Скачать презентацию Chapter 4 Linear Programming Applications Chapter Objectives Скачать презентацию Chapter 4 Linear Programming Applications Chapter Objectives

c61c0217dabb76c869f68f76f8a5e98a.ppt

  • Количество слайдов: 76

Chapter 4 Linear Programming Applications Chapter 4 Linear Programming Applications

Chapter Objectives – Tips for Building Good Models. – Illustrative Models • WINQSB, EXCEL, Chapter Objectives – Tips for Building Good Models. – Illustrative Models • WINQSB, EXCEL, LINDO • Optimal, Alternate Optimal, Unbounded, Infeasible Models. – “Real World” Applications.

4. 1 Building Good Linear Models of the problem in – Begin by listing 4. 1 Building Good Linear Models of the problem in – Begin by listing the details short phrases. – Define the decision variables precisely. – Determine the objective in general terms and then determine what is within the control of the decision maker (decision variables). – Add decision variables as needed during the course of formulation to accurately express objective function and constraints.

 • Writing constraints – Formulate a relationship / function in words before formulating • Writing constraints – Formulate a relationship / function in words before formulating it in mathematical terms. (Expression) [Has some relation to] (Another expression or constant) A + 2 B 2 A + B +10 – Bring the expression to the form: (Expression) [Relation] (Constant) -A +B 10 Keep units on both sides of the relation consistent

 • Additional variables / constraints – Use definitional variables to simplify problem formulation • Additional variables / constraints – Use definitional variables to simplify problem formulation (particularly useful in “percentage” constraints. ) – Specify variables with the following restrictions: • Non-negativity. • Upper and lower bound. • Integrality.

4. 2 Linear Programming Models • Several examples are presented to demonstrate application of 4. 2 Linear Programming Models • Several examples are presented to demonstrate application of linear programming in business and government. • Attention is given to the process of “model building”

4. 3 Galaxy Industries an Expansion Plan • Galaxy was very successful in its 4. 3 Galaxy Industries an Expansion Plan • Galaxy was very successful in its toy production line, and is now planning expansion at a new location. • Data – Up to 3000 pounds of plastic will be available. – Regular time available will be 40 hours. – Overtime available will be 32 hours. – One hour of overtime costs $180 more than

 • Data - continued – Two new products will be introduced: • Big • Data - continued – Two new products will be introduced: • Big Squirts • Soakers – Marketing requirements: • Space Rays should account for exactly 50% of total production. • No other product should account for more than 40% of total production. • Total production should increase to at least 1000 dozen per week.

 • Data - Continued • Management wants to maximize the Net Wee • • Data - Continued • Management wants to maximize the Net Wee • A weekly production schedule must be determ

SOLUTION – Decision Variables. • • X 1 = number of dozen Space Rays, SOLUTION – Decision Variables. • • X 1 = number of dozen Space Rays, X 2 = number of dozen Zapper, X 3 = number of dozen Big Squirts X 4 = number of dozen Soakers, to be produced weekly • X 5 = number of hours of overtime to be scheduled

– Objective Function • The total net weekly profit from the sale of products, – Objective Function • The total net weekly profit from the sale of products, less the extra cost of overtime, to be maximized. Maximize 16 X 1 +15 X 2 +20 X 3+22 X 4 - 180 X 5

 • Constraints • Constraints

The Definitional Variable X 6 will help in setting up the production mix X The Definitional Variable X 6 will help in setting up the production mix X 6 = total weekly production (in dozens ), constraints X 6 = X 1+X 2+X 3+X 4, or X 1+X 2+X 3+X 4 -X 6 =0

 • The Complete Mathematical Model • The Complete Mathematical Model

Maximum Profit Optimal Production and O/T plan Profit increase required for soaks to be Maximum Profit Optimal Production and O/T plan Profit increase required for soaks to be produced Optimal # of dozens produced

Ra nge LINDO Sensitivity Analysis of O ptim alit Ra ng eo f. F Ra nge LINDO Sensitivity Analysis of O ptim alit Ra ng eo f. F eas y ibil ity

