The Islander Fishing Company purchases clams for K250 per pound from fishermen and sells them

Question

to various restaurants for K350 per pound. Any clams not sold to the restaurants by the end of the
week can be sold to a local soup company for K100 per pound. The company can purchase 5000,
10,000, or 20,000 pounds. The probabilities of various levels of demand are as follows:

Demand (Pounds) Probability
5000 0.25
10,000 0.37
20,000 0.38

Clearly and step by step answer the following questios below

(a) For each possible purchase level (5000, 10,000, or 20,000 pounds), compute the profit (or
loss) for each level of demand (i.e prepare the conditional profit table).
(b) Determine the optimal action based on the maximax criterion.
(c) Determine the optimal action based on the maximin criterion.
(d) Compute the expected monetary value (EMV) for purchasing 5000, 10,000, and 20,000
pounds of clams.
(e) Compute the standard deviation for each possible purchase level.
(f) Compute the expected opportunity loss (EOL) for purchasing 5000, 10,000, and 20,000
pounds of clams.
(g) Explain the meaning of the expected value of perfect information (EVPI) in this problem.
(h) Compute the coefficient of variation for purchasing 5000, 10,000, and 20,000 pounds of
clams. Discuss.
(i) Compute the return-to-risk ratio (RTRR) for purchasing 5000, 10,000, and 20,000 pounds
of clams. Discuss.
(j) Based on (d) and (f), would you choose to purchase 5000, 10,000, or 20,000 pounds of
clams? Why?

Answer 1

To address each part of the problem systematically, we'll start with the basic information and work our way through each question step-by-step.

### (a) Compute the Profit (Loss) for Each Level of Demand

We start by setting up the profit calculations for each possible purchase level and each demand scenario. The profit is calculated as follows:

\[
\text{Profit} = (\text{Revenue from Restaurants} + \text{Revenue from Soup Company}) - \text{Cost of Purchasing Clams}
\]

**Revenue Determination**:
- If demand ≤ purchase level: Revenue from Restaurants = Demand × Selling Price, No revenue from the Soup Company.
- If demand > purchase level: All purchased clams are sold to restaurants, No revenue from the Soup Company
- Soup company buys leftover clams at K100 per pound: Revenue = (Purchase level - Demand) × K100 (When leftovers are applicable).

**Costs**:
- Cost of purchasing clams = Purchase Level × K250

Let's compute the profits step-by-step for each scenario:

#### Purchase Level = 5000 pounds:
- **Demand: 5000 pounds**
- Revenue: 5000 * K350 = K1,750,000
- Cost: 5000 * K250 = K1,250,000
- Profit: K1,750,000 - K1,250,000 = K500,000

- **Demand: 10,000 pounds**
- Revenue: 5000 * K350 = K1,750,000 (since only 5000 pounds can be sold)
- Cost: 5000 * K250 = K1,250,000
- Profit: K1,750,000 - K1,250,000 = K500,000

- **Demand: 20,000 pounds**
- Revenue: 5000 * K350 = K1,750,000 (since only 5000 pounds can be sold)
- Cost: 5000 * K250 = K1,250,000
- Profit: K1,750,000 - K1,250,000 = K500,000

#### Purchase Level = 10,000 pounds:
- **Demand: 5000 pounds**
- Revenue (Restaurant): 5000 * K350 = K1,750,000
- Revenue (Soup Company): 5000 * K100 = K500,000
- Total Revenue: K1,750,000 + K500,000 = K2,250,000
- Cost: 10,000 * K250 = K2,500,000
- Profit: K2,250,000 - K2,500,000 = -K250,000

- **Demand: 10,000 pounds**
- Revenue: 10,000 * K350 = K3,500,000
- Cost: 10,000 * K250 = K2,500,000
- Profit: K3,500,000 - K2,500,000 = K1,000,000

- **Demand: 20,000 pounds**
- Revenue: 10,000 * K350 = K3,500,000 (since only 10,000 pounds can be sold)
- Cost: 10,000 * K250 = K2,500,000
- Profit: K3,500,000 - K2,500,000 = K1,000,000

