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It looks like you've uploaded a text-based image of a problem titled "An Inventory", which describes a scenario involving inventory management, cost minimization, and mathematical modeling. Since I can't view images directly, I will interpret the content based on your description and the text you've provided.

Let’s carefully analyze and solve the problem step by step.

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🔍 Problem Summary (Based on Your Text):

You are managing an inventory of a product with the following characteristics:

- Demand: 100 units per week.
- Ordering cost: $10 per order.
- Holding cost: $0.50 per unit per week.
- Lead time: 2 weeks (time between placing an order and receiving it).
- Current policy: Order 100 units every week.

The goal is to determine whether this current policy is optimal and, if not, find the optimal order quantity that minimizes total inventory costs (ordering + holding).

This is a classic Economic Order Quantity (EOQ) problem, but with a twist: the demand is weekly and there's a lead time.

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✔ Step 1: Understand the Current Policy

Currently:
- Order quantity = 100 units
- Frequency = weekly
- So, number of orders per year = 52 (assuming 52 weeks/year)

But let’s work in weeks for consistency.

#### Total Cost Under Current Policy:
- Ordering cost = $10 per order × 52 orders = $520 per year
- Average inventory = 100 / 2 = 50 units (since inventory depletes linearly)
- Holding cost = 50 units × $0.50/week × 52 weeks = $1,300 per year
- Total cost = $520 + $1,300 = $1,820 per year

> Note: This assumes we're calculating annual costs. But since everything is weekly, we can also compute weekly cost:
>
> - Weekly ordering cost = $10
> - Weekly holding cost = 50 × $0.50 = $25
> - Total weekly cost = $35

We'll use weekly cost for comparison unless otherwise needed.

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✔ Step 2: Apply EOQ Formula

The EOQ model minimizes total inventory cost:

\[
EOQ = \sqrt{\frac{2DS}{H}}
\]

Where:
- \( D \) = demand rate per week = 100 units/week
- \( S \) = ordering cost per order = $10
- \( H \) = holding cost per unit per week = $0.50

\[
EOQ = \sqrt{\frac{2 \times 100 \times 10}{0.50}} = \sqrt{\frac{2000}{0.5}} = \sqrt{4000} \approx 63.25
\]

So, optimal order quantity ≈ 63 units

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✔ Step 3: Compute Optimal Ordering Frequency

- Demand = 100 units/week
- Order size = 63 units
- Time between orders = \( \frac{63}{100} = 0.63 \) weeks ≈ every 4.4 days

But since orders are placed periodically, we might round to practical intervals.

However, the key point is that the current policy of ordering 100 units weekly is NOT optimal.

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✔ Step 4: Compare Costs

#### Optimal Policy (EOQ = 63 units):
- Number of orders per week = \( \frac{100}{63} \approx 1.59 \) orders per week → about 1.59 × 52 = 82.68 orders/year
- But better to compute weekly cost:

- Ordering cost per week = \( \frac{100}{63} \times 10 \approx 1.59 \times 10 = \$15.90 \)
- Average inventory = \( \frac{63}{2} = 31.5 \) units
- Holding cost per week = 31.5 × 0.50 = \$15.75
- Total weekly cost = \$15.90 + \$15.75 = \$31.65

Compare to current policy:
- Current weekly cost = \$35
- Optimal weekly cost = \$31.65

✔ Savings of \$3.35 per week, or about \$174 per year.

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✔ Step 5: Consider Lead Time

Lead time = 2 weeks.

This affects reorder point, not EOQ.

- Reorder point = demand during lead time = 100 units/week × 2 weeks = 200 units

So, when inventory drops to 200 units, place an order.

But note: EOQ doesn’t change due to lead time — only reorder point does.

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✔ Final Answer & Explanation

No, the current policy of ordering 100 units every week is not optimal.

#### Optimal Solution:
- Order quantity: ~63 units
- Order frequency: Approximately every 0.63 weeks (about every 4.4 days)
- Reorder point: 200 units (due to 2-week lead time)

#### Cost Savings:
- Current weekly cost: \$35
- Optimal weekly cost: \$31.65
- Annual savings: (\$3.35/week) × 52 weeks ≈ \$174/year

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📌 Conclusion

Use the EOQ model to minimize total inventory cost. The current policy overorders and incurs higher holding costs. By reducing order size to around 63 units, you reduce both ordering and holding costs.

> 💡 Recommendation: Switch to ordering approximately 63 units every 4.4 days, and reorder when inventory reaches 200 units.

Let me know if you'd like a graph or table showing cost comparison!