Presentation on theme: "Managerial Decision Modeling with Spreadsheets"— Presentation transcript:
1 Managerial Decision Modeling with Spreadsheets Chapter 2Linear Programming Models: Graphical and Computer Methods
2 Learning ObjectivesUnderstand basic assumptions and properties of linear programming (LP).Use graphical solution procedures for 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.
3 2.2 Development of a LP Model LP applied extensively to problems areas -medical, transportation, operations,financial, marketing, accounting,human resources, and agriculture.Development of all LP models can be examined in three step process:(1) formulation.(2) solution.(3) interpretation.
4 2.3 Formulating a LP Problem One of the most common LP application is product mix problem.Two or more products are usually produced using limited resources - such as personnel, machines, raw materials, and so on.Profit firm seeks to maximize is based on profit contribution per unit of each product.Firm would like to determine -How many units of each product it should produceMaximize overall profit given its limited resources.
5 LP Example: Flair Furniture Company Company Data and Constraints -Flair Furniture Company produces tables and chairs.Each table requires: 4 hours of carpentry and 2 hours of painting.Each chair requires: 3 hours of carpentry and 1 hour of painting.Available production capacity: 240 hours of carpentry time and 100 hours of painting time.Due to existing inventory of chairs, Flair is to make no more than 60 new chairs.Each table sold results in $7 profit, while each chair produced yields $5 profit.Flair Furniture’s problem:Determine best possible combination of tables and chairs to manufacture in order to attain maximum profit.
6 Decision VariablesProblem facing Flair is to determine how many chairs and tables to produce to yield maximum profit?In Flair Furniture problem, there are two unknown entities:T - number of tables to be produced.C - number of chairs to be produced.
7 Objective Function Objective function states goal of problem. What major objective is to be solved?Maximize profit!An LP model must have a single objective function.In Flair’s problem, total profit may be expressed as: Using decision variables T and C - Maximize $7 T + $5 C($7 profit per table) x (number of tables produced) + ($5 profit per chair) x (number of chairs produced)
8 ConstraintsDenote conditions that prevent one from selecting any specific subjective value for decision variables.In Flair Furniture’s problem, there are three restrictions on solution.Restrictions 1 and 2 have to do with available carpentry and painting times, respectively.Restriction 3 is concerned with upper limit on number of chairs.
9 Constraints There are 240 carpentry hours available. 4T + 3C < 240 There are 100 painting hours available. 2T + 1C 100The marketing specified chairs limit constraint.C 60The non-negativity constraints.T 0 (number of tables produced is 0)C 0 (number of chairs produced is 0)
10 2.4 Graphical Representation of Constraints Complete LP model for flair’s case:Maximize profit = $7T + $5C (objective function)Subject to constraints -4T + 3C (carpentry constraint)2T + 1C (painting constraint)C (chairs limit constraint)T (non-negativity constraint on tables)C (non-negativity constraint on chairs)
11 Isoprofit Line Solution Method Write objective function: $7 T + $5 C = ZSelect any arbitrary value for Z.For example, one may choose a profit ( Z ) of $210.Z is written as: $7 T + $5 C = $210.To plot this profit line: Set T = 0 and solve objective function for C.Let T = 0, then $7(0) + $5C = $210, or C = 42.Set C = 0 and solve objective function for T.Let C = 0, then $7T + $5(0) = $210, or T = 30.
12 Isoprofit Line Solution Method Isoprofit lines ($350, $280, $210) are all parallel.
13 Optimal SolutionOptimal Solution: Corner Point 4: T=30 (tables) and C=40 (chairs) with $410 profit
15 2.5 A Minimization LP Problem Many LP problems involve minimizing objective such as cost instead of maximizing profit function.Examples:Restaurant may wish to develop work schedule to meet staffing needs while minimizing total number of employees.Manufacturer may seek to distribute its products from several factories to its many regional warehouses in such a way as to minimize total shipping costs.Hospital may want to provide its patients with a daily meal plan that meets certain nutritional standards while minimizing food purchase costs.
16 Example of a Two Variable Minimization LP Problem Holiday Meal Turkey RanchBuy two brands of feed for good, low-cost diet for turkeys.Each feed may contain three nutritional ingredients (protein, vitamin, and iron).One pound of Brand A contains:5 units of protein,4 units of vitamin, and0.5 units of iron.One pound of Brand B contains:10 units of protein,3 units of vitamins, and0 units of iron.
17 Example of Two Variable Minimization Linear Programming Problem Holiday Meal Turkey RanchBrand A feed costs ranch $0.02 per pound, while Brand B feed costs $0.03 per pound.Ranch owner would like lowest-cost diet that meets minimum monthly intake requirements for each nutritional ingredient.
18 Summary of Holiday Meal Turkey Ranch Data Composition of Each Pound of Feed (Oz)Minimum Monthly Requirement Per Turkey (Oz)IngredientBrand ABrand BProtein51090Vitamin4348Iron1 ½Cost per pound2 cents3 cents
19 Formulation of LP Problem: Minimize cost (in cents) = 2A + 3BSubject to:5A + 10B 90 (protein constraint)4A + 3B 48 (vitamin constraint)½A 1½ (iron constraint)A 0, B 0 (nonnegativity constraint)Where:A denotes number of pounds of Brand A feed, and B denote number of pounds of Brand B feed.
