Question

Help me answer question number 5

Quantitative Analysis for Business Decisions Number of Company B Company C consulting days Company A 0 300 550 200 400 600 800 1000 350 500 650 750 850 650 750 900 Table 15.22 Consulting charges ($'000) Expected grade for hours of study time 10 20 30 40 Course Accounting Finance Marketing Statistics | Η Η Η Ο H 0 0 0 S G H 0 GGGH Legend O = Outstanding; H = Honours; G = Good; S = Satisfactory Table 15.23 For each course 5 points are given for an O; 4 for a H; 3 for a G; and 2 for an S. Determine how Margaret should budget her studying time if she wishes to maximise her expected total grade points. If she spends less than 10 hours on a course, she will fail it. Using dynamic programming, solve this problem, assuming that Margaret must not fail any course. 5. A firm allocates 4 units of a certain resource to 3 different activi- ties yielding profits according to Table 15.24. Using dynamic programming, determine a resource alloca- tion that maximises profits.

Answer 1

5.

We define states 1, 2 and 3 as follows.

$x_1$ = units of resource associated to activity 1

$x_2$ = units of resource associated to activities 1 and 2

$x_3$ = units of resource associated to activities 1, 2 and 3

$\therefore x_3=4,x_2\leq4,x_1\leq4$

Calculation for first stage (activity 1)-

Available units of resource ($x_1$)	Total profit in activity 1
0	0
1	8
2	18
3	22
4	24

Calculation for second stage (activities 1 & 2)-

Best profit for activities 1 and 2 = Max { Profit of the feasible alternatives for activity 2 + best profit for stage 1 }

Available units of resource ($x_2$)	Total profit in activities 1 & 2	Resource allotted for activity 2
0	max {0+0}=0	0
1	max{0+8,3+0}=8	0
2	max{0+18,3+8,6+0}=18	0
3	max{0+22,3+18,6+8,9+0}=22	0
4	max{0+24,3+22,6+18,9+8,12+0}=25	1

Calculation for third stage (activities 1, 2 & 3)-

Best profit for activities 1, 2 and 3 = Max { Profit of the feasible alternatives for activity 3 + best profit for stage 2 }

Available units of resource ($x_3$)	Total profit in activities 1, 2 & 3	Resource allocated for activity 3
4	max{0+25,6+22,7+18,8+8,10+0}=28	1

So, to obtain the solution of the dynamic programming problem we observe as follows.

• For maximum profit we must allocate 1 unit of resource in 3rd activity. (Observed from third stage calculation)
• So, remaining unit of resource to be allocated in activities 1 and 2 is 4-1=3.
• For maximum profit we must allocate 0 unit of resource in 2nd activity. (Observed from second stage calculation)
• So, remaining unit of resource to be allocated in activity 1 is 3.

Our solution is to allocate 3, 0 and 1 units of resource in activities 1, 2 and 3 respectively. (Which gives maximised profit 28)