Exam 1 Study Guide PROBLEMS 1. Mabel's Ceramics spent $3000 on a new kiln last year, in the belief that it would cut energy usage 25% over the old kiln. This kiln is an oven that turns "greenware" into finished pottery. Mabel is concerned that the new kiln requires extra labor hours for its operation. Mabel wants to check the energy savings of the new oven, and also to look over other measures of their productivity to see if the change really was beneficial. Mabel has the following data to work with: Production (finished units) Labor (hrs) Capital ($) Energy (kWh) The year before 4000 350 15000 3000 Year just ended 4100 375 18000 2600 Also, suppose that the average labor cost is $12 per hour, cost of capital is 20%, and cost of energy is $0.40 per kwh. a. Were the modifications beneficial? (Compute labor, energy, and capital productivity for the two years and compare.) b. Compute percentage change in multi-factor productivity of the year just ended from that of year before. c. If the multifactor productivity for next year must be restored to what it was the year before, assuming the same output next year as the year just ended, by how the input must be reduced from what it is this year? 2. An Appliance Service company made house calls and repaired 10 lawn-mowers, 2 refrigerators, and 3 washers in an 8-hour day with his standard crew of 3 workers. The retail price for each respective service is $50, $200, and $120. The average wage for the workers is $12 per hour. The materials cost for a day was $200 while the overhead cost was $50. a. What is the company’s labor productivity? b. What is the multifactor productivity? c. How much of a reduction in input is necessary for a 5% increase in multifactor productivity? 3. Consider the tasks, durations, and predecessor relationships in the following network. Draw the AON network and answer the questions that follow. Activity Description A B C D E F G H I J Immediate Predecessor(s) --A A B D, C C F F E, G I Optimistic (Weeks) 4 2 8 1 6 2 2 6 4 1 Most Likely (Weeks) 7 8 12 2 8 3 2 8 8 2 Pessimistic (Weeks) 10 20 16 3 22 4 2 10 12 3 a. b. c. d. 4. Schedule the activities of this project and determine (i) the expected project completion time, (ii) the earliest and latest start and finish times, and the slack for all the activities, and (iii) all the critical paths. What is the probability of completion of the project before week 42? What is the probability of completion of the project before week 35? With 99% confidence what is your estimate for the project completion time. Consider the following project. All activity times are in weeks. Activity A B C D E F G H I a. b. c. d. e. Immediate Predecessor(s) A A, B B C D, E E F, G Normal Time 7 8 9 8 9 10 5 10 5 Crash time 4 5 7 8 8 8 5 8 4 Normal cost 20000 50000 80000 30000 10000 90000 25000 32000 28000 Crash cost 38000 74000 110000 30000 12000 124000 25000 40000 35000 Draw an AON network. Identify all the unique paths from start to finish and determine the critical path, normal project completion time, and normal project cost. Compute MAC, Cost of crashing/week. Which activity would you crash first and by how many weeks? Determine the project time and cost after crashing the activity selected in (d). 5. Consider the following CPM Solver model. a) Determine the successor activities in cells I2 to I10. b) Determine the Excel formulas for the following cells: F2, G2, C15, C18, D18, D21, C25, G19, G16, G15, H15, B27, B28, and B29. c) What is the Solver Target cell for minimizing the project completion time? d) What is the Solver changing cell range? e) What are the Solver constraints? 