4. 4 Jones Investment Service • Charles Jones, a financial advisor, needs to find 4. 4 Jones Investment Service • Charles Jones, a financial advisor, needs to find an optimal portfolio for Frank Baklarz, who inherited 100, 000 dollars. • Several Stocks, Bonds, and Mutual Funds are considered. • The portfolio is selected based on: –Charles evaluation of the various investments –Frank’s portfolio goals

 • Charles’ Investment Evaluation • Charles’ Investment Evaluation

 • Portfolio goals –Expected annual return of at least 7. 5%. –At least • Portfolio goals –Expected annual return of at least 7. 5%. –At least 50% invested in “A-Rated” investments. –At least 40% invested in immediately liquid investments. –No more than $30, 000 in savings accounts and certificates of deposit. • Problem summary – Determine the amount to be placed in each investment. – Minimize total overall risk. – Invest all $100, 000.

 • Variables – Xi = the amount allotted to each investment; Risk function • Variables – Xi = the amount allotted to each investment; Risk function • The Mathematical Model Minimize 25 X 3+30 X 4+20 X 5+15 X 6+65 X 7 + 40 X 8 Total investm Return ST: X 1+ X 2+ X 3+ X 4+ X 5+ X 6 + X 7+ X 8 = 100, 000. 04 X 1+. 052 X 2+. 071 X 3+. 10 X 4+. 082 X 5+. 056 X 6+. 27 X 7+. 125 X 8 ³ 7500 X 1+ X 2 + X 5 + X 7 ³ 50, 000 A - Rate Liquid Savings/Certifica

 • Recommendations: – Invest $17, 333 in savings. – Invest $12, 667 in • Recommendations: – Invest $17, 333 in savings. – Invest $12, 667 in a certificate of deposit. – Invest $22, 667 in Arkansas Reit. – Invest $47, 333 in Bedrock Insurance Annuity. – This gives an overall risk value of 1626. 667

4. 5 St. Joseph Public Utility Commission • St. Joseph Public Utility Commission needs 4. 5 St. Joseph Public Utility Commission • St. Joseph Public Utility Commission needs to inspect and report utility problems in a flood area. Experts are Three types of inspection assigned will be conducted: to inspect: • Homes • Offices • Plants • Electrical • Gas • Insulation

 • Problem Summary – St. Joseph needs to determine the number of homes, • Problem Summary – St. Joseph needs to determine the number of homes, office complexes, and plants to be inspected. – The objective is to maximize the total number of structures inspected. – Resources • • At most, 120 hours can be allocated for Data electrical inspections. – Requirementshours can be allocated for gas • At most 80 • Atinspection. offices and eight plants must be least eight inspected. 100 consulting hours can be allocated • At most

 • Variables – X 1, X 2, X 3 = number of homes, • Variables – X 1, X 2, X 3 = number of homes, office complexes, and industrial plants to be inspected, respectively. – X 4 = a definitional variable: The total number of structures to be inspected. • The Complete Mathematical Model Maximize X 4 ST: X 1 + X 2 + X 3 - X 4 = ³ X 2 X 3 Plants) X 1 -. 6 X 4 least 60% Homes) 2 X 1 + 4 X 2 + 6 X 3 (Electrical) 0 (Definitional) 8 (Min. Office) ³ 8 (Min. ³ 0 £ 120 (At

Infeasible Solution The problem is properly formulated, but there is no feasible solution to Infeasible Solution The problem is properly formulated, but there is no feasible solution to the problem. Here is why. . .

The revised problem The revised problem

4. 6 Euromerica Liquor • Euromerica Liquors purchases and distributes a number of wines 4. 6 Euromerica Liquor • Euromerica Liquors purchases and distributes a number of wines to retailers. • There are four different wines to be ordered. • Requirements – Order at least 800 of each type.

 • Data: • Management wishes to determine how many b of each type • Data: • Management wishes to determine how many b of each type to order. • The objective is to maximize the total profit fro purchasing and distributing the wine bottles.

SOLUTION • Variables – X 1 = bottles of Napa Gold purchased – X SOLUTION • Variables – X 1 = bottles of Napa Gold purchased – X 2 = bottles of Cayuga Lake purchased – X 3 = bottles of Seine Soir purchased – X 4 = bottles of Bella purchased.

