Operations and Supply Chain Management The Core

380 OPERATIONS AND SUPPLY CHAIN MANAGEMENT
NUMBER
OF UNITS
DEMANDED
PROBABILITY
OF THIS
DEMAND
CUMULATIVE
PROBABILITY
35 0.10 0.10
36 0.15 0.25
37 0.25 0.50
38 0.25 0.75
39 0.15 0.90
40 0.10 1.00
How many units should be ordered?
Solution
The cost of underestimating demand is the loss of profit, or C u = $100 − $70 = $30 per unit. The
cost of overestimating demand is the loss incurred when the unit must be sold at salvage value,
C o = $70 − $20 = $50.
The optimal probability of not being sold is
​C​ u ​
​P ≤ ​ ______
​C​ o ​ + ​C​ u ​ = ​ ______ 30
50 + 30 ​ = .375​
From the distribution data above, this corresponds to the 37th unit.
The following is a full marginal analysis for the problem. Note that the minimum cost occurs
when 37 units are purchased.
NUMBER OF UNITS PURCHASED
UNITS DEMANDED PROBABILITY 35 36 37 38 39 40
35 0.1 0 50 100 150 200 250
36 0.15 30 0 50 100 150 200
37 0.25 60 30 0 50 100 150
38 0.25 90 60 30 0 50 100
39 0.15 120 90 60 30 0 50
40 0.1 150 120 90 60 30 0
Total cost 75 53 43 53 83 125
SOLVED PROBLEM 2
Items purchased from a vendor cost $20 each, and the forecast for next year’s demand is 1,000
units. If it costs $5 every time an order is placed for more units and the storage cost is $4 per unit
per year, answer the following questions.
a. What quantity should be ordered each time?
b. What is the total ordering cost for a year?
c. What is the total storage cost for a year?
Solution
a. The quantity to be ordered each time is
____
​Q = ​ √
​ ____ 2DS
_________
H ​ = ​ √
​ _________ 2(1,000)5 ​ = 50 units​
4