Forecasting
Instructions: Use Excel to complete the problems.
Use one Excel spreadsheet file for the calculations and explanations, with one worksheet per problem. Use the problem number for each worksheet name. Cells should contain the formulas.
Problem #1: Data collected on the yearly demand for 50-pound bags of fertilizer at Wallace Garden Supply are shown in the following table. Develop a 3-year moving average to forecast sales. Then estimate demand again with a weighted moving average in which sales in the most recent year are given a weight of 2 and sales in the other 2 years are each given a weight of 1. Which method do you think is best?
DEMAND FOR FERTILIZER
YEAR (1,000S OF BAGS)
1 4
2 6
3 4
4 5
5 10
6 8
7 7
8 9
9 12
10 14
11 15
Problem #2: Sales of Cool-Man air conditioners have grown steadily during the past 5 years:
YEAR SALES
1 450
2 495
3 518
4 563
5 584
6 ?
The sales manager had predicted, before the business started, that year 1's sales would be 410 air conditioners. Using exponential smoothing with a weight of a = 0.30, develop forecasts for years 2 through 6.
Problem #3: Emergency calls to Winter Park, Florida's 911 systems, for the past 24 weeks are as follows:
WEEK CALLS WEEK CALLS WEEK CALLS
1 50 9 35 17 55
2 35 10 20 18 40
3 25 11 15 19 35
4 40 12 40 20 60
5 45 13 55 21 75
6 35 14 35 22 50
7 20 15 25 23 40
8 30 16 55 24 65
(a) Compute the exponentially smoothed forecast of calls for each week. Assume an initial forecast of 50 calls in the first week and use a=0.01. What is the forecast for the 25th week?
(b) Reforecast each period using a=0.6
(c) Actual calls during the 25th week were 85. Which smoothing constant provides a superior forecast?
Problem #4: In the past, Judy Holmes's tire dealership sold an average of 1,000 radials each year. In the past two years, 200 and 250, respectively, were sold in fall, 350 and 300 in winter, 150 and 165 in spring, and 300 and 285 in summer. With a major expansion planned, Judy projects sales next year to increase to 1,200 radials. What will the demand be each season?
Optimization Modeling
Instructions: Complete these problems manually.
Use one Excel spreadsheet file for the calculations and explanations, with one worksheet per problem. Use the problem number for each worksheet name. Cells should contain the formulas.
Problem #5: The Electrocomp Corporation manufactures two electrical products: air conditioners and large fans. The assembly process for each is similar in that both require a certain amount of wiring and drilling. Each air conditioner takes 3 hours of wiring and 2 hours of drilling. Each fan must go through 2 hours of wiring and 1 hour of drilling. During the next production period, 240 hours of wiring time are available and up to 140 hours of drilling time may be used. Each air conditioner sold yields a profit of $25. Each fan assembled may be sold for a $15 profit. Formulate and solve this LP production mix situation to find the best combination of air conditioners and fans that yields the highest profit. Use the corner point graphical approach.
Problem #6: Electrocomp's management realizes that it forgot to include two critical constraints (see Problem #5). In particular, management decides that there should be a minimum number of air conditioners produced ing of fans in the preceding period, a limit should be placed on the total number of fans produced.
(a) If Electrocomp decides that at least 20 air conditioners should be produced but no more than 80 fans should be produced, what would be the optimal solution? How much slack is there for each of the four constraints?
(b) If Electrocomp decides that at least 30 air conditioners should be produced but no more than 50 fans should be produced, what would be the optimal solution? How much slack is there for each of the four constraints at the optimal solution?
Problem #7: The dean of the Western College of Business must plan the school's course offerings for the fall semester. Student demands make it necessary to offer at least 30 undergraduate and 20 graduate courses in the term. Faculty contracts also dictate that at least 60 courses be offered in total. Each undergraduate course taught costs the college an average of $2,500 in faculty wages, and each graduate course costs $3,000. How many undergraduate and graduate courses should be taught in the fall so that total faculty salaries are kept to a minimum?
