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1 Chapter 3 Applications of Linear and Integer Programming Models - 1 Slide 2 2 3.1 The Evolution of Linear Programming Models in Business and Government Many examples are presented that demonstrate the successful application of linear and integer programming. Our goals are to: Examine potential application areas Develop good modeling skills Illustrate the use of spreadsheets to generate results Interpret properly and analyze spreadsheet results Slide 3 3 Examples of Linear Programming Models in Business and Government An optimal portfolio Optimal scheduling of personnel An optimal blend of raw crude oils A minimized cost diet An operation and shipping pattern The optimal production levels Slide 4 4 The optimal assignment of fleets to flights How to best expand a communication network An efficient air-pollution control system An agricultural resource allocation plan The set of public projects to select Examples of Linear Programming Models in Business and Government Slide 5 5 Three important factors may affect the successful process of building good models: Familiarity Simplicity Clarity 3.2 Building Good Linear and Integer Programming Models Slide 6 6 Many times the use of summation variables (representing the sum of all or of part of the decision variables) along with the summation constraints associated with them, may simplify models formulation. See the following example. Summation Variables / Constraints Slide 7 7 Example Three television models are to be produced. Each model uses 2, 3, and 4 pounds of plastic respectively. 7000 pounds of plastic are available. No model should exceed 40% of the total quantity produced. The profit per set is $23, $34, and $45 respectively. Find the production plan that maximizes the profit. Slide 8 8 Solution Max 23X 1 + 34X 2 + 45X 3 S.T. 2X 1 +3X 2 + 4X 3 Without summation variables X 1.4(X 1 + X 2 + X 3 ) X 2.4(X 1 + X 2 + X 3 ) X 3.4(X 1 + X 2 + X 3 ) X 1, X 2, X 3 With summation variables X 1 + X 2 + X 3 = X 4 X 1.4X 4 X 2.4X 4 X 3.4X 4 X 1, X 2, X 3, X 4 Summation Variables / Constraints Slide 9 9 Summation variables/constraints TV production spreadsheet =SUM(B2:D2) Total production Decision Variables Percentag e Constraint s Plastic Constraint Slide 10 10 Bring the expression to the form: (Expression) [Relation] (Constant) A checklist for building linear models A + 2B 2A + B +10 - A + B 10 Formulate a relationship / function in words before formulating it in mathematical terms. (Expression) [Has some relation to] (Another expression or constant) Slide 11 11 A checklist for building linear models Keep the units on both sides of the expressions consistent Use summation variables when appropriate Indicate which variables are Non-negative or Free Integers Binary Slide 12 12 The modeling of real problems is illustrated in this section. Examples include: Production Purchasing Finance Cash flow accounting 3.4 Applications of Linear Programming Models Slide 13 13 The modeling of real problems is illustrated in this section. Examples include: Production Purchasing Finance Cash flow accounting Emphasis is given to: Various application area, Model development, Spreadsheet design, Analysis and interpretation of the output. 3.4 Applications of Linear Programming Models Slide 14 14 These models can assist managers in making decision regarding the efficient utilization of scarce resource. Applications include: Determining production levels Scheduling shifts Using overtime The cost effectiveness of adding resources 3.4.1 Production Scheduling Models Slide 15 15 Galaxy Industries Expansion Plan Galaxy Industries is planning to increase its production and include two new products 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 one hour of regular time. Slide 16 16 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. Galaxy Industries Expansion Plan The old products are: Space rays Zappers Slide 17 17 Data - Continued PlasticProduction ProductProfit(lbs) Time (min) Space Rays $1623 Zappers $1514 Big Squirts $2035 Soakers $2246 PlasticProduction ProductProfit(lbs) Time (min) Space Rays $1623 Zappers $1514 Big Squirts $2035 Soakers $2246 Management wants to maximize the Net Weekly Profit. A weekly production schedule must be determined. Galaxy Industries Expansion Plan Slide 18 18 Decision Variables. X 1 = number of dozen Space Rays, to be produced weekly X 2 = number of dozen Zapper, to be produced weekly X 3 = number of dozen Big Squirts, to be produced weekly X 4 = number of dozen Soakers, to be produced weekly X 5 = number of hours of overtime to be scheduled weekly Galaxy Industries Expansion Plan Solution