Answer:
The number of items in each production run so that the total costs of production and storage are minimized is 8165 items/run
Step-by-step explanation:
We will use the following variables:
Q = Quantity being ordered
Q* = the optimal order Quantity: the result being sought
D = annual Demand for the item, over the year
P = unit Production cost
S = cost of setting up a production run, regardless of the number of units in the production run (fixed cost per production run)
H = annual cost to Hold one unit
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 Cost, TC = PC + SC + HC
PC = P x D : Production Cost = unit Production cost times the annual Demand
SC = (D x S)/Q : Setting up Cost = annual Demand times cost per production setup, divided by the order Quantity (number of units)
HC = (H x Q)/2: Holding Cost = annual unit Holding cost times order Quantity (number of units), divided by 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)
D = annual demand for the item = 200000
S = cost of setting up a production run, regardless of the number of units in the production run (fixed cost per production run) = $500
H = annual cost to Hold one unit = $3
Q* = √(2×200000×500/3) = 8164.97 = 8165 items.