 • The Mathematical Model Maximize 1. 75 X 1 + 1. 50 X • The Mathematical Model Maximize 1. 75 X 1 + 1. 50 X 2 + 3 X 3 + 2 X 4 ST: X 1 > 800 $4. 25 - $2. 50 = $1. 75 X 2 > 800 X 3 > 800 X 4 > 800 X 1 + X 2 - 2 X 3 - 2 X 4> 0 [Domestic wines] [ Are at least] [ Twice the imported wines] X 1, X 2, X 3, X 4 > 0

CAN EUROMERICA An unbounded MAKE AN INFINITE solution PROFIT ? ! CAN EUROMERICA An unbounded MAKE AN INFINITE solution PROFIT ? !

– The revised model: Maximize 1. 75 X 1+1. 50 X 2+3 X 3+ – The revised model: Maximize 1. 75 X 1+1. 50 X 2+3 X 3+ 2 X 4 ST: > > X 2 X 3 > X 4> X 1 + X 2 -2 X 3 - 2 X 4 > 2. 50 X 1+ 3. 00 X 2+5 X 3+ 4 X 4 < 28000(Budget) < X 1 < X 3 < Seine) > X 1 800 800 0 3600 (Napa) 2400 (

Revised Solution -Excel Revised Solution -Excel

Solution Summary Solution Summary

4. 7 Vertex Software, Inc. • Vertex Software has developed a new software product, 4. 7 Vertex Software, Inc. • Vertex Software has developed a new software product, LUMBER 2000. • A marketing plan for this product is to be developed for the next quarter. – The product will be promoted using black and white and colored full page ads. – Three publications are considered: • Building Today • Lumber Weekly • Timber World

 • Requirements – A maximum of one ad should be placed in any • Requirements – A maximum of one ad should be placed in any one issue of any of the publications during the quarter. – At least 50 full-page ads should appear during the quarter. – at least 8 color ads should appear during the quarter. – One ad should appear in each issue of Timber World. – At least 4 weeks of advertising should be placed in each of the Building Today and Lumber Weekly

 • Data Circulation and advertising costs Publication Frequency Circulat. Cost/Ad Building Today $800 • Data Circulation and advertising costs Publication Frequency Circulat. Cost/Ad Building Today $800 5 day/week Lumber Weekly 250, 000 Timber World Monthly B&W pg. : $2000 Key reader attitudes Readership Attribute Timber 400, 000 Full pg. : Half pg. : $500 Only B&W pg. : $1500 Color pg. : $4000 200, 000 Color pg. : $6000 Percentage of Rating Bldng. Lumbr

SOLUTION • The requirements can be translated as follows: – Stay within a $90, SOLUTION • The requirements can be translated as follows: – Stay within a $90, 000 budget for print advertising. – Place no more than 65 (=5 x 13 weeks) and no less than 20 ads (=5 X 4 weeks) in Building Today. – Place no more than 13 and no less than 4 ads in Lumber Weekly. – Place exactly 3 ads in Timber World. – Place at least 50 full-page ads. – Place at least 8 color ads.

 • Variables – X 1 = number of full page B&W ads placed • Variables – X 1 = number of full page B&W ads placed in Building Today – X 2 = number of half page B&W ads placed in Building Today – X 3 = number of full page B&W ads placed in Lumber Weekly – X 4 = number of full page color ads placed in Lumber Weekly – X 5 = number of full page B&W ads placed in Timber World – X 6 = number of full page color ads placed in Timber World

 • The Objective Function – The objective function measures the effectiveness of the • The Objective Function – The objective function measures the effectiveness of the promotion operation (to be maximized). – It depends on the number of ads in each publication, as well as on the relative effectiveness per ad. – A special technique (external to this problem) is applied to evaluate this relative effectiveness.

 • The Mathematical Model • The Mathematical Model

LINDO optimal solution Round off to 5 Round off to 8 LINDO optimal solution Round off to 5 Round off to 8

4. 8 United Oil Company • United Oil blends two input streams of crude 4. 8 United Oil Company • United Oil blends two input streams of crude oil – Alkylate – Catalytic Cracked. • The outputs of the blending process are – Regular gasoline. – Mid-Grade gasoline. – Premium gasoline.

