Linear Programming Models: Graphical and Computer Methods 2 Linear Programming Models: Graphical and Computer Methods

LEARNING OBJECTIVES Understand the basic assumptions and properties of linear programming (LP). Use graphical procedures to solve LP problems with only two variables to understand how LP problems are solved. Understand special situations such as redundancy, infeasibility, unboundedness, and alternate optimal solutions in LP problems. Understand how to set up LP problems on a spreadsheet and solve them using Excel’s Solver.

Introduction Management decisions involve the most effective use of resources Most widely used modeling technique is linear programming (LP) Deterministic models

Developing a LP Model All LP models can be viewed in terms of the three distinct steps Formulation of simple mathematical expressions Solution to identify an optimal (or best) solution to the model Interpretation of the results and answer “what if?” questions

Properties of a LP Model Seek to maximize or minimize some quantity (profit or cost) Restrictions or constraints Alternative courses of action Linear equations or inequalities (=, ≤, ≥)

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

Formulating a LP Model A product mix problem Flair Furniture 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

Flair Furniture Produces tables and chairs Each table takes 3 hrs of carpentry and 2 hrs of painting work Each chair 4 hrs and 1 hr, respectively 2,400 hrs of carpentry time, 1,000 hrs of painting time No more than 450 chairs At least 100 tables $7 and $5 profit for table and chair

Decision Variables What we are solving for Two variables in the Flair problem Number of tables (T, Tables or X1) Number of chairs (C, Chairs or X2) Decision variables can be in different units of measurement

The Objective Function States the goal of a problem A single objective function Objective is often to maximize profit or minimize cost

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

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

Constraints For carpentry time There are 2,400 hours of time available (3 hours per table) x (number of tables produced) + (4 hours per chair) x (number of chairs produced) There are 2,400 hours of time available 3T + 4C ≤ 2,400

Constraints All four constraints Carpentry time: 3T + 4C ≤ 2,400 Painting time: 2T + 1C ≤ 1,000 Chairs made: C ≤ 450 Tables made: T ≥ 100

Nonnegativity and Integers Decision variables must be ≥ 0, so Decision variables may have to be integers T ≥ 0, and C ≥ 0

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 100

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 (minimum tables required) T, C ≥ 0 (nonnegativity)

Graphical Representation Number of Chairs (C) Number of Tables(T) 1,000 – – 800 – 600 – 400 – 200 – 0 – | | | | | | | | | | | | 0 200 400 600 800 1,000 (T = 0, C = 600) Carpentry Constraint Line (T = 400, C = 300) (T = 800, C = 0) Figure 2.1

Graphical Representation Number of Chairs (C) Number of Tables(T) 1,000 – – 800 – 600 – 400 – 200 – 0 – | | | | | | | | | | | | 0 200 400 600 800 1,000 Region Satisfying 3T + 4C ≤ 2,400 (T = 300, C = 200) (T = 600, C = 400) Figure 2.2

Graphical Representation Number of Chairs (C) Number of Tables(T) 1,000 – – 800 – 600 – 400 – 200 – 0 – | | | | | | | | | | | | 0 200 400 600 800 1,000 (T = 0, 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) Figure 2.3

Graphical Representation Painting Constraint Number of Chairs (C) Number of Tables(T) 1,000 – – 800 – 600 – 400 – 200 – 0 – | | | | | | | | | | | | 0 200 400 600 800 1,000 Infeasible Solution (T = 50, C = 500) Minimum Tables Required Constraint Maximum Chairs Allowed Constraint Feasible Region (T = 300, C = 200) Carpentry Constraint Infeasible Solution (T = 500, C = 200) Figure 2.4

Using Level Lines 800 – – (T = 0, C = 560) 600 – 400 – | | | | | | | | | | 0 200 400 600 800 1,000 800 – – 600 – 400 – 200 – 0 – Number of Chairs (C) Number of Tables(T) (T = 0, C = 560) (T = 0, C = 420) $7T + $5C = $2,800 Feasible Region $7T + $5C = $2,100 (T = 300, C = 0) (T = 400, C = 0) Figure 2.5

Using Level Lines 0 200 400 600 800 1,000 800 – – 600 – 400 – 200 – | | | | | | | | | | 0 200 400 600 800 1,000 800 – – 600 – 400 – 200 – 0 – Number of Chairs (C) Number of Tables(T) Optimal Level Profit Line Carpentry Constraint $7T + $5C = $2,800 1 2 3 4 5 Optimal Corner Point Solution $7T + $5C = $2,100 Level Profit Line with No Feasible Points ($7T + $5C = $4,200) Painting Constraint Figure 2.6

