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Quantitative Methods for Business

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Question 1 (a)

Question 1 (b)
Revised probabilities for Survey study: | Conditional Probability | | Posterior Probability | State of Nature | P(Favourable result|State of Nature) | Prior Probability | Joint Probability | P(State of Nature|Favourable result) | Successful Market | 0.70 | × 0.50 | = 0.35 | 0.350.45=0.78 | Unsuccessful Market | 0.20 | × 0.50 | = 0.10 | 0.100.45=0.22 | P(Favourable result) = | 0.45 | 1.00 |

| Conditional Probability | | Posterior Probability | State of Nature | P(Unfavourable result |State of Nature) | Prior Probability | Joint Probability | P(State of Nature|Unfavourable result) | Successful Market | 0.30 | × 0.50 | = 0.15 | 0.150.55=0.27 | Unsuccessful Market | 0.80 | × 0.50 | = 0.40 | 0.400.55=0.73 | P(Unfavourable result) = | 0.55 | 1.00 |

Revised probabilities for Pilot study: | Conditional Probability | | Posterior Probability | State of Nature | P(Favourable result |State of Nature) | Prior Probability | Joint Probability | P(State of Nature|Favourable result) | Successful Market | 0.80 | × 0.50 | = 0.40 | 0.400.45=0.8889 | Unsuccessful Market | 0.10 | × 0.50 | = 0.05 | 0.050.45=0.1111 | P(Favourable result) = | 0.45 | 1.00 |

| Conditional Probability | | Posterior Probability | State of Nature | P(Unfavourable result |State of Nature) | Prior Probability | Joint Probability | P(State of Nature|Unfavourable result) | Successful Market | 0.20 | × 0.50 | = 0.10 | 0.100.55=0.1818 | Unsuccessful Market | 0.90 | × 0.50 | = 0.45 | 0.450.55=0.8182 | P(Unfavourable result) = | 0.55 | 1.00 |

Question 1 (c)
The best decision for Jim by using the EMV criterion is to conduct survey. As shown as the figure above, the highest EMV decision is “Conduct Survey” with EMV value of $24,160.
If the survey results are favourable, then Jim should produce the new razor or vice versa.

Utility Test By using utility theory, the best decision is also “Conduct Survey” with a utility value of 0.82.

According to the utility test, it is found that Jim is a risk avoider as the utility curve above shows his preference of risk and his decision is on the highest EMV. Question 2

To: CEO of Mt. Sinai Hospital
From: Tan Min Yee
Date: 8 November 2013
Subject: Presentation of optimal solution of 90-bed addition in the hospital

Dear Sir/ Madam,
As a business analyst, I had observed and evaluated your structure of company. As instructed by you, I hereby present my report explaining the best option of proportioning the 90-beds addition for medical and surgical patients as well as other possibilities and cost factors. Let abbreviation ‘m’= medical patients; ‘s’= surgical patients (a) Profit maximisation = 2280m + 1515s
Optimal solution: m= 2790.909, s= 2104.545
Profit maximisation = 2280(2790.909) + 1515(2104.545) = $ 9,551,659

The maximum revenue per year is $ 9,551,659 with 2790.909 medical patients/year and 2104.545 surgical patients/year.
Medical patients = 2790.909 ≈ 2791 people Total number of hospital stay = 2791 × 8 days = 22320 days Number of beds required per year = 22320 days ÷ 365 days = 61.15 ≈ 61 beds

Surgical patients = 2104.545 ≈ 2104 people Total number of hospital stay = 2104 × 5 days = 10520 days Number of beds required per year = 10520 days ÷ 365 days = 28.82 ≈ 29 beds
Hospital stay: 8m + 5s ≤ 32850
Optimal solution: m= 2790.909, s= 2104.545
Hospital stay: 8(2790.909) + 5(2104.545) = 32,850

(b) Total beds required per year = 61 medical beds + 29 surgical beds = 90 beds
Hence, there will be no empty beds available with this optimal solution. All 90 beds addition will be used up by both medical and surgical patients. There will not be a need of acquiring more beds. If more beds are acquired, Mt. Sinai Hospital will have immediate additional cost of purchasing more beds as well as additional cost in the x-ray department as serving more patients will results in handling more x-rays. This optimal solution will use up its maximum unutilized capacity of 7,000 more x-rays.

(c) Laboratory tests: 3.1m + 2.6s ≤ 15000
Optimal solution: m= 2790.909, s= 2104.545
Laboratory tests: 3.1(2790.909) + 2.6(2104.545) = 14123.63 ≈ 14124

The laboratory is not utilized to its maximum capacity. The laboratory is capable of handling another 15,000 tests per year. In this optimal solution, there will be another 14,124 lab tests only.
Hence, it is able to perform 876 more lab tests (15,000-14,124) per year.
If Mt. Sinai Hospital acquires more lab space, there will be no additional cost until it reached its maximum capacity of 15,000 additional lab tests. However, there will be additional cost if it exceeds its maximum capacity.

(d) X-ray: 1m + 2s ≤ 7000
Optimal solution: m= 2790.909, s= 2104.545
X-ray: 1(2790.909) + 2(2104.545) = 7000

The x-ray facility is being used to its maximum of 7,000 x-rays in this optimal solution. The x-ray department could not perform more than additional 7,000 x-rays per year without significant additional cost. If Mt. Sinai Hospital acquires more x-ray facilities, a significant additional cost would be incurred which is not favourable.

(e) Operating room: 0m + 1s ≤ 2800
Optimal solution: m= 2790.909, s= 2104.545
Operating room: 0(2790.909) + 1(2104.545) = 2104.545 ≈ 2105

The operating room is not being used to its maximum capacity in this optimal solution. It is possible to conduct 695 (2800-2105) more operations per year. If Mt. Sinai Hospital acquires more operating room facilities, there will be no additional operating room cost until it reached its maximum capacity of 2,800 additional operations. However, there will be additional cost if it exceeds its maximum capacity.

Thus, it would be suggested to Mt. Sinai Hospital to allocate 61 beds for medical patients and 29 beds for surgical patients in this expansion of hospital in order to maximise revenues.
As a conclusion, I hope the above clarifies the factors of expansion of hospital and helps you in your potential decision-makings.
Please do not hesitate to contact me if you require further assistance.

Prepared by,

……………..
(TAN MIN YEE)

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