

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**More General on Shadow Price:**

The shadow price of a constraint shows how much the objective function value would be better off if there were one unit increase on the RHS of the constraint. A shadow price can be negative, which shows a negative contribution (i.e., worse off) to the objective function value by an additional unit of RHS of the constraint.

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S.A. on RHS Sensitivity range for a RHS value is the range over which the RHS value can change without changing the current shadow price. Sensitivity range for a RHS value is also the range over which the RHS value can change without changing the non-zero variable mix in the solution.

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**S.A. on Objective Coefficients**

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

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S.A. on other changes To see sensitivities on following changes, one must solve the changed LP again: Changing constraint coefficients Adding a new constraint Adding a new variable

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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.

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**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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**Linear Programming: Modeling Examples**

Chapter 4 Linear Programming: Modeling Examples

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**LP Modeling To model a decision making problem with LP:**

Understand the problem thoroughly; Identify the variables and objective; Describe the problem in terms of the variables, objective function, and constraints.

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**Examples Covered: Product mix Investment Marketing Blend (?)**

Transportation (?)

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