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 = 320

Using All Corner Points Point 1 (T = 100, C = 0) Profit = $7 x 100 + $5 x 0 = $700 Point 2 (T = 100, C = 450) Profit = $7 x 100 + $5 x 450 = $2,950 Point 3 (T = 200, C = 450) Profit = $7 x 200 + $5 x 450 = $3,650 Point 4 (T = 320, C = 360) Profit = $7 x 320 + $5 x 360 = $4,040 Point 5 (T = 500, C = 0) Profit = $7 x 500 + $5 x 0 = $3,500

Extension to the Model 800 – – | | | | | | | | | | 0 200 400 600 800 1,000 800 – – 600 – 400 – 200 – 0 – Number of Chairs (C) Number of Tables(T) 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 1 5 2 3 4 6 7 Additional Constraint C – T ≥ 75 Has a Positive Slope ($7T + $5C = $2,800) This Portion of the Original Feasible Region Is No Longer Feasible Figure 2.7

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)

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) 5 10 45 Vitamin (units) 4 3 24 Iron (units) 0.5 0 1.5 Cost Per Pound $0.10 $0.15 Table 2.1

Minimization Problem 10 – 9 – Iron Constraint 8 – 7 – 6 – 5 – 4 – Pounds of Brand B (B) Pounds of Brand A (A) 10 – 9 – 8 – 7 – 6 – 5 – 4 – 3 – 2 – 1 – 0 – | | | | | | | | | | | 0 1 2 3 4 5 6 7 8 9 10 1 2 3 Feasible Region is Unbounded Iron Constraint Vitamin Constraint Protein Constraint Figure 2.8

Graphical Solution 10 – 9 – Level Cost Line for Minimum Cost 8 – 7 – Pounds of Brand B (B) Pounds of Brand A (A) 10 – 9 – 8 – 7 – 6 – 5 – 4 – 3 – 2 – 1 – 0 – | | | | | | | | | | | 0 1 2 3 4 5 6 7 8 9 10 Unbounded Feasible Region 1 2 3 Level Cost Line for Minimum Cost $0.10A + $0.15B = $1 Direction of Decreasing Cost Level Cost Line Optimal Corner Point Solution (A = 4.2, B = 2.4) Figure 2.9

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 = 4.2

Special Situations Redundant Constraints Do not affect the feasible region Changed constraint in Flair Furniture problem T ≥ 100 becomes T ≤ 100

Special Situations C ≤ 450 Number of Chairs (C) Number of Tables(T) 1,000 – – 800 – 600 – 400 – 200 – 0 – | | | | | | | | | | | | 0 200 400 600 800 1,000 C ≤ 450 Carpentry Constraint Is Redundant Feasible Region Constraint Changed to T ≤ 100 Painting Constraint Is Redundant Figure 2.10

Special Situations Infeasibility No one solution satisfies all the constraints Changed constraint in Flair Furniture problem T ≥ 100 becomes T ≥ 600

Special Situations Constraint Changed to T ≥ 600 C ≤ 450 Number of Chairs (C) Number of Tables(T) 1,000 – – 800 – 600 – 400 – 200 – 0 – | | | | | | | | | | | | 0 200 400 600 800 1,000 Region Satisfying Fourth Constraint C ≤ 450 Two Regions Do Not Overlap 3T + 4C ≤ 2,400 Region Satisfying Three Constraints 2T + C ≤ 1,000 Figure 2.11

Special Situations Alternate Optimal Solutions More than one solution satisfies all the constraints Changed objective in Flair Furniture problem $7T + $5C becomes $6T + $3C

Special Situations 0 200 400 600 800 1,000 800 – – 600 – 400 – 200 – | | | | | | | | | | 0 200 400 600 800 1,000 800 – – 600 – 400 – 200 – 0 – Number of Chairs (C) Number of Tables(T) Level Profit Line for Maximum Profit Overlaps Painting Constraint Level Profit Line Is Parallel to Painting Constraint 1 2 3 4 5 Feasible Region $6T + $3C = $2,100 Optimal Solution Consists of All Points Between Corner Points 4 and 5 Figure 2.12

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

Special Situations Pounds of Brand B (B) Pounds of Brand A (A) 10 – 9 – 8 – 7 – 6 – 5 – 4 – 3 – 2 – 1 – 0 – | | | | | | | | | | | 0 1 2 3 4 5 6 7 8 9 10 Unbounded Feasible Region Iron Constraint Direction of Increasing Value Value Can Be Increased to Infinity 8A + 12B = 100 8A + 12B = 80 Vitamin Constraint Protein Constraint Figure 2.13

Using Excel’s Solver Excel’s built-in LP solution tool for LP Commonly available and easy access Familiar software

Using Solver Screenshot 2-1A

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