10.6 A regional laboratory that performs nontraditional tests is planning to off
ID: 429410 • Letter: 1
Question
10.6 A regional laboratory that performs nontraditional tests is planning to offer new diag- nostic tests for regional hospitals. Current analyzers and staff are capable of performing these tests. The laboratory manager assessed the required staff and analyzer times, as well as the chemical materials required for a bundle of 50 vials for each type of test listed in Table EX 10.6.
a. Formulate this as a linear programming problem.
b. Solve the problem using Excel solver.
c. For the optimal solution, in terms of pro?t, which test(s) should be offered?
d. What is the optimal volume(s)?
e. What is the total expected pro?t from the new tests?
f. Which resources should be expanded?
Explanation / Answer
a. Let the no. of tests being offered be represented by A,B,C,D,E for type I to type V respectively.
Total profit = no. of tests*profit per test = A8+B10+C8+D7+E10.
This is the objective function and we have to maximize it. Constraints are:
1. Staff minutes should not exceed 3400 minutes. 15A+15B+15C+20D+25E<=3400
2. analyzer equipment minutes should not exceed 6000 minutes. 20A+40B+40C+60D+45E<=6000
3. materials should not exceed 2700. 12A+15B+16C+14D+14E<=2700
4. A,B,C,D,E>=0 (i.e. it cannot be negative)
b. The above model is created on excel and solved using solver:
c. for maximum profit: 120 tests of Type I, 56 tests of type II, 0 test for type III, 0 test for type IV, 30 tests of type V should be offered. It means that type III and IV will not be offered at all, only type I, II and V will be offered.
d. optimal volumes: of tests - 120 tests of Type I, 56 tests of type II, 0 test for type III, 0 test for type IV, 30 tests of type V should be offered. 3390 of staff minutes, 5990 minutes of equipments and 2700 materials.
e. total expected profit = $1,820
f. In the solver, i try and expand the total materials to 2800. profit in this case is 1878. so total materials can be increased. (keeping other 2 constraints same)
second, i increase the staff minutes to 3500 (keeping other 2 constraints same). This makes the profit as $1822 (increase of $2 from 1820)
thirdly, i increase the equipment minutes to 6100. This leads to a fall in profit to $1818.
Thus, materials and staff minutes should be increased.
Profit per test type I II III IV V 8 10 8 7 10 Number of tests I II III IV V 120 56 0 0 30 Total profit per test Total profit I II III IV V 960 560 0 0 300 1820 Constraints 1. staff mainutes staff minutes per test type I II III IV V 15 15 15 20 25 total staff minutes per test Total minutes I II III IV V 1800 840 0 0 750 3390 <=3400 2. equipment minutes per test I II III IV V 20 40 40 60 45 total equipment minutes Total minutes I II III IV V 2400 2240 0 0 1350 5990 <=6000 3. materials materials per test I II III IV V 12 15 16 14 14 total materials Total minutes I II III IV V 1440 840 0 0 420 2700 <=2700Related Questions
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