6. What is the forecast for May based on a 3-period MA and a weighted 3-period moving average applied to the following past demand data? Let the weights be, 3, 3, and 4 (last weight is for most recent data). Compute MAD, MSE and MAPE for both cases and compare. Nov. 37 7. Dec. 36 Jan. 40 Feb. 42 Mar. 47 April 43 Sales of music stands at the local music store over the past ten days are shown in the table below. Forecast demand using exponential smoothing with an of .6 (initial forecast = 16). a) Compute the forecast for period 6 and the MAD. b) Compute the tracking signal for periods 1 to 5. What do you recommend for this forecasting process? t Demand 1 13 2 21 3 28 4 37 5 25 8. Weekly sales of copy paper at Cubicle Suppliers are in the table below. Forecast week 8 with a trend projection model. Week Sales (cases) 1 17 2 22 3 27 4 32 5 35 6 37 7 41 9. The quarterly sales for specific educational software over the past three years are given in the following table. Compute the four seasonal indices and find forecast for Year 4 if the annual demand for year 4 is estimated to be 10% more than that of year 3. Quarter 1 Quarter 2 Quarter 3 Quarter 4 10. YEAR 1 1690 940 2625 2500 YEAR 2 1800 900 2900 2360 YEAR 3 1850 1100 2930 2615 Arnold Tofu owns and operates a chain of 12 vegetable protein "hamburger" restaurants in northern Louisiana. Sales figures and profits for the stores are in the table below. Sales are given in millions of dollars; profits are in hundred thousand dollars. Store 1 2 3 4 5 6 7 8 9 10 11 12 a. b. Sales 7 2 6 4 14 15 16 12 14 20 15 7 Profits 15 10 13 15 25 27 24 20 27 44 34 17 Calculate a regression line for the data. What is your forecast of profit for a store with sales of $24 million? $30 million? Calculate the standard error of the estimate. c. Determine the value of the coefficient of correlation between sales and profit and the value of the coefficient of determination. Answers: 1. The energy modifications did not generate the expected savings; labor and capital productivity decreased. Given data Production Labor Capital = Energy = Last year 4000 350 15000 3000 Now 4100 375 18000 2600 Labor productivity (Units/hr) = 11.4286 10.9333 Capital productivity (units/$) = Cost of capital = Energy productivity (Units/KWH) = 0.2667 3000 1.3333 0.2278 3600 1.5769 4200 15000 1200 8400 0.4762 4500 18000 1040 9140 0.4486 0.4762 8610 530 Labor cost = Hours x $12 = Capital $ = Energy $ = $0.40 x Energy = Total input $ = Multifactor productivity (Units/$) = Target productivity = Target input = 4100/0.4762 = Reduction in input needed = 9140 – 8610 = #2 Number serviced Dollar value/unit Production in $ Labor hours = 3 workers x 8 hrs. = Labor productivity = 1260/24 = Multifactor productivity Labor cost = 3x8x$12 = Material = Overhead = Total input cost = Productivity = 1260/538 = 5% improvement in MF productivity = Target productivity after 5% improvement = Input for improved productivity = Reduction in input needed = LM 10 50 500 24 52.50 $ $ $ $ 288 200 50 538 2.3420 0.1171 2.4591 512.38 25.62 Change -0.4952 Change % -4.33% -0.0389 -14.58% 0.2436 18.27% -0.0276 -5.8% R W 2 3 200 120 400 360 per day per hour of labor 1260 <-- Total $ = 288 + 200 + 50 per $ input <-- Output/Productivity = 1260/2.4591 <-- 538 – 512.38 3. (a) D B E Start A I G C J F H Task A B C D E F G H I J Task Start A B C D E F G H I J Finish a 4 2 8 1 6 2 2 6 4 1 m 7 8 12 2 8 3 2 8 8 2 t ES 7 9 12 2 10 3 2 8 8 2 0 7 7 16 19 19 22 22 29 37 39 B 10 20 16 3 22 4 2 10 12 3 EF 0 7 16 19 18 29 22 24 30 37 39 t 7 9 12 2 10 3 2 8 8 2 LS 0 8 7 17 19 24 27 31 29 37 39 Variance 1.0000 64/36 256/36 64/36 4/36 LF 0 7 17 19 19 29 27 29 39 37 39 Slack 0 1 0 1 0 5 5 9 0 0 Critical Critical Critical Critical Critical Critical path = A-C-E-I-J, Project completion time TE = 39 Variance for project completion time = 2p = 1 + 388/36 = 11.7778; p =3.4319 Fin ish b. For P(T <=42), Z = (42 – 39)/3.4319 = 0.87, Table