Optimization Modeling Applications
Instructions: Use Excel's Solver to complete the problems.
Use one Excel spreadsheet file for the calculations and explanations, with one worksheet per problem. Use the problem number for each worksheet name. Cells should contain the formulas.
Problem #8: (Restaurant work scheduling problem). The famous Y. S. Chang Restaurant is open 24 hours a day. Waiters and busboys report for duty at 3 A.M., 7 A.M., 11 A.M., 3 P.M., 7 P.M., or 11 P.M., and each works an 8-hour shift. The following table shows the minimum number of workers needed during the six periods into which the day is divided. Chang's scheduling problem is to determine how many waiters and busboys should report for work at the start of each time period to minimize the total staff required for one day's operation. (Hint: Let Xi equal the number of waiters and busboys beginning work in time period i, where i = 1, 2, 3, 4, 5, 6.)
NUMBER OF WAITERS
PERIOD TIME AND BUSBOYS REQUIRED
1 3 A.M.-7 A.M. 3
2 7 A.M.-11 A.M. 12
3 11 A.M.-3 P.M. 16
4 3 P.M.-7 P.M. 9
5 7 P.M.-11 P.M. 11
6 11 P.M.-3 A.M. 4
a) Add this additional constraint: Total Number of Workers to Start the Shifts must be less than or equal to 29.
Problem #9: (Animal feed mix problem) The Battery Park Stable feeds and houses the horses used to pull tourist-filled carriages through the streets of Charleston's historic waterfront area. The stable owner, an ex-racehorse trainer, recognizes the need to set a nutritional diet for the horses in his care. At the same time, he would like to keep the overall daily cost of feed to a minimum. The feed mixes available for the horses' diet are an oat product, a highly enriched grain, and a mineral product. Each of these mixes contains a certain amount of five ingredients needed daily to keep the average horse healthy. The table on this page shows these minimum requirements, units of each ingredient per pound of feed mix, and costs for the three mixes. In addition, the stable owner is aware that an overfed horse is a sluggish worker. Consequently, he determines that 6 pounds of feed per day are the most that any horse needs to function properly. Formulate this problem and solve for the optimal daily mix of the three feeds.
Data for #9:
FEED MIX
DIET OAT ENRICHED MINERAL MINIMUM DAILY
REQUIREMENT PRODUCT GRAIN PRODUCT REQUIREMENT
(INGREDIENTS) (UNITS/LB) (UNITS/LB) (UNITS/LB) (UNITS)
A 2 3 1 6
B 0.5 1 0.5 2
C 3 5 6 9
D 1 1.5 2 8
E 0.5 0.5 1.5 5
Cost/lb $0.09 $0.14 $0.17
Problem #10: Eddie Kelly is running for reelection as mayor of a small town in Alabama. Jessica Martinez, Kelly's campaign manager during this election, is planning the marketing campaign, and there is some stiff competition. Martinez has selected four ways to advertise: television ads, radio ads, billboards, and newspaper ads. The costs of these, the audience reached by each type of ad, and the maximum number of each is shown in the following table:
COST AUDIENCE MAXIMUM
TYPE OF AD PER AD REACHED/AD NUMBER
TV $800 30,000 10
Radio $400 22,000 10
Billboards $500 24,000 10
Newspapers $100 8,000 10
In addition, Martinez has decided that there should be at least six ads on TV or radio or some combination of those two. The amount spent on billboards and newspapers together must not exceed the amount spent on TV ads. While fundraising is still continuing, the monthly budget for advertising has been set at $15,000. How many ads of each type
should be placed to maximize the total number of people reached?
Simulation
Instructions: Use one Excel spreadsheet file for the calculations and explanations, with one worksheet per problem. Use the problem number for each worksheet name. Cells should contain the formulas.