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BM 4808

BM 4808. Linear Programming Models: Graphical Solution. Learning Objectives. Understand 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.

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BM 4808

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  1. BM 4808 Linear Programming Models: Graphical Solution

  2. Learning Objectives • Understand 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.

  3. Introduction • Management decisions in many organizations involve trying to make most effective use of resources. • Machinery, labor, money, time, warehouse space, and raw materials. • Produce products - such as computers, automobiles, or clothing or • Provide services - such as package delivery, health services, or investment decisions. • To solve problems of resource allocation one may use mathematical programming.

  4. Mathematical Programming • Mathematical programming used for resource allocation problems. • Linear programming (LP) is most common type of mathematical programming. • One assumes all relevant input data and parameters are known with certainty in models (deterministic models).

  5. 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.

  6. Three Steps of Developing LP Problem Formulation. • Process of translating problem scenario into simple LP model framework with set of mathematical relationships. Solution. • Mathematical relationships resulting from formulation process are solved to identify optimal solution. Interpretation and What-if Analysis. • Problem solver or analyst works with manager to • Interpret results and implications of problem solution. • Investigate changes in input parameters and model variables and impact on problem solution results.

  7. Properties of a LP Model • All problems seek to maximize or minimize some quantity, usually profit or cost (called objective function). • LP models usually include restrictions, or constraints, limit degree to which one can pursue objective. • Must be alternative courses of action from which to choose. • Objective and constraints in LP problems must be expressed in terms of linear equations or inequalities.

  8. Linear Equations and Inequalities • This is a linear equation: 2A + 5B = 10 • This equation is not linear: 2A2 + 5B3 + 3AB = 10 • LP uses, in many cases, inequalities like:A + B  C or A + B  C

  9. Basic Assumptions of a LP Model • Conditions of certainty exist. • Proportionality in objective function and constraints (1 unit – 3 hours, 3 units 9 hours). • Additivity (total of all activities equals sum of individual activities). • Divisibility assumption that solutions need not necessarily be in whole numbers (integers).

  10. Formulating a LP Problem • A 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 produce • Maximize overall profit given its limited resources.

  11. 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.

  12. Decision Variables • Problem facing Flair is to determine how many chairs and tables to produce to yield maximum profit? • In Flair Furniture problem, there are twounknown entities: T - number of tablesto be produced. C - number of chairs to be produced.

  13. 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

  14. Constraints • Denote 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.

  15. Constraints • There are 240 carpentry hours available. 4T + 3C < 240 • There are 100 painting hours available.  2T + 1C  100 • The marketing specified chairs limit constraint. C 60 • The non-negativity constraints. T 0 (number of tables produced is  0) C  0 (number of chairs produced is  0)

  16. Graphical Solution of a LP With Two Variables • Primary advantage of two-variable LP models (such as flair furniture’s problem) is their solution can be graphicallyillustrated using two-dimensional graph. • Allows one to provide an intuitiveexplanation of how more complexsolution procedures work for larger LP models.

  17. Graphical Representation of Constraints Complete LP model for flair’s case: Maximize profit = $7T + $5C (objective function) Subject to constraints - 4T + 3C 240 (carpentry constraint) 2T + 1C 100 (painting constraint) C 60 (chairs limit constraint) T 0 (non-negativityconstraint on tables) C 0 (non-negativity constraint on chairs)

  18. Graphical Representation of Constraints Carpentry Time Constraint 4T + 3C 240

  19. Graphical Representation of Constraints Carpentry Time Constraint Any point below line satisfies constraint.

  20. Graphical Representation of Constraints Painting Time Constraint 2T + 1C 100 Any point on line satisfies equation: 2T + 1C= 100 (30,40) yields 100. Any point below line satisfies constraint.

  21. Graphical Representation of Constraints Chair Limit Constraint and Feasible Solution Area Feasible solution area is contained by three limiting lines

  22. Corner Point Property Property states optimal solution to LP problem will alwaysoccur at a corner point.

  23. Corner Point Solution Method From figure one knows feasible region for Flair’s problem has five corner points, namely, 1, 2, 3, 4, and 5, respectively. To find point yielding maximum profit, one finds coordinates of each corner point and computes profit level at each point.

  24. Corner Point Solution Method • Point 1 (T = 0, C = 0) profit = $7(0) + $5(0) = $0 • Point 2(T = 0, C = 60) profit = $7(0) + $5(60) = $300 • Point 3(T = 15, C = 60) profit = $7(15) + $5(60) = $405 • Point 4(T = 30, C = 40) profit = $7(30) + $5(40) = $410 • Point 5(T = 50, C = 0) profit = $7(50) + $5(0) = $350 .

  25. A Minimization LP Problem Many LP problems involve minimizingobjective such as costinstead of maximizing profit function. Examples: • Restaurant may wish to develop work schedule to meet staffing needs whileminimizing total number ofemployees. • Manufacturer may seek to distribute its products from several factories to its many regional warehouses in such a way as tominimize 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.

  26. Example of a Two Variable Minimization LP Problem Holiday Meal Turkey Ranch • Buy 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, and • 0.5 units of iron. • One pound of Brand B contains: • 10 units of protein, • 3 units of vitamins, and • 0 units of iron.

  27. Example of Two Variable Minimization Linear Programming Problem Holiday Meal Turkey Ranch • Brand 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 meetsminimum monthly intake requirementsfor each nutritionalingredient.

  28. Summary of Holiday Meal Turkey Ranch Data

  29. Formulation of LP Problem: Minimize cost (in cents) = 2A + 3B Subject 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.

  30. Graphical Solution of Holiday Meal Turkey Ranch Problem Drawing Constraints: ½A 1½ 4A + 3B 48 5A + 10B 90 Nonnegativity ConstraintA 0, B 0

  31. 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 cents • Point 3 - coordinates (A = 18, B = 0) • cost of (2)(18) + (3)(0) = 36 cents. • Optimal minimal cost solution: Corner Point 2, cost = 31.2 cents

  32. Summary of Graphical Solution Methods • Graph each constraint equation. • Identify feasible solution region, that is, area that satisfies all constraints simultaneously. • Use Corner Point Method and proceed to solve problem.

  33. Corner Point Method Determine 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).  Summary of Graphical Solution Methods (Continued)

  34. Special Situations in Solving LP Problems 1. Redundancy: A redundant constraint is constraint that does not affect feasible region in any way. Maximize Profit = 2X + 3Y Subject to: X + Y20 2X + Y 30 X 25 X, Y 0

  35. Special Situations in Solving LP Problems 2. Infeasibility: A condition that arises when an LP problem has no solution that satisfies all of its constraints. X + 2Y  6 2X + Y  8 X 7

  36. Special Situations in Solving LP Problems 3. Unboundedness: Sometimes an LP model will not have a finite solution Maximize profit = $3X + $5Y Subject to: X 5 Y 10 X + 2Y 10 X, Y 0

  37. 4. Alternate Optimal Solutions Special Situations in Solving LP Problems • An LP problem may have more than oneoptimal solution. • In other words, when they have same slope.

  38. Maximize profit = $3x + $2y Subject to: 6X + 4Y 24 X 3 X, Y  0 Example: Alternate Optimal Solutions

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