Question

The A&M Hobby Shop carries a line of radio-controlled model racing cars. Demand for the cars is assumed to be constant at a rate of 40 cars per month. The cars cost $60 each, and ordering costs are approximately $15 per order, regardless of the order size. The annual holding cost rate is 20%.

Determine the economic order quantity and total annual cost under the assumption that no backorders are permitted.

Answer 1

Economic order quantity = 34.64 cars per order ≈ 35 cars per order.

Total Annual Cost = $29,215.7

Explanation:

We will use the following variables:

Q = Quantity being ordered

Q* = the economic order Quantity: the result being sought

D = annual Demand for the item, over the year = 40 × 12 = 480

P = unit Production cost = $60

S = Ordering Cost regardless of the number of units in the order (fixed cost per production order) = $15 per order

H = annual cost to Hold one unit = 20% of $60 = $12

It is important to note which variables are annualized, which are per-order and which are per-unit.

Using the variables, here are the components of the first equation

Total Annual Cost, TC = PC + SC + HC

PC = P x D

Production Cost = (unit Production cost) × (the annual Demand)

SC = (D x S)/Q

Ordering Cost = (annual Demand) × (cost per production setup) / (the order Quantity)

HC = (H x Q)/2

Holding Cost = (annual unit Holding cost) × (order Quantity) / 2 (because throughout the year, on average the warehouse is half full).

So TC = PC + SC + HC = (P x D) + ((D x S)/Q) + ((H x Q)/2) = PD + (DS/Q) + HQ/2

To obtain the optimal order quantity, Q* that minimizes TC, at the minimum TC, dTC/dQ = 0

dTC/dQ = (H/2) – (D x S)/(Q²) = 0

(H/2) – (D x S)/(Q²) = 0

Solving for Q, which is Q* at this point.

(Q*)² = 2DS/H

Q* = √(2DS/H)

a) Economic order quantity

Q* = √(2DS/H)

D = 480 cars

S = $15

H = $12

Q* = √(2×480×15/12) = 34.64 cars per order ≈ 35 cars per order.

b) Total Annual Cost with no backorders

TC = PC + SC + HC = (P x D) + [(D x S)/Q] + [(H x Q)/2]

P = $60

D = 480

S = $15

H = $12

Q = 34.64 cars per order (Using the exact value obtained)

Q = 35 cars per order (Using the realistic approximation for the quantity of cars per order)

Total Annual Cost using the exact value for quantity of cars per order obtained.

= (60 × 480) + [(480 × 15)/34.64] + [(12×34.64)/2]

= 28,800 + 207.85 + 207.84 = $29,215.69

Total Annual Cost Using the realistic quantity of cars per order

= (60 × 480) + [(480 × 15)/35] + [(12×35)/2]

= 28,800 + 205.71 + 210 = $29,215.71

Hope this Helps!!!