 • Restrictions – Weekly supply of Crude oil is limited. – Contracted weekly • Restrictions – Weekly supply of Crude oil is limited. – Contracted weekly demand for commercial gasoline has to be met. – To classify gasoline as Regular, Mid-Grade, or Premium, certain levels of octane and vapor pressure must be met. – Profit per barrel of each type of commercial

 • Data Crude Oil Product Data Blended Gasoline Data • Data Crude Oil Product Data Blended Gasoline Data

SOLUTION • Problem Summary – Determine how many barrels of catalytic cracked to blend SOLUTION • Problem Summary – Determine how many barrels of catalytic cracked to blend into regular, mid--grade and premium each week. – Maximize total weekly profit. – Remain within raw gas availability. – Meet contract requirements. – Produce gasoline blends that meet the octane

 • Decision Variables X 1, X 2, X 3 = number of barrels • Decision Variables X 1, X 2, X 3 = number of barrels of Alkylate blended each week into Regular, Mid-Grade, and Premium gas respectively. Y 1, Y 2, Y 3 = number of barrels of Catalytic Cracked blended each week into Regular Mid-Grade, and Premium respectively. R, M, P = barrels of Regular, Mid-Grade, Premium respectively, produced weekly (definitional

 • The Mathematical Model • The Mathematical Model

Solution continued next. . . Solution continued next. . .

4. 9 The Powers Group • The Power Group has decided to make a 4. 9 The Powers Group • The Power Group has decided to make a short term (three months) investment of $9 million. • Possible investments are – Two month term accounts. – Three month construction loans. – Passbook savings accounts. • Powers investment strategy is one of diversity and caution. • Powers seeks to maximize the interest

 • Requirements – Invest a total of $9 million during January. – Keep • Requirements – Invest a total of $9 million during January. – Keep at least $2 million liquid in passbook savings at all times. – Have at least $5 million liquid for the venture capital investment with Gramm Crackers on April 1. – Have a maximum of $4 million in either of the

 • Decision variables – The decision variables are the amount of new investment • Decision variables – The decision variables are the amount of new investment in each option during each month.

 • Objective function • Data – Maximize the interest earned during the quarter. • Objective function • Data – Maximize the interest earned during the quarter. are $9 million available for investment. – There – Interest earned on unmatured investments is calculated – Interest earned on each investment is: proportionately • Constraints to the term rate. • 0. 7% over two months for two month term account. – Amount invested in January – • 1. 5% over three months for three months Amount available for investment in construction loan. succeeding months – • 0. 2% for a one-month period for passbook saving Minimum investment in passbook account – Maximum investment in term account and account. construction loans.

 • Building the Mathematical Model Full interest paid by the end of March • Building the Mathematical Model Full interest paid by the end of March One half of the two month interest is paid by the end of

 • The Complete Mathematical Model Amount invested in January Minimum investment in passbook • The Complete Mathematical Model Amount invested in January Minimum investment in passbook acc Needed amount Amount available for investment in succeeding months on April 1 Maximum investment in term accounts and constructio

A friendly Excel spreadsheet presentation of the LP optimal solution A friendly Excel spreadsheet presentation of the LP optimal solution

4. 10 Mobile Cabinet Company • The Mobile Cabinet Company produces cabinets used in 4. 10 Mobile Cabinet Company • The Mobile Cabinet Company produces cabinets used in mobile and motor homes. • Summer production quotas has just been distributed to Mobile’s plants. • Some information about the plant in Lexington, Kentucky. – Part time workers can be employed. – Both regular time and over time labor can be used. – The Lexington plant can store up to 300 cabinets.

 • Management at the Lexington plant would like to devise a monthly production • Management at the Lexington plant would like to devise a monthly production schedule that will minimize their costs over the quarter. • Data

– Unit production costs = Material cost + labor cost 146 + 3(14) = – Unit production costs = Material cost + labor cost 146 + 3(14) = 188 More data * See definitions for these variables later.

 • Data - continued – Initial stock: • 25 Motor home cabinets. • • Data - continued – Initial stock: • 25 Motor home cabinets. • 20 Mobile home cabinets. – Stock required at the end of September: • At least 10 motor home cabinets. • At least 25 mobile home cabinets. – Maximum of 300 cabinets can be stored in any one month. – Inventory holding cost per month: • $6 for Motor home cabinet. • $9 for Mobile home cabinets.

