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INTRODUCTION TO OPERATIONS RESEARCH Linear Programming WHAT IS LINEAR PROGRAMMING? 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 EXAMPLE - WYNDOR GLASS CO. The company produces glass products and owns 3 plants. Management decides to produce two new products. Product 1 Product 2 1 hour production time in Plant 1 3 hours production time in Plant 3 $3,000 profit per batch Production time available each week Plant 1: Plant 2: Plant 3: 4 hours 12 hours 18 hours 2 hours production time in Plant 2 2 hours production time in Plant 3 $5,000 profit per batch WYNDOR GLASS CO. Plant Product 1 Product 2 Production Time 1 1 0 4 2 0 2 12 3 3 2 18 $3,000 $5,000 Profit Maximize Z = 3x1 + 5x2 Subject to: x1 ≤ 4 2x2 ≤ 12 3x1 + 2x2 ≤ 18 x1 ≥ 0, x2 ≥ 0 WYNDOR GLASS CO. Graph the equations to determine relationships Maximize Z = 3x1 + 5x2 Subject to: x1 ≤ 4 2x2 ≤ 12 3x1 + 2x2 ≤ 18 x1 ≥ 0, x2 ≥ 0 GENERAL LINEAR PROGRAMMING Allocating resources to activities General Example Resources Production capacities of plants m resources 3 plants Activities Production of products n activities 2 Products Level of activity j, xj Production rate of product j, xj Overall measure of performance Z Profit Z 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 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 Functional Constraints GENERAL LINEAR PROGRAMMING Resource 1 2 … m Contribution to Z Resource Usage per Unit of Activity Activity 1 2 … n a11 a12 … a1n a21 a22 … a2n … … … … am1 am2 … amn c1 c2 … cn Amount of Resource Available b1 b2 … bm LINEAR PROGRAMMING SOLUTIONS 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 No Feasible Solution Multiple Optimal Solutions No Optimal Solution Corner-point Feasible (CPF) Solution LINEAR PROGRAMMING ASSUMPTIONS 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