Fluent Essays

Work individually on solving the remaining application cases Media Selection (Case 2), Portfolio Selection (Case 3), and Workforce Assignment (Case 4) using Excel Solver.  Submit a SINGLE Excel file and solve every case in a separate sheet. Marketing Applications- Media Selection One application of linear programming in marketing is media selection. LP can be used to help marketing managers allocate a fixed budget to various advertising media. The objective is to maximize reach, frequency, and quality of exposure. Restrictions on the allowable allocation usually arise during consideration of company policy, contract requirements, and media availability. Example: SMM Company recently developed a new instant salad machine, has $282,000 to spend on advertising. The product is to be initially test marketed in the Dallas area. The money is to be spent on a TV advertising blitz during one weekend (Friday, Saturday, and Sunday) in November. The three options available are: daytime advertising, evening news advertising, and Sunday game-time advertising. A mixture of one-minute TV spots is desired. Here is the information regarding the estimated audience reached and cost per ad type: Ad Type Estimated Audience Reached with each Ad Cost per Ad Daytime 3,000 $5,000 Evening News 4,000 $7,000 Sunday Game 75,000 $100,000 SMM wants to take out at least one ad of each type (daytime, evening-news, and game-time). Further, there are only two game-time ad spots available. There are ten daytime spots and six evening news spots available daily. SMM wants to have at least 5 ads per day, but spend no more than $50,000 on Friday and no more than $75,000 on Saturday. Considering the above limitations, write a linear programming formulation to help SMM determine the number for each ad type on the weekend so as to maximize the total audience reached. Hint 1: You need to define 7 decision variables for daytime and evening news ad on Friday, Saturday and Sunday as well as a decision variable for Sunday game ad. Hint 2: Maximize the total audience reached: Max (audience reached per ad of each type) x (number of ads used of each type) Hint 3: You need to define A SET of constraints for each of the followings: - Take out at least one ad of each type - Ten daytime spots available daily - Six evening news spots available daily - At least 5 ads per day Hint 4: Finally, you need to define a constraint for each of the followings: - Only two Sunday game-time ad spots available - Spend no more than $50,000 on Friday - Spend no more than $75,000 on Saturday - Spend no more than $282,000 in total Hint 5: Can you make negative number of ads? Marketing Application-Media Selection 1. Define the decision variables DFR = number of daytime ads on Friday DSA = number of daytime ads on Saturday DSU = number of daytime ads on Sunday EFR = number of evening ads on Friday ESA = number of evening ads on Saturday ESU = number of evening ads on Sunday GSU = number of game-time ads on Sunday 2. Define the Objective Function Maximize the total audience reached: Max (audience reached per ad of each type) x (number of ads used of each type) Max 3000DFR +3000DSA +3000DSU +4000EFR+4000ESA +4000ESU +75000GSU 3. Define the Constraints Take out at least one ad of each type: (1) DFR + DSA + DSU > 1 (2) EFR + ESA + ESU > 1 (3) GSU > 1 Ten daytime spots available: (4) DFR < 10 (5) DSA < 10 (6) DSU < 10 Six evening news spots available: (7) EFR < 6 (8) ESA < 6 (9) ESU < 6 At least 5 ads per day: (10) DFR + EFR > 5 (11) DSA + ESA > 5 (12) DSU + ESU + GSU > 5 Only two Sunday game-time ad spots available: Marketing Application-Media Selection (13) GSU < 2 Spend no more than $50,000 on Friday: (14) 5000DFR + 7000EFR < 50000 Spend no more than $75,000 on Saturday: (15) 5000DSA + 7000ESA < 75000 Spend no more than $282,000 in total: (16) 5000DFR + 5000DSA + 5000DSU + 7000EFR + 7000ESA + 7000ESU + 100000GSU7 < 282000 Non-negativity: DFR, DSA, DSU, EFR, ESA, ESU, GSU > 0 Financial Applications- Portfolio Selection LP can be used in financial decision-making that involves capital budgeting, make-or-buy, asset allocation, portfolio selection, financial planning, and more. Portfolio selection problems involve choosing specific investments – for example, stocks and bonds – from a variety of investment alternatives. This type of problem is faced by managers of banks, mutual funds, and insurance companies. The objective function usually is maximization of expected return or minimization of risk. Example: Winslow Savings has $20 million available for investment. It wishes to invest over the next four months in such a way that it will maximize the total interest earned over the four-month period as well as have at least $10 million available at the start of the fifth month for a high rise building venture in which it will be participating. For the time being, Winslow wishes to invest only in 2-month government bonds (earning 2% over the 2- month period) and 3-month construction loans (earning 6% over the 3-month period). Each of these is available each month for investment. Funds not invested in these two investments are liquid and earn 0.75% per month when invested locally. Formulate a linear program that will help Winslow Savings determine how to invest over the next four months. Hint 1: You need 12 decision variables for 3 types of investment (government, construction, local) in 4 months. Use index for the month. Hint 2: Objective function is: Maximize total interest earned in the 4-month period: Max (interest rate on investment) x (amount invested) Hint 3: one constraint for each of the followings: 1. Month 1's total investment limited to $20 million 2. Month 2's total investment limited to principle and interest invested locally in Month 1 3. Month 3's total investment amount limited to principle and interest invested in government bonds in Month 1 and locally invested in Month 2 4. Month 4's total investment limited to principle and interest invested in construction loans in Month 1, government bonds in Month 