# 2 Linear Questions 1

I’m working on a linear algebra multi-part question and need an explanation to help me study.

Question 1

NewCrest is looking to invest at most \$500,000. The investment can be divided among three options. The return on each option depends on the economic outlook, which is shown in Table 1. For example, if the economic outlook is bright, investing \$1 in option I will generate a gain of \$3, and investing \$1 in option II will result in a loss of \$5.

Table 1: Return per dollar invested in the different options

Return per dollar invested in option

Economic Outlook

I

II

III

Bright

3

-5

7

Gloomy

-2

6

-3

(a) Axel, a business executive, proposed two alternatives to NewCrest. One is investing \$100,000 in option I, \$200,000 in II and \$150,000 in III. The other is investing \$60,000 in option I, \$100,000 in II and \$125,000 in III. NewCrest takes a conservative approach to decision-making. That is, it chooses the alternative that maximizes the minimum return produced by the two outcomes of the economic outlook.

Discuss among the two alternatives proposed by Axel, which one NewCrest will choose. Please write clearly alternatives, states of nature and payoffs in a payoff table. Please limit the answer to within one page.

(10 marks)

(b) Ben, another business executive, pointed out that NewCrest has an unlimited number of alternatives, not just the two proposed by Axel. Which alternative is best if NewCrest takes a conservative approach to decision-making? To answer that question, Ben drafted,

“Define x1 = investment in option I,

x2 = investment in option II, and

x3 = investment in option III.

The objective is to

Maximize min {3×1 – 5×2 + 7×3, – 2×1 + 6×2- 3×3}”.

The objective function is not linear.

Please search online “tricks to maximize the min of objective functions” or “transformations into linear programs”. Then help Ben develop a complete Linear Programming (LP) model that represents the problem of deciding how much to invest in each option. Please limit the answer to within one page.

(10 marks)

(c) Use the Excel Solver to find the optimal solution to the problem in Part (b) and summarise how NewCrest should invest.

(1) the problem worksheet just before clicking the Solver command,

(2) the Solver Parameters dialog just before clicking Solve and

(20 marks)Question 2

This question consists of three parts. Parts (a) and (b) are about the forecast of newborns in Singapore and please find the number of babies born in Singapore for 1986 1Q through 2020 2Q in the Excel file “GBA_Q2_InputFile”. Part (c) is about general social science forecasts.

(a)Comment on the underlying pattern(s) in the historical data. Please limit the answer to within one page.

(10 marks)

(b) Using a suitable method, forecast the number of newborns in Singapore for 2020 3Q through 2021 4Q. Please provide justification for the method you choose. Please limit the answer to within two pages.

(10 marks)

(c) For social science forecasts, there are a few famous examples of forecasts with large errors, including

“There’s no chance that the iPhone is going to get any significant market share.”

Steve Ballmer, CEO of Microsoft 2007

“I think there’s a world market for maybe five computers.”

Thomas Watson, Chairman of IBM 1943

Infer two (2) reasons why in the social sciences, the forecast errors can be large. Please limit the answer to within one page.

(40 marks)