Math Problem Statement

A company needs to decide between expanding its existing plant, building a new one, or subcontracting to meet rising demand for compact discs. The net present values (NPVs) for each alternative are provided under three demand scenarios: rising faster, continuing at the current rate, or increasing slower/falling. Probabilities for these scenarios and a forecast with conditional probabilities are also provided. Determine the best course of action, calculate the expected value of perfect information (EVPI), and assess whether the decision changes in light of the forecast.

Solution

To solve this problem step by step:

Part (a)

Step 1: Calculate the expected net present value (NPV) for each alternative.

Given probabilities for the demand scenarios are:

  • Demand rising faster: 60%60\% or 0.60.6
  • Demand rising at the current rate: 30%30\% or 0.30.3
  • Demand increasing at a slower rate or falling: 10%10\% or 0.10.1

We calculate the expected NPV for each alternative:

  1. Expand: Expected NPV=(500×0.6)+(400×0.3)+(150×0.1)\text{Expected NPV} = (500 \times 0.6) + (400 \times 0.3) + (-150 \times 0.1) =300+12015=405= 300 + 120 - 15 = 405

  2. Build a new plant: Expected NPV=(700×0.6)+(200×0.3)+(300×0.1)\text{Expected NPV} = (700 \times 0.6) + (200 \times 0.3) + (-300 \times 0.1) =420+6030=450= 420 + 60 - 30 = 450

  3. Subcontract: Expected NPV=(200×0.6)+(150×0.3)+(50×0.1)\text{Expected NPV} = (200 \times 0.6) + (150 \times 0.3) + (-50 \times 0.1) =120+455=160= 120 + 45 - 5 = 160

Step 2: Determine the best course of action.

The highest expected NPV is 450450 for "Build a new plant." Therefore, the company should choose to build a new plant.


Part (b)

Step 1: Use the forecast probabilities to adjust the decision.

The forecast suggests demand will rise at the current rate. The revised probabilities (from the forecast) are:

  • Probability of actual demand rising faster: P=0.3P = 0.3
  • Probability of actual demand continuing at the current rate: P=0.7P = 0.7
  • Probability of actual demand increasing slower/falling: P=0.4P = 0.4

Now, we re-calculate the expected NPV based on these probabilities.

  1. Expand: Expected NPV=(500×0.3)+(400×0.7)+(150×0.4)\text{Expected NPV} = (500 \times 0.3) + (400 \times 0.7) + (-150 \times 0.4) =150+28060=370= 150 + 280 - 60 = 370

  2. Build a new plant: Expected NPV=(700×0.3)+(200×0.7)+(300×0.4)\text{Expected NPV} = (700 \times 0.3) + (200 \times 0.7) + (-300 \times 0.4) =210+140120=230= 210 + 140 - 120 = 230

  3. Subcontract: Expected NPV=(200×0.3)+(150×0.7)+(50×0.4)\text{Expected NPV} = (200 \times 0.3) + (150 \times 0.7) + (-50 \times 0.4) =60+10520=145= 60 + 105 - 20 = 145

Step 2: Determine the new best course of action.

The highest expected NPV is now 370370 for "Expand the existing plant." Based on the revised probabilities, the company should choose to expand the existing plant.


Summary of Decisions:

  1. Part (a): Build a new plant (Expected NPV: 450).
  2. Part (b): Expand the existing plant (Revised Expected NPV: 370).

Let me know if you'd like detailed explanations for any part of the calculations.


Questions to Explore Further:

  1. How would changes in the probabilities impact the expected NPV for each alternative?
  2. What are the risks of overestimating or underestimating demand scenarios in this context?
  3. How might additional forecast data (e.g., economic trends) refine the decision-making process?
  4. Would including a risk premium or discount rate alter the recommended decision?
  5. How sensitive is the optimal choice to variations in the estimated NPVs?

Tip:

Always validate decision-making models with real-world data and sensitivity analysis to account for uncertainty and changing market conditions.

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Math Problem Analysis

Mathematical Concepts

Expected Value
Decision Theory
Probability

Formulas

Expected Value (EV) = Σ (Probability × Outcome)
Expected Value of Perfect Information (EVPI) = EV with perfect information - EV without perfect information

Theorems

Law of Total Probability

Suitable Grade Level

Undergraduate or MBA Level