LEARNING OBJECTIVES. Introduction 1/4/2013 CHAPTER. Linear Programming Models: Graphical and Computer Methods
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1 CHAPTER 2 Linear Programming Models: Graphical and Computer Methods PowerPoint presentation to accompany Balakrishnan/Render/Stair Managerial Decision Modeling with Spreadsheets, 3/e 2-1 LEARNING OBJECTIVES 1. Understand the basic assumptions and properties of linear programming (LP). 2. Use graphical procedures to solve LP problems with only two variables to understand how LP problems are solved. 3. Understand special situations such as redundancy, infeasibility, unboundedness, and alternate optimal solutions in LP problems. 4. Understand how to set up LP problems on a spreadsheet and solve them using Excel s Solver. 2-2 Introduction Management decisions involve the most effective use of resources Most widely used modeling technique is linear programming (LP) Deterministic models 2-3 1
2 Developing a LP Model All LP models can be viewed in terms of the three distinct steps 1. Formulation of simple mathematical expressions 2. Solution to identify an optimal (or best) solution to the model 3. Interpretation of the results and answer what if? questions 2-4 Properties of a LP Model 1. Seek to maximize of minimize a some quantity 2. Restrictions or constraints 3. Alternative courses of action 4. Linear equations or inequalities (=,, ) 2-5 LP Characteristics Feasible Region The set of points that satisfies all constraints Corner Point Property An optimal solution must lie at one or more corner points Optimal Solution The corner point with the best objective function value is optimal 2-6 2
3 Formulating a LP Model A product mix problem Decide how much to make of two or more products Objective is to maximize profit Limited resources Flair Furniture Best combination of tables and chairs 2-7 Decision Variables What we are solving for Two variables in the Flair problem Number of tables (T, Tables or X 1 ) Number of chairs (C, Chairs or X 2 ) Decision variables can be in different units of measurement 2-8 The Objective Function States the goal of a problem A single objective function Objective is often to maximize profit or minimize cost 2-9 3
4 The Objective Function For Flair Furniture Profit = ($7 profit per table) x (number of tables produced) + ($5 profit per chairs) x (numbers of chairs produced) Using decision variables T and C Maximize $7T + $5C 2-10 Constraints Restrictions or limits on our decisions As many as necessary Can be independent Flair has four constraints Carpentry time Painting time Number of chairs to make Number of tables to make 2-11 Constraints For carpentry time (3 hours per table) x (number of tables produced) + (3 hours per chair) x (number of chairs produced) There are 2,400 hours of time available 3T + 4C 2,
5 Constraints All four constraints Carpentry time 3T + 4C 2,400 Painting time 2T + 1C 1,000 Chairs sold C 450 Tables sold T 100 Interactions exist between variables 2-13 Nonnegativity and Integers Decision variables must be 0, so T 0, and C 0 Decision variables may have to be integers 2-14 Flair Model Matrix TABLES (T) CHAIRS (C) LIMIT Profit Contribution $7 $5 Carpentry 3 hrs 4 hrs 2,400 Painting 2 hrs 1 hr 1,000 Chairs 0 unit 1 unit 450 Tables 1 unit 0 unit
6 Guidelines Recognizing and defining decision variables Different variables, different units Use only the decision variables in the model Difficulties may point to a need for more variables or better definitions 2-16 Guidelines One expression, one entity One unit of measurement per expression Constraints are separate Translate expressions into words 2-17 Graphical Solution Complete model Maximize profit = $7T + $5C Subject to 3T + 4C 2,400(carpentry time) 2T + 1C 1,000(painting time) C 450 (maximum chairs allowed) T 100 (maximum tables allowed) T, C 0 (nonnegativity)
7 Graphical Representation Number of Chairs (C) 1, (T = 0, C = 600) 600 Carpentry Constraint Line 400 (T = 400, C = 300) 200 (T = 800, C = 0) ,000 Number of Tables(T) Figure Graphical Representation Number of Chairs (C) 1, Region Satisfying 3T + 4C 2,400 (T = 300, C = 200) ,000 Number of Tables(T) (T = 600, C = 400) Figure Graphical Representation (T = 0, C = 1,000) Number of Chairs (C) 1,000 (T = 100, C = 700) (T = 0, C = 600) Painting Constraint (T = 300, C = 200) Carpentry Constraint (T = 500, C = 200) (T = 500, C = 0) (T = 800, C = 0) ,000 Number of Tables(T) Figure
