Below is a Sensitivity Report for a Maximization Problem.
Adjustable Cells:
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Cells
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Name
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Final Value
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Reduced Cost
|
Objective Coefficient
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Allowable Increase
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Allowable Decrease
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$B$8
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Number of water bottle packs
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30
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0
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7
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3
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0.33
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$C$8
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40
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0
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5
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0.25
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1.5
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|
Cells
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Name
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Final Value
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Reduced Cost
|
Objective Coefficient
|
Allowable Increase
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Allowable Decrease
|
|
$D$5
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Space
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240
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1.5
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260
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60
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40
|
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$D$6
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Weight
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100
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0.5
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100
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20
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20
|
1. The objective function is: ____________________________
2. The two decision variables are: _____________________ and _____________________
3. The constraints are: ___________________________________
4. If the manager carries out the decision as laid out in this report. What can s/he expect for the decisionvariables and the objective function?
5. What would happen, if all else remained constant, but Number of Water Bottles Packs changed to 10?
6. What would happen, if all else remained constant, but Number of Dry Packaged Food Packs changed to 6?
7. Do you see slack anywhere?
Yes ___________ No _________ Can't be determined _________
8. What would happen if everything else remained constant but Space changed to 300?
9. What would happen if everything else remained constant but Weight changed to 80?
LP MODEL:
Cohen Chemicals, Inc., produces two types of photo-developing fluids. The first, a black-and-white picture chemical, costs Cohen $2,500 per ton to produce. The second, a color photo chemical, costs $3,000 per ton. Based on an analysis of current inventory levels and outstanding orders, Cohen's production manager has specified that at least 30 tons of the black-and-white chemical and at least 20 tons of the color chemical must be produced during the next month. In addition, the manager notes that an existing inventory of a highly perishable raw material needed in both chemicals must be used within 30 days. To avoid wasting the expensive raw material, Cohen must produce a total of at least 60 tons of the photo chemicals in the next month.
You have to develop a complete LP model based on the information above. In order to complete the model, you have to answer 3 questions - Question 10 pertains to the Decision Variables; Question 11 to the Objective Function; and Question 12 to the Constraints. You may choose as many of the options provided below in order to completely answer each of the Questions.
10. The Decision Variables in this LP model are:?(1) X1 = number of tons of black-and-white picture chemical to be produced (2) X2 = number of tons of picture chemical to be produced?(3) X3 = number of tons of expensive raw material
11. The Objective Function is:?(1) MIN: 2500X1 + 3000X2 + 60X3 (2) MIN: 3000X1 + 2500X2 + 60 X3 (3) MAX: 2500X1 + 3000X2 + 60X3
(4) MIN: 2500X1 + 3000X2 (5) MAX: 2500X1 + 3000X2
12. The constraints are:
(1) X1 + X2 >=60
(2) X2 >= 30
(3) X1 >= 20
(4) X1 = 2500
(5) X2 = 3000
(6) X3 >=60
(7) X3 <= 30
(8) X1 >= 30
(9) X2 >= 20
(10) X1 >= 0
(11) X2 >= 0
(12) X3 >= 0
Arlington Bank of Commerce and Industry is a busy bank that has requirements for between 10 and 18 tellers depending on the time of day. Lunchtime, from noon to 2 P.M., is usually heaviest. The table below indicates the workers needed at various hours that the bank is open.
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TIME PERIOD
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Number of Tellers Required
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10 A.M.-11 A.M.
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10
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Noon - 1 P.M.
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12
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2 P.M. - 3 P. M
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14
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4 P.M. - 5 P.M
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16
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9 A.M.-10 A.M.
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18
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11 A.M. - 12 Noon
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17
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1 P.M. - 2 P.M.
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15
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3 P.M - 4 P. M.
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10
|
The bank now employs 12 full-time tellers, but many people are on its roster of available part-time employees. A part-time employee must put in exactly 4 hours per day but can start anytime between 9 A.M. and 1 P.M. Part-timers are a fairly inexpensive labor pool because no retirement or lunch benefits are provided them. Full-timers, on the other hand, work from 9 A.M.to5 P.M. but are allowed 1 hour for lunch. (Half the full-timers eat at 11 A.M., the other half at noon.) Full-timers thus provide 35 hours per week of productive labor time. By corporate policy, the bank limits part-time hours to a maximum of 50% of the day's total requirement. Part-timers earn $6 per hour (or $24 per day) on average, whereas full-timers earn $75 per day in salary and benefits on average. The bank would like to set a schedule that would minimize its total manpower costs. It will release 1 or more of its full-time tellers if it is profitable to do so.
