Answer :
Sure! Let's go through the problem step-by-step:
### Given Data:
- Production rate per day: 1200 yards
- Operating days per year: 260 days
- Annual carrying cost: \$0.63 per yard
- Order cost: \$425
- Lead time: 7 days
- We assume a 30-day month for September.
### Step 1: 3-Month Moving Average
To calculate the 3-month moving average for Month 9, we average the demands of Months 6, 7, and 8:
Month 6: 11,000 yards
Month 7: 10,000 yards
Month 8: 12,000 yards
3-Month Moving Average for Month 9:
[tex]\[ \text{Forecast for Month 9} = \frac{(11,000 + 10,000 + 12,000)}{3} = \frac{33,000}{3} = 11,000 \text{ yards} \][/tex]
This is the forecasted demand for Month 9.
### Step 2: Daily Demand
To find the daily demand for September (a 30-day month):
[tex]\[ \text{Daily Demand} = \frac{\text{Forecasted Monthly Demand (Month 9)}}{30} = \frac{11,000 \text{ yards}}{30 \text{ days}} = 366.67 \text{ yards per day} \][/tex]
### Step 3: Annual Demand
Given the daily demand, we can calculate the annual demand assuming 260 operating days:
[tex]\[ \text{Annual Demand} = \text{Daily Demand} \times \text{Operating Days} = 366.67 \text{ yards/day} \times 260 \text{ days} = 95,333.33 \text{ yards} \][/tex]
### Step 4: Optimal Order Quantity (EOQ)
The Economic Order Quantity (EOQ) is given by the formula:
[tex]\[ \text{EOQ} = \sqrt{\frac{2DS}{H}} \][/tex]
where:
- \(D\) is the annual demand,
- \(S\) is the order cost (\$425),
- \(H\) is the annual carrying cost (\$0.63).
Plugging in the values:
[tex]\[ \text{EOQ} = \sqrt{\frac{2 \times 95,333.33 \text{ yards} \times \[tex]$425}{\$[/tex]0.63}} \approx 11,341.27 \text{ yards}
\][/tex]
### Step 5: Total Inventory Cost
The total annual inventory cost consists of the ordering cost and the carrying cost.
1. Ordering Cost:
[tex]\[ \text{Ordering Cost} = \frac{D}{\text{EOQ}} \times S = \frac{95,333.33 \text{ yards}}{11,341.27 \text{ yards}} \times \[tex]$425 \approx \$[/tex]3,572.49
\][/tex]
2. Carrying Cost:
[tex]\[ \text{Carrying Cost} = \frac{\text{EOQ}}{2} \times H = \frac{11,341.27 \text{ yards}}{2} \times \[tex]$0.63 \approx \$[/tex]3,572.49
\][/tex]
The total annual inventory cost:
[tex]\[ \text{Total Annual Cost} = \text{Ordering Cost} + \text{Carrying Cost} \approx \[tex]$3,572.49 + \$[/tex]3,572.49 = \$7,145
\][/tex]
The total monthly cost:
[tex]\[ \text{Monthly Cost} = \frac{\text{Total Annual Cost}}{12} \approx \frac{\[tex]$7,145}{12} \approx \$[/tex]595.42
\][/tex]
### Step 6: Reorder Point
The reorder point is calculated by multiplying the daily demand by the lead time:
[tex]\[ \text{Reorder Point} = \text{Daily Demand} \times \text{Lead Time} = 366.67 \text{ yards/day} \times 7 \text{ days} \approx 2,566.67 \text{ yards} \][/tex]
### Results Summarized:
- Forecasted demand for Month 9: 11,000 yards
- Daily demand: 366.67 yards/day
- Optimal order quantity (EOQ): 11,341.27 yards
- Total annual inventory cost: \$7,145
- Total monthly inventory cost: \$595.42
- Reorder point: 2,566.67 yards
### Given Data:
- Production rate per day: 1200 yards
- Operating days per year: 260 days
- Annual carrying cost: \$0.63 per yard
- Order cost: \$425
- Lead time: 7 days
- We assume a 30-day month for September.
### Step 1: 3-Month Moving Average
To calculate the 3-month moving average for Month 9, we average the demands of Months 6, 7, and 8:
Month 6: 11,000 yards
Month 7: 10,000 yards
Month 8: 12,000 yards
3-Month Moving Average for Month 9:
[tex]\[ \text{Forecast for Month 9} = \frac{(11,000 + 10,000 + 12,000)}{3} = \frac{33,000}{3} = 11,000 \text{ yards} \][/tex]
This is the forecasted demand for Month 9.
### Step 2: Daily Demand
To find the daily demand for September (a 30-day month):
[tex]\[ \text{Daily Demand} = \frac{\text{Forecasted Monthly Demand (Month 9)}}{30} = \frac{11,000 \text{ yards}}{30 \text{ days}} = 366.67 \text{ yards per day} \][/tex]
### Step 3: Annual Demand
Given the daily demand, we can calculate the annual demand assuming 260 operating days:
[tex]\[ \text{Annual Demand} = \text{Daily Demand} \times \text{Operating Days} = 366.67 \text{ yards/day} \times 260 \text{ days} = 95,333.33 \text{ yards} \][/tex]
### Step 4: Optimal Order Quantity (EOQ)
The Economic Order Quantity (EOQ) is given by the formula:
[tex]\[ \text{EOQ} = \sqrt{\frac{2DS}{H}} \][/tex]
where:
- \(D\) is the annual demand,
- \(S\) is the order cost (\$425),
- \(H\) is the annual carrying cost (\$0.63).
Plugging in the values:
[tex]\[ \text{EOQ} = \sqrt{\frac{2 \times 95,333.33 \text{ yards} \times \[tex]$425}{\$[/tex]0.63}} \approx 11,341.27 \text{ yards}
\][/tex]
### Step 5: Total Inventory Cost
The total annual inventory cost consists of the ordering cost and the carrying cost.
1. Ordering Cost:
[tex]\[ \text{Ordering Cost} = \frac{D}{\text{EOQ}} \times S = \frac{95,333.33 \text{ yards}}{11,341.27 \text{ yards}} \times \[tex]$425 \approx \$[/tex]3,572.49
\][/tex]
2. Carrying Cost:
[tex]\[ \text{Carrying Cost} = \frac{\text{EOQ}}{2} \times H = \frac{11,341.27 \text{ yards}}{2} \times \[tex]$0.63 \approx \$[/tex]3,572.49
\][/tex]
The total annual inventory cost:
[tex]\[ \text{Total Annual Cost} = \text{Ordering Cost} + \text{Carrying Cost} \approx \[tex]$3,572.49 + \$[/tex]3,572.49 = \$7,145
\][/tex]
The total monthly cost:
[tex]\[ \text{Monthly Cost} = \frac{\text{Total Annual Cost}}{12} \approx \frac{\[tex]$7,145}{12} \approx \$[/tex]595.42
\][/tex]
### Step 6: Reorder Point
The reorder point is calculated by multiplying the daily demand by the lead time:
[tex]\[ \text{Reorder Point} = \text{Daily Demand} \times \text{Lead Time} = 366.67 \text{ yards/day} \times 7 \text{ days} \approx 2,566.67 \text{ yards} \][/tex]
### Results Summarized:
- Forecasted demand for Month 9: 11,000 yards
- Daily demand: 366.67 yards/day
- Optimal order quantity (EOQ): 11,341.27 yards
- Total annual inventory cost: \$7,145
- Total monthly inventory cost: \$595.42
- Reorder point: 2,566.67 yards
