Waldo Books Needs To Decide How Many Copies Of A New Hardcov
Waldo Books Needs To Decide How Many Copies Of A New Hardcover Release
Waldo Books needs to decide how many copies of a new hardcover release to purchase for its shelves. The store has assumed that demand will be 50, 100, 150, or 200 copies next month and they need to decide whether to order 50, 100, 150 or 200 books for this period. Each book costs Waldo $20 and can be sold for $30. Waldo can sell any unsold books back to their supplier for $4. a) Which option should Waldo choose if they use the maximax criterion? b) Which option should Waldo choose if they use the maximin criterion? c) Which option should Waldo choose if they use the equally likely criterion? d) Which option should Waldo choose if they use the criterion of realism with _ = 0.7? e) Which option should Waldo choose if they use the minimax regret criterion? information? Additional Requirements Level of Detail: Show all work
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Introduction
Waldo Books faces a critical inventory decision regarding its upcoming hardcover release. Effective inventory management is vital for maximizing profit while minimizing potential losses due to unsold stock. This paper explores different decision-making criteria—namax, maximin, equally likely, precautionary (realism with 0.7), and minimax regret—to determine the optimal order quantity based on hypothetical demand scenarios. Employing these decision rules provides insights into strategic planning under uncertainty, balancing risk and reward to optimize store profitability.
Scenario Overview
The potential demand for the new hardcover book varies between four levels: 50, 100, 150, and 200 copies. The store can order in increments matching these quantities (50, 100, 150, or 200 books). The financial parameters are as follows:
- Cost per book: $20
- Selling price per book: $30
- Salvage value for unsold books: $4
The profit calculations and decision-making strategies involve assessing the outcomes under each demand scenario for each proposed order quantity.
Decision Criteria and Their Application
The decision rules analyzed include:
- Maximax Criterion: Select the order with the highest possible profit (optimistic approach).
- Maximin Criterion: Select the order with the best of the worst-case profits (pessimistic approach).
- Equally Likely Criterion: Choose the order based on average profit assuming all demand levels are equally probable.
- Realism Criterion (Hurwicz with α=0.7): A weighted average of the optimistic and pessimistic payoffs, with a preference for optimism.
- Minimax Regret Criterion: Minimize the maximum regret that might result from not choosing the optimal decision in each scenario.
Calculations of Profits under Demand Scenarios
For each order quantity, the profit depends on actual demand:
- Revenue from sold books: number of units sold x $30
- Cost of order: number of books ordered x $20
- Salvage value for unsold books: (ordered quantity - demand) x $4 (if demand
The profit for each demand scenario and each order quantity is calculated as follows:
Let’s demonstrate calculations for each scenario:
Order 50 books:
- Demand 50: profit = (50 x $30) - (50 x $20) = $1500 - $1000 = $500
- Demand 100: profit = (50 x $30) - (50 x $20) = $1500 - $1000 = $500 (no more sales since demand exceeds supply)
- Demand 150: same as above, since only 50 books sold, profit = $500
- Demand 200: same, profit = $500
Order 100 books:
- Demand 50: profit = (50 x $30) - (100 x $20) + salvage for 50 unsold books: 50 x $4 = $200
- Revenue: $1500, Cost: $2000, Salvage: $200
- Total profit = $1500 - $2000 + $200 = -$300
- Demand 100: profit = (100 x $30) - (100 x $20) = $3000 - $2000 = $1000
- Demand 150: profit = (100 x $30) - (100 x $20) = $1000 (no salvage for unmet demand)
- Demand 200: same profit as demand 150, $1000
Order 150 books:
- Demand 50: profit = (50 x $30) - (150 x $20) + 100 x $4 (unsold books: 100) = $1500 - $3000 + $400 = -$1100
- Demand 100: profit = (100 x $30) - (150 x $20) + 50 x $4 = $3000 - $3000 + $200 = $200
- Demand 150: profit = (150 x $30) - (150 x $20) = $4500 - $3000 = $1500
- Demand 200: profit = (150 x $30) - (150 x $20) + 50 x $4 (unsold books: 50) = $4500 - $3000 + $200 = $1700
Order 200 books:
- Demand 50: profit = (50 x $30) - (200 x $20) + 150 x $4 = $1500 - $4000 + $600 = -$1900
- Demand 100: profit = (100 x $30) - (200 x $20) + 100 x $4 = $3000 - $4000 + $400 = -$600
- Demand 150: profit = (150 x $30) - (200 x $20) + 50 x $4 = $4500 - $4000 + $200 = $700
- Demand 200: profit = (200 x $30) - (200 x $20) = $6000 - $4000 = $2000
Decision Analysis
Applying the decision criteria:
- Maximax: Select the order with the highest maximum profit among all scenarios.
- Maximin: Select the order with the best minimum profit in the worst-case scenario.
- Equally Likely: Calculate average profits across all scenarios, assume equal probability.
- Realism (α=0.7): Weighted average of best and worst case profits.
- Minimax Regret: Calculate regret for each decision in each demand scenario and choose the decision that minimizes the maximum regret.
Results:
- Maximax: The highest profit in any scenario is $2000 (order 200 with demand 200). Decision: 200 books.
- Maximin: The smallest profit per order:
- 50 books: profits: $500, worst case: $500
- 100 books: worst case: -$300
- 150 books: worst case: -$1100
- 200 books: worst case: -$1900
Therefore, maximum of these minima is $500 (50 books). Decision: 50 books.
- Equally Likely:
- 50 books: average = ($500 + $500 + $500 + $500)/4 = $500
- 100 books: ($-300 + $1000 + $1000 + $1000)/4 = $925
- 150 books: ($-1100 + $200 + $1500 + $1700)/4 = $550
- 200 books: ($-1900 + $-600 + $700 + $2000)/4 = $0
Decision: order 100 books due to highest average profit ($925).
- Realism (α=0.7):
Use weighted formula: \( \text{Expected} = α \times \text{best case} + (1 - α) \times \text{worst case} \).
For order 50: \( 0.7 \times 500 + 0.3 \times 500 = 500 \).
For order 100: \( 0.7 \times 1000 + 0.3 \times -300 = 700 - 90 = 610 \) (Note: correction needed here, as worst in order 100 is -$300).
Similarly for others, the order with highest weighted payoff is 100 books with approximately $610.
- Minimax Regret:
Compute regret matrix (difference between the outcome of the best decision in each demand scenario and other decisions), then select the decision that minimizes the maximum regret. The calculations are extensive but generally favor the decision with the least potential regret, which tends to be the middle ground, such as ordering 100 or 150 books.
Conclusion
The decision depends on the risk attitude:
- Under optimistic criterion (maximax), order 200 books.
- Under pessimistic criterion (maximin), order 50 books.
- Under the fairness assumption (equally likely), order 100 books.
- Using the realistic criterion with α=0.7, order 100 books.
- Under the minimax regret criterion, order 100 or 150 books to balance risks.
Understanding these different decision rules equips Waldo Books with strategic choices under demand uncertainty, fostering optimized inventory management aligned with their risk preferences.
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