Question

An engineering company is about to undertake a major overhaul of a factory's machinery for a customer. The overhaul will be carried out on a Saturday and Sunday but, it's not completed by the Monday morning, the factory will experience serious production losses. In this event, the engineering company has agreed to compensate the customer by paying a penalty of $20,000.

The manager of the engineering company has to decide how many engineers to include in the overhaul team. Each engineer in the team will be paid $480 for working over the weekend, but because the nature of the work, only teams of 10, 15 or 20 engineers can be considered. The manager estimates that the chances of a 10-person team completing the overhaul by Monday morning is only 0.4. A 15-person team has, he estimates, a 0.6 probability of meeting the deadline, while a 20-person team has a 0.9 probability of completing the work in time for Monday morning.

(a) Assuming that the manager wants to minimize expected costs, how large a team should he choose?

(b) Having mad a provisional decision about the size of the team, the manager hears that a piece of specialized equipment will be available for hire on the Sunday, at short notice. The cost of hiring this equipment would be $4400, and it would require at least 15 engineers to operate it. However, it's virtually certain that the overhaul would be completed on time if the equipment was used. Before making a decision on whether to hire the equipment, the manager will review the progress that has been made on Saturday evening. He reckons that there is a 0.5 probability that a 15-person team would be behind schedule by Saturday evening, while there is only a 0.2 probability that a 20-person team will be in this position. He then calculates the probabilities of the overhaul overrunning the deadline if the equipment is not hired, given the position on Saturday evening. These probabilities are shown below:

15-person team on Saturday evening: p(overhaul exceeds deadline if equipment not hired) - Behind schedule:0.6 -Not behind schedule:0.2

20-person team position on Saturday evening: p(overhaul exceeds deadline if equipment not hired) - Behind schedule:0.2 -Not behind schedule:0.075

How many people should the manager now include in the team and should he hire the equipment on Saturday evening?

Answer 1

Answer a)

To begin with the solution, it is necessary first to present a summary of the data provided by the statement:
- Compensation for non-compliance = $ 20,000 (additional cost, in case of not being successful in the completion of the revision of the machine)
- Payment to each Engineer of the work team = $ 480
- Possible sizes of the work team = 10, 15, 20.

In addition, the following given probabilities should be considered:

P (with a work team of size 10 complete the revision of the machine) = 0.4
P (with a work team of size 15 complete the revision of the machine) = 0.6
P (with a work team of size 20 complete the revision of the machine) = 0.9

Considering the given probabilities, it can also be considered that:

P (with a work team of size 10 no complete the revision of the machine) = 0.6
P (with a work team of size 15 no complete the revision of the machine) = 0.4
P (with a work team of size 20 no complete the revision of the machine) = 0.1

You can define the Random Variable X as the number of engineers that work in the revision of the equipment, thus, the Cost of the revision of the machinery is defined as follows:

C = 480X, if the revision work of the machinery ends on time;
C = 480X + 20,000, if the revision work of the machinery is delayed.

With all the above, you can build the following table, which shows the value of the Cost function for the different values of X (10, 15, 20) and the two situations that can occur: complete the revision of the machinery to time (success), or have delay in the revision of the machinery (failure):

	Situación
Número de Ingenieros	Éxito	Fracaso
10	480x10=4800	480x10+20000=24800
15	480x15=7200	480x15+20000=27200
20	480x20=9600	480x20+20000=29600

The calculation of the Expected Value of the Cost for each quantity of contracted engineers is:

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The recommendation to minimize the Expected Costs, is to hire a team of 20 Engineers to perform the review of the machinery.

Answer b)

To answer the question: How many people should the manager now include in the team and should he hire the team on Saturday night? We will base the conclusions on the calculation of probabilities, but also on the Expected Value.

Initially, it was suggested to work with a team of 20 Engineers, which resulted in an Expected Value for the Cost of $ 11,600, in comparison with the $ 15,200 that the Expected Cost gives if working with 15 Engineers.

Now, hiring a team on Sunday increases the Costs by $ 4400, according to the problem statement.

We will analyze the situation for each case (X = 15 and X = 20)

For X = 15 (15 contracted Engineers):

P (being retracted on Saturday night) = 0.5
P (NOT to be retracted on Saturday night) = 0.5

Since it is proposed not to hire the team, then:
P (the review exceeds the deadline if you are late on Saturday night) = 0.6
P (the review does not exceed the deadline if you are late on Saturday night) = 0.4
P (the review exceeds the deadline if they are not late on Saturday night) = 0.2
P (the review does not exceed the deadline if they are not late on Saturday night) = 0.8

So you can determine the followingProbabilities:
P (exceeding the delivery deadline) = 0.5 x 0.6 + 0.5 x 0.2 = 0.4
P (Do not exceed the deadline for delivery) = 0.5 x 0.4 + 0.5 x 0.8 = 0.6

For X = 20 (20 Contracted Engineers)

P (being retracted on Saturday night) = 0.2
P (NOT to be retracted on Saturday night) = 0.8

Since it is proposed not to hire the team, then:
P (the review exceeds the deadline if you are late on Saturday night) = 0.2
P (the review does not exceed the deadline if you are late on Saturday night) = 0.8
P (the review exceeds the deadline if they are not late on Saturday night) = 0,075
P (the review does not exceed the deadline if they are not late on Saturday night) = 0.925

So you can determine the followingProbabilities:
P (exceeding the delivery deadline) = 0.2 x 0.2 + 0.8 x 0.075 = 0.1
P (Do not exceed the deadline for delivery) = 0.2 x 0.8 + 0.8 x 0.925 = 0.9

The calculations of the Probabilities P (exceeding the deadline for delivery) and P (Do not exceed the deadline for delivery) for both X = 15 and X = 20 coincide exactly with those provided to make the calculations in the previous question, for What the recommendation is still to hire 20 Engineers to do the review to the client's equipment.

I also recommend not to hire a team on Sunday, since having 20 Engineers hired, the Expected Cost Value is $ 11600. If a team is hired on Sunday, what generates is increase in costs but does not generate changes in the probabilities that allow to diminish the expected value of the Costs.