Decision Analysis

Decisions often involve a choice among a small set of alternatives with some degree of uncertainty. Decision problems can be formulated by defining:

$-$ The decision alternatives that can be chosen

$-$ The uncertain events that may occur after

Payoffs can be summarized in a payoff table which helps the decision maker to determine a strategy for the decision. For example, when buying a house, one can choose several mortgage options with various interest charges depending on the future change in interest rates:

Outcome

Decision Rates Rise Rates Stable Rates Fall

$1$ year variable rate $ $61,134$ $ $46,443$ $ $40,161$

$3$ year variable rate $ $56,901$ $ $51,075$ $ $46,721$

$30$ year fixed $ $54,658$ $ $54,658$ $ $54,658$

Decision Strategies when outcome probabilities are unknown:

$1.$ Aggressive $($Optimistic$)$ Strategy: decision maker might choose the option that minimizes the potential loss. This approach asks $$What is the best that could result from each decision?$$ and chooses the best.

$2.$ Conservative Strategy: for this type, the decision maker would the best option in the worst scenario.

$3.$ Opportunity Loss Strategy: this approach considers the opportunity loss, which represents the regret one may feel after making a suboptimal decision. It calculates the absolute difference between the best decision for a particular outcome and the payoff that was chosen. The decision maker then chooses the option that minimizes the largest opportunity loss. When the goal is to minimize something, choose the option that minimizes the worst possible $($largest$)$ regret. When it is to maximize, choose the option that maximizes the worst possible $($least$)$ gain.

See Excel example for the decision under each strategy.

Question $1$

A company is a leading manufacturer of robotic arms. All robot arms are manufactured in their plant in Ontario, and shipped to distribution centers or major customers. The company recently acquired a direct competitor in BC$,$ and is considering moving its operations to the BC plant. Considerations in this decision are the transportation, labor, and production costs at the two plants. Marketing is also predicting a decline in the demand. The company developed three scenarios:

$1.$ Demand falls slightly, with no noticeable effect on production.

$2.$ Demand and production decline $20\%.$

$3.$ Demand and production decline $40\%.$

The following table shows the total costs under each decision and scenario:

Slight Decline $20\%$ Decline $40\%$ Decline

Stay in Ontario $ $1,000,000$ $ $900,000$ $ $840,000$

Move to BC $ $1,200,000$ $ $915,000$ $ $800,000$

a$)$ What decisions should the company make using each strategy?

$-$ Aggressive strategy

$-$ Conservative strategy

$-$ Opportunity loss strategy

b$)$ If the probabilities are estimated to be $0\mathrm{.}15,0\mathrm{.}4,$ and $0\mathrm{.}45$ respectively, what is the optimal decision based on expected value