Slide 19 19 Objective Function The total net weekly profit from the sale of products, less the extra cost of overtime, to be maximized. One hour of overtime costs $180 more than one hour of regular time. Maximize 16X 1 +15X 2 +20X 3 +22X 4 - 180X 5 Galaxy Industries Expansion Plan Solution Slide 20 20 Constraints Galaxy Industries Expansion Plan Solution Slide 21 21 Introduce the summation Variable X 6, that helps in setting up the production mix constraints X6 = total weekly production (in dozens ), X6 = X1+X2+X3+X4, or X1+X2+X3+X4 -X6 =0 Galaxy Industries Expansion Plan Solution Slide 22 22 Galaxy Industries Expansion Plan Slide 23 23 The Complete Mathematical Model Max 16X 1 + 15X 2 + 20X 3 + 22X 4 180X 5 S.T. 2X 1 + 1X 2 + 3X 3 + 4X 4 3X 1 + 4X 2 + 5X 3 + 6X 4 60X 5 2400 X 5 32 1X 2 200 X 1 + X 2 +X 3 + X 4 - - X 6 = 0 X 1 -.5X 6 =0 X 2 -.4X 6 =0 X 3 -.4X 6 = 0 X 4 -.4X 6 =0 X 6 1000 X j are non-negative Galaxy Industries Expansion Plan Slide 24 24 =SUM(B4:E4) Percentage Constraints SUMPRODUCT($B$4:$F$4,B6,F6) Drag to G7:G10 Galaxy Industries Expansion Plan Slide 25 25 Galaxy Industries Expansion Plan D E F G H 7 8 9 10 11 16 19 21 22 23 24 Slide 26 26 Galaxy Industries Expansion Plan ModeloProduo (Dzias)Lucro Bruto% Total Space Rays5659,04050 Zappers2003,00017.7 Big Squirts3657,30032.3 Soakers000 Total113019,340 $ Horas-extras5,760 Lucro Lquido13,580 Foram utilizados todos os minutos de hora regular (2,400) e hora extra (32) Restries binding; 2,425 das 3,000 libras disponveis de plstico foram utilizadas Restrio nonbinding; Total produzido (1130) excedeu o mnimo em 130 dzias Restrio nonbinding. Slide 27 27 Galaxy Industries Expansion Plan Intervalos de Otimalidade ModeloLucro/DziaLucro MnimoLucro Mximo Space Rays164 = 16 - 1220 = 16 + 4 Zappers15No h mnimo15.5 = 15 + 0.5 Big Squirts2019.43 = 20 - 0.57No h Mximo Soakers22No h mnimo24.5 = 22 + 2.5 Slide 28 28 Galaxy Industries Expansion Plan - Outras constataes Soluo permanece tima enquanto custo H-E < $270 (H11); Para produzir Soakers seu lucro dever aumentar de $2.5 (E10); Horas Regulares adicionais melhoram o lucro total em $4.5/min ou $270/h (E21), sem passar de 920 minutos ou 15 1/3 h (G21); Horas-Extras adicionais (ou a menos) acima ou abaixo de 32 melhoram (ou pioram) o lucro total de $90 (E22), se total de H-E ficar no intervalo de 23 1/3 = 32 8 2/3 (H22) e 47 1/3 = 32 + 15 1/3 (G22); Cada dzia de Zapper adicional ao contrato, at 280 dzias (G23), subtrai $0.5 (E23) do lucro total. Redues no contrato melhoram o lucro total em $0.5/dzia desde que no excedam 89.23 dzias (H23); Cada Dzia de Space Rays que seja permitido produzir acima de 50% do total produzido melhora o lucro total em $5 (E19), at 486 2/3 dzias a mais que 50% (G19). Slide 29 29 Galaxy Industries Expansion Plan Recomendaes ao Gerente Autorizar mais H-E; Aumentar a % de Space Rays produzidos; Reduzir o contrato dos Soakers ou melhorar o lucro unitrio Slide 30 30 3.4.2 Portfolio Models Portfolio models are usually designed to: Maximized return on investment, Minimize risk. Factors considered include: Liquidity requirements, Long and short term investment goals, Funds available. Slide 31 31 Jones Investment Charles Evaluation Slide 32 32 Jones Investment 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. Meet the investor goals (diversify). Slide 33 33 Variables X i = the amount allotted to each investment; The Mathematical Model Minimize 25X 3 +30X 4 +20X 5 +15X 6 +65X 7 + 40X 8 ST: X 1 + X 2 + X 3 + X 4 + X 5 + X 6 + X 7 + X 8 = 100,000.04X 1 +.052X 2 +.071X 3 +.10X 4 +.082X 5 +.056X 6 +.27X 7 +.125X 8 7500 X 1 + X 2 + X 5 + X 7 50,000 X 1 + X 3 + X 4 + X 7 40,000 X 1 +X 2 30,000 All the variables are non-negative Risk function Total investment Return A - Rate Liquid Savings/ Certificate Jones Investment Solution Slide 34 34 =SUM(B5:B12) =SUMPRODUCT(B5,B12,C5:C12) =SUMIF(E5:E12,"A",B5:B12) =SUMIF(F5:F12,"Immediate",B5:B12) =SUMPRODUCT(B5,B12,D5:D12) =B5+B6 Jones Investment - Spreadsheet Slide 35 35 Jones Investment - Spreadsheet Slide 36 36 Jones Investment - Spreadsheet Slide 37 37 Jones Investment Restries Binding: Retorno mdio anual de $7,500 Aplicar mnimo de $40,000 em Investimentos com liquidez Aplicar no mximo $30,000 em poupana e certificado de depsito bancrio Restrio Nonbinding: Aplicar no mnimo $50,000 em investimentos com ranking A Slide 38 38 Jones Investment Recomendaes: Aplicar $17,333 em poupana Aplicar $12,667 em Certificados de Depsito Aplicar $22,667 em Arkansas REIT Aplicar $47,333 em Bedrock Insurance Annuity Risco total = 1,626,667 ou seja fator mdio de risco de 16.27 por dolar aplicado Slide 3

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