2, and locally invested in Month 3 5. $10 million must be available at start of Month 5 Hint 4: Can you invest negative amount of money? Financial Applications- Portfolio Selection 1. Define the Decision Variables 𝐺𝑖= amount of new investment in government bonds in month i (for i = 1, 2, 3, 4) 𝐶𝑖= amount of new investment in construction loans in month i (for i = 1, 2, 3, 4) 𝐿𝑖= amount invested locally in month i, (for i = 1, 2, 3, 4) 2. Define the Objective Function Maximize total interest earned in the 4-month period: Max (interest rate on investment) x (amount invested) Max .02G1 + .02G2 + .02G3 + .02G4 + .06C1 + .06C2 + .06C3 + .06C4 + .0075L1 + .0075L2 + .0075L3 + .0075L4 3. Define the Constraints Month 1's total investment limited to $20 million: (1) G1 + C1 + L1 = 20,000,000 Month 2's total investment limited to principle and interest invested locally in Month 1: (2) G2 + C2 + L2 = 1.0075L1 or G2 + C2 - 1.0075L1 + L2 = 0 Month 3's total investment amount limited to principle and interest invested in government bonds in Month 1 and locally invested in Month 2: (3) G3 + C3 + L3 = 1.02G1 + 1.0075L2 or - 1.02G1 + G3 + C3 - 1.0075L2 + L3 = 0 Month 4's total investment limited to principle and interest invested in construction loans in Month 1, government bonds in Month 2, and locally invested in Month 3: (4) G4 + C4 + L4 = 1.06C1 + 1.02G2 + 1.0075L3 or - 1.02G2 + G4 - 1.06C1 + C4 - 1.0075L3 + L4 = 0 $10 million must be available at start of Month 5: (5) 1.06C2 + 1.02G3 + 1.0075L4 > 10,000,000 Non-negativity: Gi, Ci, Li > 0 for i = 1, 2, 3, 4 Operations Management Application- Workforce Assignment LP can be used in operations management to aid in decision-making about product mix, production scheduling, staffing, inventory control, capacity planning, and other issues. An important application of LP is multi-period planning such as production scheduling. Usually the objective is to establish an efficient, low-cost production schedule for one or more products over several time periods. Typical constraints include limitations on production capacity, labor capacity, storage space, and more. Example: National Wing Company (NWC) is gearing up for the new B-48 contract. NWC has agreed to produce 20 wings in April, 24 in May, and 30 in June. Currently, NWC has 100 fully qualified workers. A fully qualified worker can either be placed in production or can train new recruits. A new recruit can be trained to be an apprentice in one month. After another month, the apprentice becomes a qualified worker. Each trainer can train two recruits. The production rate and salary per employee type is listed below. Type of Employee Production Rate (wings/month) Wage per month Production 0.6 $3,000 Trainer 0.3 $3,300 Apprentice 0.4 $2,600 Recruit 0.05 $2,200 At the end of June, NWC wishes to have no recruits or apprentices, but have at least 140 full-time workers. Considering the above limitations, write a linear programming model to help NWC determine the number of employees in each type at each month (April, May, June) so as to minimize total wedge cost. Hint 1: you need to define 9 decision variables; - 3 decision variables for the number of producers in April, May, and June, - 2 decision variables for the number of trainers in April and May (Why not June?) - 2 decision variables for the number of apprentices in May and June (Why not April?) - 2 decision variables for the number of recruits in April and May (Why not June?) Hint 2: The objective function is: Minimize total wage cost for producers, trainers, apprentices, and recruits for April, May, and June Hint 3: Constraints: - Total production in Month 1 (April) must equal or exceed contract for Month 1 - Total production in Months 1-2 (April, May) must equal or exceed total contracts for Months 1-2 - Total production in Months 1-3 (April, May, June) must equal or exceed total contracts for Months 1-3 - In April there are 100 employees that can be producers or trainers: - At the end of June, there are to be at least 140 employees - 2 constraints for each trainer can train at most two recruits in April and May - 2 constraints for the number of producers and trainers in a month must equal the number of producers, trainers, and apprentices in the previous month - 2 constraints for the number of apprentices in a month must equal the number of recruits in the previous month Hint 4: Can you consider negative number of employees at each month? Operations Management Application- Workforce Assignment 1. Define the Decision Variables Pi = number of producers in month i (where i = 1, 2, 3 for April, May, June) Ti = number of trainers in month i (where i = 1, 2 for April, May) Ai = number of apprentices in month i (where i = 2, 3 for May, June) Ri = number of recruits in month i (where i = 1, 2 for April, May) 2. Define the Objective Function Minimize total wage cost for producers, trainers, apprentices, and recruits for April, May, and June: Min 3000P1 + 3300T1 + 2200R1 + 3000P2 + 3300T2 + 2600A2+2200R2 + 3000P3 + 2600A3 3. Define the Constraints Total production in Month 1 (April) must equal or exceed contract for Month 1: (1) .6P1 + .3T1 +.05R1 > 20 Total production in Months 1-2 (April, May) must equal or exceed total contracts for Months 1-2: (2) .6P1 + .3T1 + .05R1 + .6P2 + .3T2 + .4A2 + .05R2 > 44 Total production in Months 1-3 (April, May, June) must equal or exceed total contracts for Months 1-3: (3) .6P1+.3T1+.05R1+.6P2+.3T2+.4A2+.05R2+.6P3+.4A3 > 74 In April there are 100 employees that can be producers or trainers: (4) P1 + T1 = 100 At the end of June, there are to be at least 140 employees: (5) P3 + A3 > 140 Each trainer can train two recruits: (6) 2T1 > R1 (7) 2T2 > R2 The number of producers and trainers in a month must equal the number of producers, trainers, and apprentices in the previous month (8) P1 + T1 = P2 + T2 (9) P2 + T2 + A2 = P3 The number of apprentices in a month must equal the number of recruits in the previous month: (10) A2 = R1 (11) A3 = R2 Non-negativity: P1, T1, R1, P2, T2, A2, R2, P3, A3 > 0