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Chapter 8 Slides by John Loucks St. Edward’s University ‹#› © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. ‹#› ‹#› Exam Time Management Three hour exam 3 Cases Allocate time for each section Allow time at the end to read over your answers Never leave early … there is always more that you can write 2 ‹#› © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. ‹#› Exam Format The final Exam is worth 40 %. The exam paper is made of the following: Case A: Linear programming and graphical presentation Objective function and constraints Graphical representation of optimal solution and feasible area 3 questions 10,20,10 marks. Total 40 marks Case B: Sensitivity analysis Deacreas/increase in objective function and its impact on optimal solution marks 3 ‹#› © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. ‹#› Binding/non-binding constraints Shadow price Allowable increase/decrease 5 questions 6 marks each. Total 30 marks Case C:Network model Develop the network model for a big question 1 question 30 marks ‹#› © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. ‹#› Mathematical Models Objective Function – a mathematical expression that describes the problem’s objective, such as maximizing profit or minimizing cost Consider a simple production problem. Suppose x denotes the number of units produced and sold each week, and the firm’s objective is to maximize total weekly profit. With a profit of $10 per unit, the objective function is 10x. ‹#› © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. ‹#› Mathematical Models Constraints – a set of restrictions or limitations, such as production capacities To continue our example, a production capacity constraint would be necessary if, for instance, 5 hours are required to produce each unit and only 40 hours are available per week. The production capacity constraint is given by 5x < 40.="" the="" value="" of="" 5x="" is="" the="" total="" time="" required="" to="" produce="" x="" units;="" the="" symbol="" indicates="" that="" the="" production="" time="" required="" must="" be="" less="" than="" or="" equal="" to="" the="" 40="" hours="" available.="" ‹#›="" ©="" 2016="" cengage="" learning.="" all="" rights="" reserved.="" may="" not="" be="" copied,="" scanned,="" or="" duplicated,="" in="" whole="" or="" in="" part,="" except="" for="" use="" as="" permitted="" in="" a="" license="" distributed="" with="" a="" certain="" product="" or="" service="" or="" otherwise="" on="" a="" password-protected="" website="" for="" classroom="" use.="" ‹#›="" ‹#›="" mathematical="" models="" uncontrollable="" inputs="" –="" environmental="" factors="" that="" are="" not="" under="" the="" control="" of="" the="" decision="" maker="" in="" the="" preceding="" mathematical="" model,="" the="" profit="" per="" unit="" ($10),="" the="" production="" time="" per="" unit="" (5="" hours),="" and="" the="" production="" capacity="" (40="" hours)="" are="" environmental="" factors="" not="" under="" the="" control="" of="" the="" manager="" or="" decision="" maker.="" ‹#›="" ©="" 2016="" cengage="" learning.="" all="" rights="" reserved.="" may="" not="" be="" copied,="" scanned,="" or="" duplicated,="" in="" whole="" or="" in="" part,="" except="" for="" use="" as="" permitted="" in="" a="" license="" distributed="" with="" a="" certain="" product="" or="" service="" or="" otherwise="" on="" a="" password-protected="" website="" for="" classroom="" use.="" ‹#›="" ‹#›="" decision="" variables="" –="" controllable="" inputs;="" decision="" alternatives="" specified="" by="" the="" decision="" maker,="" such="" as="" the="" number="" of="" units="" of="" a="" product="" to="" produce.="" in="" the="" preceding="" mathematical="" model,="" the="" production="" quantity="" x="" is="" the="" controllable="" input="" to="" the="" model.="" mathematical="" models="" ‹#›="" ©="" 2016="" cengage="" learning.="" all="" rights="" reserved.="" may="" not="" be="" copied,="" scanned,="" or="" duplicated,="" in="" whole="" or="" in="" part,="" except="" for="" use="" as="" permitted="" in="" a="" license="" distributed="" with="" a="" certain="" product="" or="" service="" or="" otherwise="" on="" a="" password-protected="" website="" for="" classroom="" use.="" ‹#›="" ‹#›="" a="" complete="" mathematical="" model="" for="" our="" simple="" production="" problem="" is:="" maximize="" 10x="" (objective="" function)="" subject="" to:="" 5x="">< 40="" (constraint)="" x=""> 0 (constraint) [The second constraint reflects the fact that it is not possible to manufacture a negative number of units.] Mathematical Models ‹#› © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. ‹#› ‹#› Mathematical Models Uncontrollable Inputs (Environmental Factors) Controllable Inputs (Decision Variables) Output (Projected Results) Mathematical Model ‹#› © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. ‹#› Example: Iron Works, Inc. Iron Works, Inc. manufactures two products and just received this month's allocation of b pounds of steel. It takes a1 pounds of steel to make a unit of product 1 and a2 pounds of steel to make a unit of product 2. Let x1 and x2 denote this month's production level of product 1 and product 2, respectively. Denote by p1 and p2 the unit profits for products 1 and 2, respectively. Iron Works has a contract calling for at least m units of product 1 this month. At most u units of product 2 may be produced monthly. ‹#› © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. ‹#› Example: Iron Works, Inc. Mathematical Model The total monthly profit = (profit per unit of product 1) x (monthly production of product 1) + (profit per unit of product 2) x (monthly production of product 2) = p1x1 + p2x2 We want to maximize total monthly profit: Max p1x1 + p2x2 ‹#› © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. ‹#› Example: Iron Works, Inc. Mathematical Model (continued) The total amount of steel used during monthly production equals: (steel required per unit of product 1) x (monthly production of product 1) + (steel required per unit of product 2) x (monthly production of product 2) = a1x1 + a2x2 This quantity must be less than or equal to the allocated b pounds of steel: a1x1 + a2x2 < b="" ‹#›="" ©="" 2016="" cengage="" learning.="" all="" rights="" reserved.="" may="" not="" be="" copied,="" scanned,="" or="" duplicated,="" in="" whole="" or="" in="" part,="" except="" for="" use="" as="" permitted="" in="" a="" license="" distributed="" with="" a="" certain="" product="" or="" service="" or="" otherwise="" on="" a="" password-protected="" website="" for="" classroom="" use.="" ‹#›="" example:="" iron="" works,="" inc.="" mathematical="" model="" (continued)="" the="" monthly="" production="" level="" of="" product="" 1="" must="" be="" greater="" than="" or="" equal="" to="" m="" :="" x1=""> m The monthly production level of product 2 must be less than or equal to u : x2 < u="" however,="" the="" production="" level="" for="" product="" 2="" cannot="" be="" negative:="" x2=""> 0 ‹#› © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. ‹#› Example: Iron Works, Inc. Mathematical Model Summary Max p1x1 + p2x2 s.t. a1x1 + a2x2 < b="" x1=""> m