Question

The Gap is considering 3 options for managing its data processing operations: continue with its own staff, hire an outside vendor to do the managing (outsourcing), or use a combination of its own staff and an outside vendor. The cost of operation depends on future processing demand.

	Demand
Staffing Options	High	Medium	Low
Own Staff	650	650	600
Outsource	900	600	300
Combination	800	650	500

(e) Suppose the probability demand will be high is 0.2 but the other probabilities are unknown. Determine the range of the probability of the demand being medium for which it is best for the GAP to outsource.

Answer 1

Let H = Event that the demand will be high

M = Event that the demand will be medium

L = Event that the demand will be low

Then,

P(H) = Probability that the demand will be high = 0.2 (Given)

P(M) = Probability that the demand will be medium

P(L) = Probability that the demand will be low

And,

C(H) = Cost when demand is high

C(M) = Cost when demand is medium

C(L) = Cost when demand is low

Now, the cost involved in an option = P(H)*C(H) + P(M)*C(M) + P(L)*C(L) ......................(1)

To determine: Range of probability of the demand being medium for which it is best for the GAP to outsource

Solution:

The cost involved in each option is (using (1)):

Own Staff: Cost = C(Own)

= 0.2*650 + P(M)*650 + P(L)*600

= 130 + P(M)*650 + (1-0.2-P(M))*600 (H, M, L are exclusive and exhaustive events)

= 130 + 650x + 0.8*600 - 600x (Let P(M) = x hereafter)

= 130 + 50x + 480

= 610 + 50x

Outsource: Cost = C(Out)

= 0.2*900 + P(M)*600 + P(L)*300

= 180 + P(M)*600 + (1-0.2-P(M))*300

= 180 + 600x + 0.8*300 - 300x (P(M) = x)

= 180 + 300x + 240

= 420 + 300x

Combination: Cost = C(Comb)

= 0.2*800 + P(M)*650 + P(L)*500

= 160 + P(M)*650 + (1-0.2-P(M))*500

= 160 + 650x + 0.8*500 - 500x (P(M) = x)

= 160 + 150x + 400

= 560 + 150x

We have to find the P(M) (=x) when

and

Let us consider these two situations:

Case 1:

or,

or, .............................(2(a))

Case 2:

or,

or,    .............................(2(b))

But, since H, M, L are exclusive and exhaustive events,

P(H) + P(M) + P(L) = 1

or,

or,

or, ..............................(2(c))

Hence, from equations 2(a), 2(b), 2(c), for C(Out) to be minimum among the three option,

.

Also, a probability cannot be negative.

Hence, the range of probability of the demand being medium for which it is best for the GAP to outsource is [0, 0.76].