Chapter 6 Supplement Linear Programming

Linear Programming Linear Programming (LP) deals with the problems of allocating limited resources among competing activities in the best possible way (optimal) A linear program consist of a linear objective function and a set of linear constraints

Linear Programming Model Objective: the goal of an LP model is maximization or minimization Decision variables: amounts of either inputs or outputs Constraints: limitations that restrict the available alternatives Parameters: numerical values

Linear Programming Assumptions Linearity: the impact of decision variables is linear in constraints and objective function Divisibility: noninteger values of decision variables are acceptable Certainty: values of parameters are known and constant Nonnegativity: negative values of decision variables are unacceptable

Linear Programming Application Procedure Parameter Estimation Problem Formulation Optimal Solution Graphical Method Simplex Method Computer Solution Other Methods Sensitivity Analysis

Linear Programming Application Areas Production Inventory Financial Marketing Distribution Sports Agriculture

Linear Programming: Some Definitions Solution: A solution is a set of values of the decision variables Feasible Solution: A feasible solution is a solution for which all the constraints are satisfied Optimal Solution: An optimal solution is a feasible solution which optimizes the objective function

Linear Programming: Types of Solutions Single Optimal Solution Multiple Optimal Solutions No Optimal Solution

Graphical Linear Programming Set up objective function and constraints in mathematical format Plot the constraints Identify the feasible solution space Plot the objective function Determine the optimum solution

Graphical Linear Programming Maximize Z = 4X1 + 5X2 Subject to X1 + 3X2 < 12 (constraint 1) 4X1 + 3X2 < 24 (constraint 2) X1 > 0 X2 > 0

Linear Programming Example Plot Constraint 1 X1 + 3X2 = 12

Linear Programming Example Add Constraint 2 4X1 + 3X2 = 24 Constraint 1 X1 + 3X2 = 12 Solution space

Linear Programming Example Z = 60 Z = 40 Z = 20 X1

LP Formulation and Computer Solution: Problem 1

Linear Programming Problem 1: Formulation Let Xi be the number of units of product type i to be produced per week, i = 1, 2, 3 Maximize Z = 30X1 + 12X2 + 15X3 Subject to 9X1 + 3X2+ 5X3 < 500 (Milling) 5X1 + 4X2 < 350 (Lathe) 3X1 + 2X3 < 150 (Drill) X3 < 20 (Sales Potential) X1 > 0, X2 > 0, X3 > 0

Slack and Surplus Binding constraint: a constraint that forms the optimal corner point of the feasible solution space Slack: when the optimal values of decision variables are substituted into a less than or equal to constraint and the resulting value is less than the right side value Surplus: when the optimal values of decision variables are substituted into a greater than or equal to constraint and the resulting value exceeds the right side value

Linear Programming Problem 1: Solution Using LINGO Software Objective value: 1742.857 Variable Value Reduced Cost X1 26.19048 0.0000000 X2 54.76190 0.0000000 X3 20.00000 0.0000000 Row Slack or Surplus Dual Price PROFIT 1742.857 1.000000 MILLING 0.0000000 2.857143 LATHE 0.0000000 0.8571429 DRILL 31.42857 0.0000000 SALESPOT 0.0000000 0.7142857

Sensitivity Analysis Range of optimality: the range of values for which the solution quantities of the decision variables remains the same Range of feasibility: the range of values for the fight-hand side of a constraint over which the shadow price (dual price) remains the same Shadow prices: negative values indicating how much a one-unit decrease in the original amount of a constraint would decrease the final value of the objective function

Linear Programming Problem 1: Solution Using LINGO Software Ranges in which the basis is unchanged: Objective Coefficient Ranges Current Allowable Allowable Variable Coefficient Increase Decrease X1 30.00000 0.7500000 15.00000 X2 12.00000 12.00000 0.6000000 X3 15.000 INFINITY 0.7142857 Righthand Side Ranges Row Current Allowable Allowable RHS Increase Decrease MILLING 500.0000 55.00000 137.5000 LATHE 350.0000 183.3333 73.33334 DRILL 150.0000 INFINITY 31.42857 SALESPOT 20.00000 27.50000 20.00000

Linear Programming Problem 1: Solution Using EXCEL (a)

Linear Programming Problem 1: Solution Using EXCEL Software (b)

Linear Programming Problem 1: Solution Using EXCEL Software (c)

Linear Programming Problem 1: Solution Using EXCEL Software (d)

LP Formulation And Computer Solution: Problem 2

Linear Programming Problem 2: Formulation Let X1 X2 X3 be the kilograms of corn, tankage, and alfalfa, respectively. Minimize Z = 21X1 + 18X2 + 15X3 Subject to 90X1 + 20X2+ 40X3 > 200 (Carbo) 30X1 + 80X2 + 60X3 > 180 (Protein) 10X1 + 20X2 + 60X3 > 150 (Vitamin) X1 > 0, X2 > 0, X3 > 0

Linear Programming Problem 2: Solution Using LINGO Software Objective value: 60.42857 Variable Value Reduced Cost X1 1.142857 0.0000000 X2 0.0000000 4.428571 X3 2.428571 0.0000000 Row Slack or Surplus Dual Price COST 60.42857 1.000000 CARBOHY 0.0000000 -0.1928571 PROTEIN 0.0000000 -0.1214286 VITAMIN 7.142857 0.0000000

Linear Programming Problem 2: Solution Using LINGO Software Ranges in which the basis is unchanged: Objective Coefficient Ranges Current Allowable Allowable Variable Coefficient Increase Decrease X1 21.00000 12.75000 9.299998 X2 18.00000 INFINITY 4.428571 X3 15.00000 2.818181 5.666667 Righthand Side Ranges Row Current Allowable Allowable RHS Increase Decrease CARBOHY 200.0000 25.00000 80.00000 PROTEIN 180.0000 120.0000 6.000000 VITAMIN 150.0000 7.142857 INFINITY

Linear Programming Problem 2: Solution Using EXCEL Software (a)

Linear Programming Problem 2: Solution Using EXCEL Software (b)

Linear Programming Problem 2: Solution Using EXCEL Software (c)

Linear Programming Problem 2: Solution Using EXCEL Software (d)