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Introduction to Operations Research

Linear Programming. Introduction to Operations Research. Linear Programming provides methods for allocating limited resources among competing activities in an optimal way. Linear → All mathematical functions are linear Programming → Involves the planning of activities

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Introduction to Operations Research

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  1. Linear Programming Introduction to Operations Research

  2. Linear Programming provides methods for allocating limited resources among competing activities in an optimal way. • Linear → All mathematical functions are linear • Programming → Involves the planning of activities • A linear program is a mathematical optimization model that has a linear objective function and a set of linear constraints What is Linear Programming?

  3. The company produces glass products and owns 3 plants. Management decides to produce two new products. Example - Wyndor Glass Co. Product 1 1 hour production time in Plant 1 3 hours production time in Plant 3 $3,000 profit per batch Product 2 2 hours production time in Plant 2 2 hours production time in Plant 3 $5,000 profit per batch Production time available each week Plant 1: 4 hours Plant 2: 12 hours Plant 3: 18 hours

  4. Subject to: x1 ≤ 4 2x2 ≤ 12 3x1 + 2x2 ≤ 18 x1 ≥ 0, x2 ≥ 0 Wyndor Glass Co. Maximize Z = 3x1 + 5x2

  5. Graph the equations to determine relationships Maximize Z = 3x1 + 5x2 Subject to: x1 ≤ 4 2x2 ≤ 12 3x1 + 2x2 ≤ 18 x1 ≥ 0, x2 ≥ 0 Wyndor Glass Co.

  6. Allocating resources to activities General Linear Programming

  7. Z = Value of overall measure of performance xj = Level of activity j = Decision variables cj = Increase in Z resulting from increase in j = Parameters bi = Amount of available resources = Parameters aij = Amount of resource i consumed by each unit of j = Parameters General Linear Programming Objective Function c1 x1 +   c2 x2 + ... +   cn xn Constraints a11 x1 + a12 x2 + ... + a1n xn  ≤ b1 a21 x1 + a22 x2 + ... + a2n xn  ≤ b2 ..... am1 x1 + am2 x2 + ... + amn xn  ≤ bm x1 ≥ 0, x2 ≥ 0, ..., xn ≥ 0 , Non-negativity Constraints Functional Constraints

  8. General Linear Programming

  9. Solution – Any specification of values for the decision variables (xj) • Feasible solution – A solution for which all constraints are satisfied • Infeasible solution – A solution for which at least one constraint is violated • Feasible region – The collection of all feasible solutions • Optimal solution – A feasible solution that has the most favorable value of the objective function Linear Programming Solutions

  10. No Feasible Solution • Multiple Optimal Solutions • No Optimal Solution • Corner-point Feasible (CPF) Solution Linear Programming Solutions

  11. Proportionality – The contribution of each activity to Z or a constraint is proportional to the level of activity xj • Z = 3x1 + 5x2 • Additivity – Every function is the sum of the individual contributions of the activities • Divisibility – Decision variables are allowed to have any value, including non-integer values • Certainty – The value assigned to each parameter is assumed to be a known constant Linear Programming Assumptions

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