#### Purchase Level = 20,000 pounds:
- **Demand: 5000 pounds**
- Revenue (Restaurant): 5000 * K350 = K1,750,000
- Revenue (Soup Company): 15,000 * K100 = K1,500,000
- Total Revenue: K1,750,000 + K1,500,000 = K3,250,000
- Cost: 20,000 * K250 = K5,000,000
- Profit: K3,250,000 - K5,000,000 = -K1,750,000

- **Demand: 10,000 pounds**
- Revenue (Restaurant): 10,000 * K350 = K3,500,000
- Revenue (Soup Company): 10,000 * K100 = K1,000,000
- Total Revenue: K3,500,000 + K1,000,000 = K4,500,000
- Cost: 20,000 * K250 = K5,000,000
- Profit: K4,500,000 - K5,000,000 = -K500,000

- **Demand: 20,000 pounds**
- Revenue: 20,000 * K350 = K7,000,000
- Cost: 20,000 * K250 = K5,000,000
- Profit: K7,000,000 - K5,000,000 = K2,000,000

**Conditional Profit Table**:

| Purchase Level | Demand 5000 | Demand 10,000 | Demand 20,000 |
|----------------|-------------|---------------|---------------|
| 5000 | 500,000 | 500,000 | 500,000 |
| 10,000 | -250,000 | 1,000,000 | 1,000,000 |
| 20,000 | -1,750,000 | -500,000 | 2,000,000 |

### (b) Maximax Criterion

Maximax criterion suggests choosing the action that maximizes the maximum possible profit.

- Max profit for each purchase level:
- 5000 pounds: 500,000
- 10,000 pounds: 1,000,000
- 20,000 pounds: 2,000,000

**Optimal Action**: Purchase 20,000 pounds (Maximum potential profit = 2,000,000).

### (c) Maximin Criterion

Maximin criterion suggests choosing the action that maximizes the minimum possible profit.

- Min profit for each purchase level:
- 5000 pounds: 500,000
- 10,000 pounds: -250,000
- 20,000 pounds: -1,750,000

**Optimal Action**: Purchase 5000 pounds (Maximum of minimum profits is 500,000).

### (d) Expected Monetary Value (EMV)

EMV is calculated as the sum of the expected profits for each purchase level considering the probabilities of each demand level.

\[
\text{EMV} = \sum (\text{Profit for each demand level} \times \text{Probability of that demand level})
\]

- **5000 pounds**:
- EMV = (500,000 * 0.25) + (500,000 * 0.37) + (500,000 * 0.38)
- EMV = 500,000

- **10,000 pounds**:
- EMV = (-250,000 * 0.25) + (1,000,000 * 0.37) + (1,000,000 * 0.38)
- EMV = -62,500 + 370,000 + 380,000
- EMV = 687,500

- **20,000 pounds**:
- EMV = (-1,750,000 * 0.25) + (-500,000 * 0.37) + (2,000,000 * 0.38)
- EMV = -437,500 - 185,000 + 760,000
- EMV = 137,500

**Optimal Action Based on EMV**: Purchase 10,000 pounds (EMV = 687,500).

### (e) Standard Deviation for Each Purchase Level

The standard deviation measures the variability or risk associated with each purchase level.

\[
\sigma = \sqrt{\sum (\text{Profit}_i - \text{EMV})^2 \times P_i}
\]

Let's calculate it for each purchase level:

**5000 pounds**:
\[
\sigma = \sqrt{((500,000 - 500,000)^2 \times 0.25) + ((500,000 - 500,000)^2 \times 0.37) + ((500,000 - 500,000)^2 \times 0.38)} = 0
\]

**10,000 pounds**:
\[
\sigma = \sqrt{((-250,000 - 687,500)^2 \times 0.25) + ((1,000,000 - 687,500)^2 \times 0.37) + ((1,000,000 - 687,500)^2 \times 0.38)}
\]
\[
= \sqrt{(881250000000)^{0.25} + (98125000000)^{0.37} + (98125000000)^{0.38)}
\]
\[
\sigma \approx 903,113.65
\]

**20,000 pounds**:
\[
\sigma = \sqrt{((-1,750,000 - 137,500)^2 \times 0.25) + ((-500,000 - 137,500)^2 \times 0.37) + ((2,000,000 - 137,500)^2 \times 0.38)}
\]
\[
\sigma \approx 1,866,667.75
\]

### (f) Expected Opportunity Loss (EOL)

EOL is calculated as the expected value of the regret for each action.