 • Problem Summary • Schedule no more than the maximum • Minimize the • Problem Summary • Schedule no more than the maximum • Minimize the total cost over the quarter. number • Determine the number of Motor home of regular hours and over time hours. and Mobile home cabinets to produce each mo • Meet the minimum shipping requirements. • Store no more than 300 cabinets in any • Meet the minimum be produced in regular Octob one • How many are toin-stock requirement for time, a how month. many in overtime?

 • Decision variables: SXJ, SXA, SXS = the number of Motor home cabinets • Decision variables: SXJ, SXA, SXS = the number of Motor home cabinets store end of July, August, and September respe SYJ, SYA, SYS = the number of Mobile home cabinets store end of July, August, and September respe

 • Objective Function: Minimize total costs of production (in regular time and in • Objective Function: Minimize total costs of production (in regular time and in overtime) plus inventory costs. Min 188 XJRu+l 209 XJO +August + 218 XAO + 200 XSR + 194 XAR JJ u lyy August September Production costs of producing Motor home cabi in regular time and overtime for the month of. . . + 280 YJR ul lyy 290 YART JJu+ 315 YJO + August + 330 YAO + 300 YSR August September Production costs of producing Mobile home cab in Regular time and overtime for the month of. . . + J+ 6 SXJ u l y 9 SYJ+ August + 6 SXA September 9 SYA + 6 SXS Inventory costs of Motor homes and Mobile homes

 • Constraints: – Monthly production: Beginning inventory + monthly production = = Shipping • Constraints: – Monthly production: Beginning inventory + monthly production = = Shipping quota + Amount stored Jul 25 + XJR + XJO = 250 + SXJ y Monthly production August of Motor home cabi SXJ + XAR + XAO = 250 + SXA Septemberfor the month of. . . SXA + XSR + XSO = 150 + SXS Ju 20 + YJR + YJO = 100 + SYJ l y Monthly production August of Mobile home cab SYJ + YAR + YAO = 300 + SYA for September the month of. . . SYA + YSR + YSO = 400 + SYS

October in-stock requirement Maximum Storage Limit SXS$10 Production hours Motor home cabinet stored + October in-stock requirement Maximum Storage Limit SXS$10 Production hours Motor home cabinet stored + Motor home cabinets Mobile home cabinet stored 3 XJR + 5 YJR 2100 (July - SYS$25 Regular time) Mobile home cabinets #300 # 3 XJO + 5 YJO #1050 (July Overtime)SXJ + SYJ # 300 (July limit) SXA + SYA # 300 (August 3 XAR + 5 YAR #1500 (August limit) Regular time) SXS + SYS # 300 ( 3 XAO + 5 YAO # September limit) 750 (August -

Optimal solution - part 1 Optimal solution - part 1

Optimal solution - Continued Optimal solution - Continued

Application of Linear Models in Business and Government 4. 11 – Linear programming has Application of Linear Models in Business and Government 4. 11 – Linear programming has been successfully applied in many areas. The following is a brief list of typical applications: • Aircraft fleet assignments. • Telecommunications network expansion. • Air pollution control. • Health care applications: allocating blood to hospitals. • Bank portfolios. • Agriculture: allocating areas for planting various crops.

 • Defense / Aerospace contractors: determining production schedules and hiring and layoff policies • Defense / Aerospace contractors: determining production schedules and hiring and layoff policies at minimum cost. • Land--use planning: resolve conflicts between community concerns, developer interests, and environmental issues. • The dairy industry: determining optimal production schedules under equipment capacity limitations, demand requirements, supply restrictions, product flow, and process compression ratios. • Solid waste management. • The military strategic deployment of aircraft and sea lift forces.

 • Copyright ã 1998 John Wiley & Sons, Inc. All rights reserved. Reproduction • Copyright ã 1998 John Wiley & Sons, Inc. All rights reserved. Reproduction or translation of this work beyond that named in Section 117 of the United States Copyright Act without the express written consent of the copyright owner is unlawful. Requests for further information should be addressed to the Permissions Department, John Wiley & Sons, Inc. The purchaser may make back-up copies for his/her own use only and not for distribution or resale. The Publisher assumes no responsibility for errors, omissions, or damages, caused by 42