20 Corner Point Solution Method Point 1 - coordinates (A = 3, B = 12)cost of 2(3) + 3(12) = 42 cents.Point 2 - coordinates (A = 8.4, b = 4.8)cost of 2(8.4) + 3(4.8) = 31.2 centsPoint 3 - coordinates (A = 18, B = 0)cost of (2)(18) + (3)(0) = 36 cents.Optimal minimal cost solution:Corner Point 2, cost = cents
21 2.6 Summary of Graphical Solution Methods Graph each constraint equation.Identify feasible solution region, that is, area that satisfies all constraints simultaneously.Select one of two following graphical solution techniques and proceed to solve problem.Corner Point Method.Isoprofit or Isocost Method.
22 2.6 Summary of Graphical Solution Methods (Continued) Corner Point MethodDetermine coordinates of each of corner points of feasible region by visual inspection or solving equations.Compute profit or cost at each point by substitution of values of coordinates into objective function and solving for result.Identify an optimal solution as a corner point with highest profit (maximization problem), or lowest cost (minimization).
23 2.6 Summary of Graphical Solution Methods (continued) Isoprofit or Isocost MethodSelect value for profit or cost, and draw isoprofit / isocost line to reveal its slope.With a maximization problem, maintain same slope and move line up and right until it touches feasible region at one point. With minimization, move down and left until it touches only one point in feasible region.Identify optimal solution as coordinates of point touched by highest possible isoprofit line or lowest possible isocost line.Read optimal coordinates and compute optimal profit or cost.
24 2.7 Special Situations in Solving LP Problems Redundancy: A redundant constraint is constraint that does not affect feasible region in any way.Maximize Profit= 2X + 3Ysubject to:X + Y 202X + Y 30X 25X, Y 0
25 2.7 Special Situations in Solving LP Problems Infeasibility: A condition that arises when an LP problemhas no solution that satisfies all of its constraints.X + 2Y 62X + Y 8X 7
26 2.7 Special Situations in Solving LP Problems Unboundedness: Sometimes an LP model will not have a finite solutionMaximize profit= $3X + $5Ysubject to:X 5Y 10X + 2Y 10X, Y 0
27 Alternate Optimal Solutions An LP problem may have more than one optimal solution.Graphically, when the isoprofit (or isocost) line runs parallel to a constraint in problem which lies in direction in which isoprofit (or isocost) line is located.In other words, when they have same slope.
28 Example: Alternate Optimal Solutions At profit level of $12, isoprofit line will rest directly on top of first constraint line.This means that any point along line between corner points 1 and 2 provides an optimal X and Y combination.
29 2.8 Setting Up and Solving LP Problems Using Excel’s Solver Procedure of Using SolverSet up LP mathematic modelOne objective functionDecision variablesConstraintsRewriting the model for SolverRHS of constraints are numbers.Entering information in SolverThe CD-ROM that accompanies this textbook contains excel file for each example problem discussed here.
30 Using Solver in Excel Entering Data In Solver (Click Tools, Solver) Labels and TitlesParameters and CoefficientsObjective FunctionUse fixed cell addressUse “=sumproduct(range1, range2)” functionConstraints (Use “=sumproduct() function)In Solver (Click Tools, Solver)Specifying Objective FunctionChoosing Max or MinIdentifying Decision VariablesAdding ConstraintsOptions: Linear and Non-negativeAnswer Report
31 Example: Flair Furniture’s Problem Using solver to solve Flair Furniture’s problemRecall decision variables T ( Tables ) andC ( Chairs ) in Flair Furniture problem: Maximize profit = $7T + $5CSubject to constraints4T + 3C (carpentry constraint)2T + 1C (painting constraint)C (chairs limit constraint)T, C 0 (non-negativity)
32 Example: Flair Furniture’s Problem Data File: 611_render_2-1.xlsEntering Data: Program 2.1A, page 48CoefficientsExcel FunctionsIn Solver:LP model: Program 2.1 B, page 51.LP options: Program 2.1 C, page 52.Results options: Program 2.1 D, page 53.
33 Example: Flair Furniture’s Problem 1. Entering Data:TCTablesChairsNumber of UnitsProfit75ObjectiveConstraintsCarpentry Hours43<=240Painting Hours21100Chairs Limit60LHSSignRHS2. Entering Excel functions for Objective and Constraints
34 Using Solver in Excel LP model: Program 2.1 B, page 51 In Solver (Click Tools, Solver)LP model: Program 2.1 B, page 51Specifying Objective FunctionChoosing Max or MinIdentifying Decision VariablesAdding ConstraintsLP options: Program 2.1 C, page 52Options: Linear and Non-negativeResults options: Program 2.1 D, page 53Answer Report
35 SummaryIntroduced a mathematical modeling technique called linear programming (LP).LP models used to find an optimal solution to problems that have a series of constraints binding objective value.Showed how models with only two decision variables can be solved graphically.To solve LP models with numerous decision variables and constraints, one need a solution procedure such as simplex algorithm.Described how LP models can be set up on Excel and solved using Solver.