ISE 330 Introduction to Operations Research: Deterministic Models. What is Linear Programming? www-scf.usc.edu/~ise330/2007. August 29, 2007 Lecture 2

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1 ISE 330 Introduction to Operations Research: Deterministic Models www-scf.usc.edu/~ise330/007 August 9, 007 Lecture 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 Any problem whose model fits the format for the linear programming model is a linear programming problem. Linear Programming (Sec ) The company produces glass products owns 3 plants. Management decides to produce two new products. Product hour production time in Plant 3 hours production time in Plant 3 $3,000 profit per batch Product hours production time in Plant hours production time in Plant 3 $5,000 profit per batch Production time available each week Plant : 4 hours Plant : hours Plant 3: 8 hours

2 Plant 3 Profit per batch Production Time per Batch, Hours 0 3 $3,000 Product 0 $5,000 Production Time Available per Week, Hours 4 8 Maximize Z = 3 + 5x Subject to: 4 x 3 + x 8 0, x 0 Maximize 3 x + x Z x = 4 3x 8 + 5x General Linear Programming Problems Maximize Z = 3 + 5x Subject to: 4 x 3 + x 8 0, x 0 x = 4 Z = 36 Optimal: = x = x = 8 Allocating resources to activities Example Production capacities of plants 3 plants Production of products Products Production rate of product j, x j Profit Z General Resources m resources Activities n activities Level of activity j, x j Overall measure of performance Z Z = 0 Z = 0

3 General Linear Programming Problems General Linear Programming Problems Objective Function Z = c + c x + + c n x n a + a x + + a n x n b a + a x + + a n x n b a m + a m x + + a mn x n b m 0, x 0,, x n 0 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 b i = Amount of available resources a ij = Amount of resource i consumed by each unit of j Functional Non-negativity Resource m Contribution to Z per unity of activity Resource Usage per Unit of Activity Activity n a a a a m c a a m c a n a n a mn c n Amount of Resource Available b b b m Other Forms of Linear Programming Problems Minimize rather than maximize objective function Minimize Z = c + c x + + c n x n Some function constraints with greater-than-or-equal-to ( ) a i + a i x + + a in x n b i for some value of i Some functional constraints in equation form a i + a i x + + a in x n = b i for some value of i Deleting non-negativity constraints x j unrestricted in sign for some value of j 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

4 x = 4 = 9 x = x = x = 8 No Feasible Solution Multiple Optimal Solutions x = 4 x = x = x = 8 No Optimal Solution Corner-point Feasible (CPF) Solution

5 Linear Programming Assumptions Proportionality The contribution of each activity to Z or a constraint is proportional to the level of activity x j Z = 3 + 5x Additivity Every function is the sum of the individual contributions of the activities Z = 3 + 5x + x 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 In-class Example 3.-6 The Whitt Window Company is a company with only three employees which makes two different kinds of h-crafted windows: a wood-framed an aluminum-framed. They earn $60 profit for each wood-framed window $30 for each aluminum-framed window. Doug makes the wood frames, can make 6 per day. Linda makes the aluminum frames, can make 4 per day. Bob forms cuts the glass, can make 48 square feet of glass per day. Each wood-framed window uses 6 square feet of glass each aluminum-framed window uses 8 square feet of glass. How many windows of each type should be produced per day to maximize profit? Homework (Due September 5, 007) points points points points points points points points points points

Linear programming I João Carlos Lourenço

Decision Support Models Linear programming I João Carlos Lourenço joao.lourenco@ist.utl.pt Academic year 2012/2013 Readings: Hillier, F.S., Lieberman, G.J., 2010. Introduction to Operations Research, 9th