area = 0.80785, Probability = 0.80785 c. For P(T <=35), Z = (35 – 39)/3.4319 = -1.17, Table area = .879; Probability = 1 - .879 = 0.121 d. Z for 99% confidence = 2.325, T = 39 + 2.325(3.4319) = 46.98 4. C A F I Finish Start G B D H E Normal Time Crash time Normal cost Crash cost MCA Crashing cost/week A 7 4 20000 38000 3 6000 B 8 5 50000 74000 3 8000 C 9 7 80000 110000 2 15000 D 8 8 30000 30000 0 E 9 8 10000 12000 1 2000 F 10 8 90000 124000 2 17000 G 5 5 25000 25000 0 H 10 8 32000 40000 2 4000 I 5 4 28000 35000 1 7000 Sum = 365,000 Activity Paths A-C-F-I A-D-G-I B-D-G-I B-E-G-I B-E-H Path time 31 25 26 27 27 Predecessor(s) A A, B B C D, E E F, G Critical path Normal project time = 31 weeks Normal project cost = 365,000 Activity to crash = A – among the critical activities (A, C, F, I) the crashing cost/week for A is the smallest. Weeks to crash = Minimum{MTR of A, Project time – time of second longest path} i.e. = Minimum{3, 31-27} = 3 Project time after crashing A 3 weeks = 31 – 3 = 28 weeks Project cost after crashing A = 365,000 + 3 x 6,000 = 383,000 5. (a) Activity Successors(s) A C, D B D, E C F D G E G, H F I G I H Finish I Finish (b) F2 G2 C15 D18 E18 D21 C25 G19 F19 G16 G15 H15 B27 B2-C2 (E2-D2)/F2 B2-B15 Max(E15,E16) D18+C18 Max(E18,E19) Max(E22,E23) Min(F21,F22) G19-C19 Min(F18,F19) Min(F17,F18) F15-D15 or G15-E15 Sum(D2:D10) B28 B29 Sumproduct(BG15:B23,G2:G10) B27+B28 (c) (d) (e) Solver Target cell for minimizing the project completion time = C25 Changing cell range = B15:B23 What are the Solver constraints? B15:B23 <= F2:F10 B15:B23 = Integer (Optional) 6. Month Nov. Demand (At) 37 3-MA Forecast |Et| |Et|/At Et2 Weight 3 Weighted 3-MA |Et| |Et|/At Dec. Jan. Feb. Mar. April 36 40 42 47 43 3 4 Forecast = 37.67 4.33 0.1031 18.7489 37.90 4.1 0.0976 39.33 7.67 0.1632 58.8289 39.60 7.4 0.1574 43.00 0 MAD = 4 0.0000 MAPE = 8.88% 0 MSE =25.8593 43.40 0.4 MAD = 3.97 0.0093 MAPE = 8.81% 44.00 Forecast = 43.90 7. Period 1 2 3 4 5 Demand 13 21 28 37 25 Ft Et |Et| 16.00 -3.00 3.00 14.20 6.80 6.80 18.28 9.72 9.72 24.11 12.89 12.89 31.84 -6.84 6.84 F6 = 27.74 MAD = 7.85 Period 1 2 3 4 5 Demand 13 21 28 37 25 Ft Et 16.00 -3.00 3.00 14.20 6.80 6.80 3.80 18.28 9.72 9.72 13.52 24.11 12.89 12.89 26.41 31.84 -6.84 6.84 19.57 Week 1 2 3 4 5 6 7 Sales 17 22 27 32 35 37 41 Tracking signal |Et| CFEt CAEt MADt TS -3.00 3.00 3 -1 9.80 4.9 0.78 19.52 6.51 2.08 32.41 8.1 3.26 39.25 7.85 2.49 8. 28 b 211 XY n X Y X nX 2 a Y bX 2 b XY X2 17 1 n= 7 X2 = 140 44 4 X = 28 954 81 9 Y = 211 XY = b= 128 16 = 4.0000 175 25 = 30.14 222 36 287 954 49 140 954 7(4)(30.14) 3.9286 140 7(4) 2 a 30.14 3.9286(4) 14.4286 a= 3.9286 14.4286 Regression equation: Ŷ = 14.4286 + 3.9286t F8 = 14.4286 + 3.9286(8) = 45.8571 9. Quarter 1 2 3 4 Year 1 Demand Year 2 Year 3 1690 940 2625 2500 1800 900 2900 2360 1850 1100 2930 2615 Overall average = Average Index 1780.00 0.8823 980.00 0.4857 2818.33 1.3969 2491.67 2017.50 1.2350 Year 3 sum = 8495 Annual demand for year 4 = 1.1 x 8495 = 9345 Demand/season = 2336 Forecast for year 4 Quarter 1 2 3 4 Average demand 2336 2336 2336 2336 Seasonal Index 0.8823 0.4857 1.3969 1.2350 Forecast 2061 1135 3263 2885 10. Store 1 2 3 4 5 6 7 8 9 10 11 12 Sum = X 24 30 (b)√ Sales (X) 7 2 6 4 14 15 16 12 14 20 15 7 132 Profits (Y) 15 10 13 15 25 27 24 20 27 44 34 17 271 Y 43.2923 52.8503 Estimated profit $ 4,329,230 $ 5,285,030 7159−5.0601(271)−1.593(3529) 12−2 XY 105 20 78 60 350 405 384 240 378 880 510 119 3529 = 4.0738 X2 49 4 36 16 196 225 256 144 196 400 225 49 1796 Y2 225 100 169 225 625 729 576 400 729 1936 1156 289 7159 n= X = Y = = = 12 1796 X2 = 132 XY = 3529 271 7159 Y2 = 11 b = 1.5930 22.5833 a = 5.0601 Ft = 5.0601 + 1.593 X (c) 12(3529)−(132)(271) √[12(1796)−1322 ][12(7159− 2712 ] r2 = 0.8403 = 6576 7173.8258 =0.9167