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

LP-2 Lecture Outline   What is LP?   Where is LP used?   LP Assumptions   Model Formulation   Examples Solving Solving

LP-3 A model consisting of linear relationships representing a firm’s objective and resource constraints Linear Programming (LP) 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

LP-4 Types of LP

LP-5 Types of LP (cont.)

LP-6 Types of LP (cont.)

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

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

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

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

LP-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

LP-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

LP-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

LP-14 LP Formulation: Example LaborClayRevenue PRODUCT(hr/unit)(lb/unit)($/unit) Bowl1440 Mug2350 There are 40 hours of labor and 120 pounds of clay available each day Formulate this problem as a LP model RESOURCE REQUIREMENTS

LP-15 LP Formulation: Example  Indices p = products {b, m} p = products {b, m}  Data REVENUE p = $ revenue per unit of p made REVENUE p = $ revenue per unit of p made LABOR p = # of hours to produce a unit of p LABOR p = # of hours to produce a unit of p CLAY p = lbs of clay to produce a unit of p CLAY p = lbs of clay to produce a unit of p TOTLABOR = total hours available TOTLABOR = total hours available TOTCLAY = total lbs of clay available TOTCLAY = total lbs of clay available

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

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

LP-18 Bowls and Mugs Solved  Use OMTools > Linear Programming

LP-19 Another Example  Joe’s Woodcarving, Inc. manufactures two types of wooden toys: soldiers and trains. Sale $ Raw Cost Labor / Overhead Finishing Labor Carpentry Labor Soldier$27$10$14 2 hr 1 hr Train$21$9$10  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

LP-20 Wooden Toys Example  Indices ??? ???  Data ??? ???  Variables ??? ???  OF ??? ???  Constraints ??? ???

LP-21 Wooden Toys Solved