Linear Programming

Chapter 13 Supplement Linear Programming Operations Management - 5th 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) 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

Types of LP

Types of LP (cont.)

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.

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

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

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

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

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

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

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

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

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

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

Bowls and Mugs Solved • Use OMTools > Linear Programming

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

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

Wooden Toys Solved

Linear Programming