Consider the Avionic Manufacturing Company that wishes to meet a demand of 10 units per month by purchasing the items from a vendor with a lead time of three-quarters of a month. The item cost is $2.50 per unit, replenishment cost is $3.50 per order, and the holding cost is $0.50 per unit per month. The company estimates the shortage cost to be $0.20 per unit per month. By letting the period be one month, one has D = 100, Cu = 2.5, Ch= 0.50, Cr = 3.50, Cs= 0.20 and L = 0.75. What is the optimal inventory policy and the total cost of the system under the optimal policy?

To find the optimal inventory policy and the total cost of the system under the optimal policy, we can use the Economic Order Quantity (EOQ) model. The EOQ formula is:

EOQ = √((2 * D * S) / H)

Where:
D = demand per period
S = setup or replenishment cost per order
H = holding cost per unit per period

In this case, D = 10 (demand per month), S = $3.50 (replenishment cost per order), and H = $0.50 (holding cost per unit per month). Plugging in these values:

EOQ = √((2 * 10 * 3.50) / 0.50)
EOQ = √(70 / 0.50)
EOQ = √140
EOQ ≈ 11.83

Since we cannot order a fraction of a unit, we should round the EOQ value to the nearest whole number. Therefore, the optimal inventory policy would be to order 12 units per order.

To calculate the total cost of the system under the optimal policy, we can use the total cost formula:

Total Cost = (D * Cu) + ((D * S) / EOQ) + ((EOQ * H) / 2) + ((D * L) * Cs)

Where:
Cu = item cost per unit
L = lead time in months
Cs = shortage cost per unit per month

In this case, Cu = $2.50 (item cost per unit), L = 0.75 (lead time in months), and Cs = $0.20 (shortage cost per unit per month). Plugging in these values:

Total Cost = (10 * 2.50) + ((10 * 3.50) / 12) + ((12 * 0.50) / 2) + ((10 * 0.75) * 0.20)
Total Cost = 25 + (35 / 12) + (6) + (7.50 * 0.20)
Total Cost ≈ 25 + 2.92 + 6 + 1.50
Total Cost ≈ $35.42

Therefore, the optimal inventory policy is to order 12 units per order, and the total cost of the system under the optimal policy is approximately $35.42.

To determine the optimal inventory policy and the total cost of the system under the optimal policy, we can make use of the Economic Order Quantity (EOQ) model.

The EOQ formula is given by:
EOQ = √((2 * D * S) / H)

Where:
D = Demand per period
S = Setup cost per order
H = Holding cost per unit per period

Let's calculate the EOQ:

EOQ = √((2 * 100 * 3.50) / 0.50)
= √(700 / 0.50)
= √1400
≈ 37.42

The EOQ represents the order quantity that minimizes the total cost of inventory management. However, since the demand rate is 10 units per month, we need to determine the Reorder Point (ROP) and the Order Quantity (Q) based on this demand rate and the lead time (L).

ROP = D * L
= 10 * 0.75
= 7.5
≈ 8 (rounding up)

The ROP represents the inventory level at which a replenishment order should be placed to avoid stockout.

Since the EOQ is larger than the reorder point, we can choose to order EOQ units when the inventory level reaches the reorder point. Therefore, the optimal inventory policy is to order 37.42 units when the inventory level reaches 8 units.

Now, let's calculate the total cost of the system under the optimal policy:

Ordering cost per period (Co) = (D / Q) * S
= (100 / 37.42) * 3.50
≈ 9.43

Holding cost per period (Ch) = (Q / 2) * H
= (37.42 / 2) * 0.50
≈ 9.35

Shortage cost per period (Cs) = (Q - D) * Cs
= (37.42 - 10) * 0.20
≈ 5.88

Total cost (Ct) = Co + Ch + Cs
= 9.43 + 9.35 + 5.88
≈ $24.66

Therefore, the optimal inventory policy is to order 37.42 units when the inventory level reaches 8 units, and the total cost of the system under the optimal policy is approximately $24.66.