For questions 13 through 15 you have to indicate True or False, based on the information above on Arlington Bank
13. The following are decision variables - indicate True or False against each option.
(a) P = number of part-time workers ________
(b) F = number of full-time workers ________
(c) P1= part-time workers starting at 9 AM _______
(d) P2 = part-time workers starting at 10 AM ________
(e) P3 = part-time workers starting at 11 AM _______
(f) P4 = part-time workers starting at 12 Noon _________
(g) P5= part-time workers starting at 1 PM _______
(h) P6= part-time workers starting at 2 PM ________
14. Indicate whether the objective functions below are True or False. If all four options are false, write down the True objective function.
(a) MIN: 75F + 6(P+P1+P2+P3+P4+P5+P6) _______ (b) MIN: 75F + 6(P+P1+P2+P3+P4+P5+P6) ________ (c) MIN: 75F + 24 (P1+P2+P3+P4+P5) _________?(d) MIN: 75F + 6 (P1+P2+P3+P4+P5) _________
(e) ___________________________
15. Below are a list of constraints for scheduling tellers at Arlington Bank. Indicate True or False against each option. If all the constraints are not modeled completely by the True constraints, complete the model by filling in the blanks below.
(a) F + P >= 10 _______
(b) F + P1 >=10 ________
(c) F + P1 + P2 >= 12 _______
(d) 0.5F + P1 + P2 + P3 >= 14 _______
(e) 0.5F + P + P2 + P3 >= 14 ________
(f) F + P1 + P2 + P3 >= 14 ________
(g) F + P1 + P2 + P3 + P4 >= 16 _________
(h) 0.5 F + P1 + P2 + P3 + P4 >= 16 ________
(i) F + P2 + P3 + P4 + P5 >= 18 _________
(j) F + P3 + P4 + P5 >= 17 _________
(k) F + P4 + P5 >= 15 _________
(l) F + P5 >= 10 __________
(m) F <= 12 _________
(n) ______________________________
(o) __________________________
(p) _________________________
Problem 16: Odor Sensitivity
Sulfur odor compounds cause "off-odors" in wine, so winemakers want to know the odor threshold, the lowest concentration of a compound that the human nose can detect. The odor threshold for dimethyl sulfide (DMS) in trained wine tasters is about 25 micrograms per liter of wine (μg/l). The untrained noses of consumers may be less sensitive, however. In a SRS of 30 college students, the mean (x-bar) threshold was 30.4 μg/l. Assume that the standard deviation (σ) of the odor threshold for untrained noses is known to be 7 μg/l.
(a) Construct a 96% confidence interval for the mean DMS odor threshold among the student population.
(b) You want to estimate the mean DMS threshold among the student population to within ±3 μg/l with 99% confidence. How large a sample do you need?
Problem 17: Hotel Male Managers
Successful hotel managers must have personality characteristics often thought of as feminine (such as "compassionate") and traits often associated with men (such as "forceful"). The Bem Sex-Role Inventory (BSRI) is a personality test that gives separate ratings for female and male stereotypes, both on a scale of 1 to 7. A sample of 148 male general managers of three-star and four-star hotels had mean BSRI femininity score x-bar = 5.29. The standard deviation of femininity score for the population of all hotel managers is σ is 0.78.
(a) Construct a 88% confidence interval for the mean femininity score of all male hotel managers.
(b) You would be satisfied to estimate the BSRI femininity score of hotel managers to within ± 0.2 with 95% confidence. How large a sample of hotel managers do you need?
Problem 18: Nursing Mothers
Breast-feeding mothers secrete calcium into their milk; some of which may come from their bones. So it is possible that lactating mothers may lose bone mineral. Researchers measured the percent change in mineral content of the spines of 47 mothers during three months of breast-feeding, and found the mean change (x-bar) to be -3.4%. Suppose that the percent change in this population has a standard deviation, σ of 2.5%.
(a) Construct a 92% confidence interval for the mean percent change in this population.
(b) You would be satisfied to estimate the percent change amongst lactating mothers to within ± 0.5% with 99% confidence. How large a sample of nursing mothers do you need?