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

The company produces glass products and owns 3 plants. Management decides to produce two new products. 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

PlantProduct 1Product 2Production Time Profit$3,000$5,000 Subject to: x 1 ≤ 4 2x 2 ≤ 12 3x 1 + 2x 2 ≤ 18 x 1 ≥ 0, x 2 ≥ 0 Maximize Z = 3x 1 + 5x 2

Graph the equations to determine relationships Maximize Z = 3x 1 + 5x 2 Subject to: x 1 ≤ 4 2x 2 ≤ 12 3x 1 + 2x 2 ≤ 18 x 1 ≥ 0, x 2 ≥ 0

Allocating resources to activities GeneralExample ResourcesProduction capacities of plants m resources3 plants ActivitiesProduction of products n activities2 Products Level of activity j, x j Production rate of product j, x j Overall measure of performance ZProfit Z

Z = Value of overall measure of performance x j = Level of activity j = Decision variables c j = Increase in Z resulting from increase in j = Parameters b i = Amount of available resources = Parameters a ij = Amount of resource i consumed by each unit of j = Parameters Objective Function c 1 x 1 + c 2 x c n x n Constraints a 11 x 1 + a 12 x a 1n x n ≤ b 1 a 21 x 1 + a 22 x a 2n x n ≤ b a m1 x 1 + a m2 x a mn x n ≤ b m x 1 ≥ 0, x 2 ≥ 0,..., x n ≥ 0, Functional ConstraintsNon-negativity Constraints

Resource Resource Usage per Unit of Activity Amount of Resource Available Activity 12…n 1a 11 a 12 …a 1n b1b1 2a 21 a 22 …a 2n b2b2 ……………… ma m1 a m2 …a mn bmbm Contribution to Z c1c1 c2c2 …cncn

 Solution – Any specification of values for the decision variables ( x j )  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

 No Feasible Solution  Multiple Optimal Solutions  No Optimal Solution  Corner-point Feasible (CPF) Solution

 Proportionality – The contribution of each activity to Z or a constraint is proportional to the level of activity xj  Z = 3x 1 + 5x 2  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