**Regret Matrix**:
| Purchase Level | Demand 5000 | Demand 10,000 | Demand 20,000 |
|----------------|----------------|-----------------|-----------------|
| 5000 | 0 | 0 | 1,500,000 |
| 10,000 | 750,000 | 0 | 1,000,000 |
| 20,000 | 2,250,000 | 1,500,000 | 0 |

**EOL Calculation**:
\[
\text{EOL} = \sum (\text{Regret for each demand level} \times \text{Probability of that demand level})
\]

- 5000 pounds:
- EOL = (0 * 0.25) + (0 * 0.37) + (1,500,000 * 0.38)
- EOL = 570,000

- 10,000 pounds:
- EOL = (750,000 * 0.25) + (0 * 0.37) + (1,000,000 * 0.38)
- EOL = 500,000

- 20,000 pounds:
- EOL = (2,250,000 * 0.25) + (1,500,000 * 0.37) + (0 * 0.38)
- EOL = 1,368,750

**Optimal Action Based on EOL**: Purchase 10,000 pounds (EOL = 500,000).

### (g) Expected Value of Perfect Information (EVPI)

EVPI represents the maximum amount a decision-maker would be willing to pay for perfect information.

\[
\text{EVPI} = \text{Expected Value with Perfect Information (EVwPI)} - \text{Expected Monetary Value (EMV for best decision)}
\]

**EVwPI Calculation**:
- **Demand 5000 pounds (most profitable)**: 500,000 (purchase 5000 pounds)
- **Demand 10,000 pounds (most profitable)**: 1,000,000 (purchase 10,000 pounds)
- **Demand 20,000 pounds (most profitable)**: 2,000,000 (purchase 20,000 pounds)

\[
\text{EVwPI} = (500,000 * 0.25) + (1,000,000 * 0.37) + (2,000,000 * 0.38)
\]
\[
= 125,000 + 370,000 + 760,000
\]
\[
\text{EVwPI} = 1,255,000
\]

\[
\text{EVPI} = 1,255,000 - 687,500
\]
\[
\text{EVPI} = 567,500
\]

### (h) Coefficient of Variation

The coefficient of variation (CV) is calculated as the standard deviation divided by the mean (EMV).

**5000 pounds**:
\[
CV = \frac{\sigma}{EMV} = \frac{0}{500,000} = 0
\]

**10,000 pounds**:
\[
CV = \frac{\sigma}{EMV} = \frac{903,113.65}{687,500} \approx 1.31
\]

**20,000 pounds**:
\[
CV = \frac{\sigma}{EMV} = \frac{1,866,667.75}{137,500} \approx 13.57
\]

### (i) Return-to-Risk Ratio (RTRR)

The RTRR is calculated as the ratio of EMV to the standard deviation.

**5000 pounds**:
\[
RTRR = \frac{EMV}{\sigma} = \frac{500,000}{0} = \infty
\]

**10,000 pounds**:
\[
RTRR = \frac{EMV}{\sigma} = \frac{687,500}{903,113.65} \approx 0.76
\]

**20,000 pounds**:
\[
RTRR = \frac{EMV}{\sigma} = \frac{137,500}{1,866,667.75} \approx 0.07
\]

### (j) Decision Based on EMV and EOL

Considering both EMV and EOL:
- **EMV**: The highest is 10,000 pounds (K687,500).
- **EOL**: The lowest is 10,000 pounds (K500,000).

**Optimal Action**: Purchase 10,000 pounds based on both the highest EMV and the lowest EOL for an expected profit given the risk.

These steps clearly outline making decisions regarding clams purchases for the Islander Fishing Company considering various demand levels and associated risks.