8 Graphical Representation Painting Constraint Number of Chairs (C) 1, Feasible Infeasible Solution (T = 50, C = 500) Maximum Tables Required Constraint Maximum Chairs Allowed Constraint (T = 300, C = 200) Carpentry Constraint Infeasible Solution (T = 500, C = 200) Region ,000 Number of Tables(T) Figure Using Level Lines Number of Chairs (C) (T = 0, C = 560) (T = 0, C = 420) Feasible Region (T = 300, C = 0) (T = 400, C = 0) ,000 Number of Tables(T) Figure Using Level Lines 800 Optimal Level Profit Line 600 Carpentry Constraint Number of Chairs (C) Optimal Corner Point Solution Level Profit Line with No Feasible Points ($7T + $5C = $4,200) Painting Constraint ,000 Number of Tables(T) Figure
9 Calculating a Solution Optimal point 4 is the intersection of two constraints, carpentry and painting Solving simultaneously 6T + 8C = 4,800 (6T + 3C = 3,000) 5C = 1,800 implies C = 360 and T = Using All Corner Points Point 1 (T = 100, C = 0) Profit = $7 x $5 x 0 = $700 Point 2 (T = 100, C = 450) Profit = $7 x $5 x 450 = $2,950 Point 3 (T = 200, C = 450) Profit = $7 x $5 x 450 = $3,650 Point 4 (T = 320, C = 360) Profit = $7 x $5 x 360 = $4,040 Point 5 (T = 500, C = 0) Profit = $7 x $5 x 0 = $3, Extension to the Model Number of Chairs (C) Optimal Level Profit Line for Revised Problem (T = 300, C = 375) is the New Optimal Corner Point Solution (T = 320, C = 360) is No Longer Feasible 7 Additional Constraint C T 75 4 Has a Positive Slope ($7T + $5C = $2,800) This Portion of the Original Feasible Region Is No Longer Feasible ,000 Number of Tables(T) Figure
10 Minimization Problem Minimize cost Holiday Meal Turkey Ranch Two types of feed Minimize cost = $0.10A + $0.15B subject to 5A + 10B 45 (protein required) 4A + 3B 24 (vitamin required) 0.5A 1.5 (iron required) A,B 0 (nonnegativity) 2-28 Minimization Problem Data for Holiday Meal Turkey Ranch NUTRIENTS PER POUND OF FEED MINIMUM REQUIRED PER TURKEY PER NUTRIENT BRAND A FEED BRAND B FEED MONTH Protein (units) Vitamin (units) Iron (units) Cost Per Pound $0.10 $0.15 Table Minimization Problem Pounds of Brand B (B) Iron Constraint Feasible Region is Unbounded Vitamin Constraint 2 Protein Constraint Pounds of Brand A (A) Figure
11 Graphical Solution Pounds of Brand B (B) 10 Level Cost 9 Line for 8 Minimum Cost Unbounded Feasible Region Level Cost Line 2 Optimal Corner 1 Point Solution (A = 4.2, B = 2.4) Pounds of Brand A (A) Figure Calculating a Solution Optimal point 2 is the intersection of two constraints, vitamin and protein Solving simultaneously 4(5A + 10B = 45) implies 20A + 40B = 180 5(4A + 3B = 24) implies (20A + 15B = 120) 25B = 60 implies B = 2.4 and A = Special Situations Redundant Constraints Do not affect the feasible region Changed constraint in Flair Furniture problem T 100 becomes T
12 Special Situations Number of Chairs (C) 1, C 450 Carpentry Constraint Is Redundant 400 Constraint Changed to T Painting Constraint Is Redundant ,000 Feasible Region Number of Tables(T) Figure Special Situations Infeasibility No one solution satisfies all the constraints Changed constraint in Flair Furniture problem T 100 becomes T Special Situations Number of Chairs (C) Constraint Changed to T 600 1,000 C Two Regions Do Not Overlap Region 200 Satisfying Three Constraints 3T + 4C 2,400 2T + C 1,000 Region Satisfying Fourth Constraint ,000 Number of Tables(T) Figure
13 Special Situations Alternate Optimal Solutions More than one solution satisfies all the constraints Changed objective in Flair Furniture problem $7T + $5C becomes $6T + $3C 2-37 Special Situations 800 Level Profit Line for Maximum Profit Overlaps Painting Constraint Number of Chairs (C) Feasible Region Level Profit Line Is Parallel to Painting Constraint $6T + $5C = $2,100 Optimal Solution Consists of All Points Between Corner Points 4 and ,000 Number of Tables(T) Figure Special Situations Unbounded Solution May or may not have a finite solution Usually improper formulation Changed objective in Holiday Meal problem Minimize = $0.10A + $0.15B becomes Maximize = 8A + 12B
14 Special Situations Pounds of Brand B (B) 10 9 Iron Constraint 8 7 Unbounded Feasible Region Vitamin Constraint 1 Protein Constraint Pounds of Brand A (A) Figure Using Excel s Solver Excel s built-in LP solution tool for LP Commonly available and easy access Familiar software 2-41 Screenshot 2-1A
15 Screenshot 2-1B 2-43 Screenshot 2-1B 2-44 Screenshot 2-1C
16 Screenshot 2-1D 2-46 Screenshot 2-1E 2-47 Screenshot 2-1F
17 Screenshot Screenshot 2-3A
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