Chapter 4 Linear Programming Applications
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 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 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 (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 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 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 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 • A weekly production schedule must be determ
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, 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
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
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 eas y ibil ity
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
• 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 • 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 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 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, 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, 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 the problem. Here is why. . .
The revised problem
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 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 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 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 ? !
– 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
Solution Summary
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 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 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, 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 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 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
LINDO optimal solution Round off to 5 Round off to 8
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 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
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 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
Solution continued next. . .
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 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 in each option during each month.
• 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 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 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
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 schedule that will minimize their costs over the quarter. • Data
– 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. • 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 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 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 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 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 + 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 - Continued
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 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.
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