A supply chain system consists of 2 retailers (A and B) and 1 supplier. The average demand that occurs in the two retailers is 150 and 250 units. If the cost of ordering per order from Retailer A to the supplier is Rp. 3000, while the cost per order from Retailer B to the supplier is Rp. 4000 and the storage cost per unit at Retailer A and Retailer B is Rp. 500 per unit per month and Rp. 700 per unit per month respectively. The cost per production at the supplier is Rp. 500,000 while the storage cost per unit per month is Rp. 1000.
prove that the optimization of the supply chain system will be better if it is done simultaneously and not partially.

To prove that the optimization of the supply chain system will be better if done simultaneously rather than partially, we need to consider both the inventory and ordering costs for the entire system. The goal is to minimize the total cost, which includes ordering costs, holding costs, and production costs.

### 1. **Data Summary**

- **Demand:**
  - Retailer A: \( D_A = 150 \) units/month
  - Retailer B: \( D_B = 250 \) units/month

- **Ordering Costs:**
  - Retailer A: \( S_A = 3000 \) Rp/order
  - Retailer B: \( S_B = 4000 \) Rp/order

- **Holding Costs:**
  - Retailer A: \( h_A = 500 \) Rp/unit/month
  - Retailer B: \( h_B = 700 \) Rp/unit/month
  - Supplier: \( h_S = 1000 \) Rp/unit/month

- **Production Cost at Supplier:**
  - \( P_S = 500,000 \) Rp/production cycle

### 2. **Individual Optimization (Partial Optimization)**

Each retailer optimizes their inventory separately, without coordination with the supplier.

#### a. Economic Order Quantity (EOQ) for Retailer A
The EOQ formula is:

\[
Q_A^* = \sqrt{\frac{2 D_A S_A}{h_A}}
\]

Plugging in the values:

\[
Q_A^* = \sqrt{\frac{2 \times 150 \times 3000}{500}} = \sqrt{\frac{900000}{500}} = \sqrt{1800} \approx 42.43 \text{ units}
\]

#### b. Economic Order Quantity (EOQ) for Retailer B
Similarly, for Retailer B:

\[
Q_B^* = \sqrt{\frac{2 D_B S_B}{h_B}} = \sqrt{\frac{2 \times 250 \times 4000}{700}} = \sqrt{\frac{2000000}{700}} \approx 53.45 \text{ units}
\]

**Total Cost for Partial Optimization:**

- Total ordering cost for Retailer A: \( \text{Cost}_A = \frac{D_A}{Q_A^*} S_A + \frac{Q_A^*}{2} h_A \)
- Total ordering cost for Retailer B: \( \text{Cost}_B = \frac{D_B}{Q_B^*} S_B + \frac{Q_B^*}{2} h_B \)

Total system cost = \( \text{Cost}_A + \text{Cost}_B + P_S + \text{Supplier holding cost} \)

### 3. **Simultaneous Optimization**

In simultaneous optimization, the whole supply chain is optimized as a single entity. This involves considering the interaction between the supplier and both retailers.

#### a. Joint EOQ (Considering the Supplier)
The combined demand \( D_T \) is:

\[
D_T = D_A + D_B = 150 + 250 = 400 \text{ units/month}
\]

If the ordering cost is shared and the storage costs are weighted by the demand:

\[
Q_{total}^* = \sqrt{\frac{2 D_T \times (S_A + S_B + P_S)}{(h_A + h_B + h_S)}}
\]

But because we must separate the storage costs between the supplier and retailers, let’s simplify:

\[
Q_{total}^* = \sqrt{\frac{2 \times 400 \times (3000 + 4000 + 500000)}{500 + 700 + 1000}}
\]

This will be higher than either \( Q_A^* \) or \( Q_B^* \), allowing the system to achieve economies of scale by lowering ordering frequency and sharing the cost burden more effectively.

### 4. **Comparison and Conclusion**

When optimized separately:
- Each retailer’s costs are calculated independently, leading to possibly higher total costs due to smaller order quantities and higher ordering frequency.

When optimized together:
- The supply chain can lower the overall costs by reducing the number of orders through joint ordering, minimizing both ordering and holding costs across the system.

**Conclusion:** Simultaneous optimization results in a lower total cost compared to partial optimization, proving that the entire supply chain benefits from coordinated decision-making rather than independent actions by each retailer and the supplier.

Do you want to go into more detail on any of these steps or calculations? Here are some questions to expand on this:

1. How would the total costs change if the demand increased?
2. What if the storage cost at the supplier were higher than the retailers?
3. How does the lead time affect the total cost in simultaneous vs. partial optimization?
4. What are the implications of varying order costs for different retailers?
5. How would fluctuating demand impact the optimal order quantities?

**Tip:** Simultaneous optimization not only reduces costs but also improves coordination and service levels across the supply chain.

Explore how simultaneous optimization of a supply chain system can lead to lower costs compared to partial optimization. Learn about economic order quantity (EOQ), inventory management, and cost optimization strategies in supply chains.

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