To solve this problem, we will use the Economic Order Quantity (EOQ) model with shortages. Here's how we can calculate each requested item step-by-step.

(i) EOQ with Shortages

The EOQ with shortages is given by the formula:

$Q^* = \sqrt{\frac{2DSH}{H+P}}$

Where:

• $D$ = Annual demand = 10,000 units
• $S$ = Ordering cost = 50
• $H$ = Holding cost per unit per year = 2
• $P$ = Shortage cost per unit per year = 5

Let's calculate $Q^*$:

$Q^* = \sqrt{\frac{2 \times 10,000 \times 50 \times 2}{2 + 5}} = \sqrt{\frac{2,000,000}{7}} \approx \sqrt{285,714.29} \approx 534.52 \text{ units}$

(ii) Optimal Cycle Time (T)

The optimal cycle time $T$ is given by:

$T^* = \frac{Q^*}{D} \times 365$

Where $Q^*$ is the EOQ with shortages. Let's calculate $T^*$:

$T^* = \frac{534.52}{10,000} \times 365 \approx 19.52 \text{ days}$

(iii) Plots for the Costs

The costs include ordering cost, holding cost, and shortage cost. We plot these costs as functions of $Q$.

• Ordering Cost: $\text{Ordering Cost} = \frac{DS}{Q}$
• Holding Cost: $\text{Holding Cost} = \frac{HQ^2}{2D}$
• Shortage Cost: $\text{Shortage Cost} = \frac{P(D-Q)}{Q}$

To generate these plots, we would plot these cost functions against $Q$, the order quantity.

(iv) Equation for the Holding Cost Curve

The holding cost per cycle is given by:

$\text{Holding Cost per Cycle} = \frac{HQ^2}{2D}$

This is a parabolic function where $H$ and $D$ are constants, and $Q$ is the variable.

(v) Quantity when Holding Cost and Ordering Cost are the Same

The holding cost and ordering cost are equal when:

$\frac{HQ^2}{2D} = \frac{DS}{Q}$

Multiplying both sides by $2Q$ to clear the fractions:

$HQ^3 = 2DS^2$

Solving for $Q$:

[ Q = \sqrt[3]{\frac{2DS^2}{H}} = \sqrt[3]{\frac{2 \times 10,000 \times 50^2}{2}} \approx \sqrt[3]{25,000,000} \approx 135.72 \text{ units} ]

This is the quantity where holding cost equals ordering cost.

Summary

1. EOQ with Shortages (Q*): Approximately 534.52 units.
2. Optimal Cycle Time (T*): Approximately 19.52 days.
3. Plots: These include the ordering cost, holding cost, and shortage cost functions.
4. Holding Cost Curve: $\text{Holding Cost per Cycle} = \frac{HQ^2}{2D}$.
5. Quantity When Holding Cost Equals Ordering Cost: Approximately 135.72 units.

Would you like details on the cost plots or further explanations?

Relative Questions:

1. How do changes in shortage costs affect the EOQ with shortages?
2. What impact does an increase in demand have on the optimal cycle time?
3. How does the EOQ change if the holding cost per unit increases?
4. What is the significance of balancing holding cost and ordering cost?
5. How do different values of $S$ and $H$ affect the total cost curve?

Tip:

When managing inventory, it's crucial to regularly update parameters like holding costs and ordering costs, as they directly influence the optimal order quantity and total costs.