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Linear Programming

Chapter 13 Supplement. Linear Programming. Operations Management - 5 th Edition. Roberta Russell & Bernard W. Taylor, III. Lecture Outline. What is LP? Where is LP used? LP Assumptions Model Formulation Examples Solving. Linear Programming (LP).

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Linear Programming

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  1. Chapter 13 Supplement Linear Programming Operations Management - 5th Edition Roberta Russell & Bernard W. Taylor, III

  2. Lecture Outline • What is LP? • Where is LP used? • LP Assumptions • Model Formulation • Examples • Solving

  3. Linear Programming (LP) A model consisting of linear relationships representing a firm’s objective and resource constraints LP is a mathematical modeling technique used to determine a level of operational activity in order to achieve an objective, subject to restrictions called constraints

  4. Types of LP

  5. Types of LP (cont.)

  6. Types of LP (cont.)

  7. Common Elements to LP • Decision variables • Should completely describe the decisions to be made by the decision maker (DM) • Objective Function (OF) • DM wants to maximize or minimize some function of the decision variables • Constraints • Restrictions on resources such as time, money, labor, etc.

  8. LP Assumptions • OF and constraints must be linear • Proportionality • Contribution of each decision variable is proportional to the value of the decision variable • Additivity • Contribution of any variable is independent of values of other decision variables

  9. LP Assumptions, cont’d. • Divisibility • Allow both integer and non-integer (real numbers) • Certainty • All coefficients are known with certainty • We are dealing with a deterministic world

  10. LP Model Formulation (NPS format) • Indices • Domains and fundamental dimensions of the model • Examples: products, time period, region, … • Data • Input to the model – given in the problem • Indexed using indices • Convention is UPPERCASE

  11. LP Model Formulation • Decision variables • Mathematical symbols representing levels of activity of an operation • The quantities to be determined, indexed using indices • Convention is lowercase

  12. LP Model Formulation, cont’d. • Objective function (OF) • The quantity to be optimized • A linear relationship reflecting the objective of an operation • Most frequent objective of business firms is to maximize profit • Most frequent objective of individual operational units (such as a production or packaging department) is to minimize cost

  13. LP Model Formulation, cont’d. • Constraint • A linear relationship representing a restriction on decision making • Binding relationships • Attach a word description to each set of constraints • Include bounds on variables

  14. RESOURCE REQUIREMENTS Labor Clay Revenue PRODUCT (hr/unit) (lb/unit) ($/unit) Bowl 1 4 40 Mug 2 3 50 There are 40 hours of labor and 120 pounds of clay available each day Formulate this problem as a LP model LP Formulation: Example

  15. LP Formulation: Example • Indices • p = products {b, m} • Data • REVENUEp = $ revenue per unit of p made • LABORp = # of hours to produce a unit of p • CLAYp = lbs of clay to produce a unit of p • TOTLABOR = total hours available • TOTCLAY = total lbs of clay available

  16. LP Formulation: Example • Variables • nump = units of p to produce • totrev = total revenue • Objective Function • Max totrev = • Constraints (labor constraint) (clay constraint) (non-negativity)

  17. LP Formulation: Example Maximize totrev = 40 numb + 50 numm Subject to numb + 2numm40 (labor constraint) 4numb + 3numm120 (clay constraint) numb , numm0 Solution is: numb = 24 bowls numm = 8 mugs totrev = $1,360

  18. Bowls and Mugs Solved • Use OMTools > Linear Programming

  19. Another Example • Joe’s Woodcarving, Inc. manufactures two types of wooden toys: soldiers and trains. • Unlimited supply of raw material, but only 100 finishing hours and 80 carpentry hours • Demand for trains unlimited, but at most 40 soldiers can be sold each week

  20. Wooden Toys Example • Indices • ??? • Data • ??? • Variables • ??? • OF • ??? • Constraints • ???

  21. Wooden Toys Solved

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