Optimal Order Quantity and Total Cost in Inventory Control

Month 1 2 3 4 5 6 7 8 9 10 11 12
Demand 300 700 800 900 3300 200 600 900 200 300 1000 800
Question 1 Annual demand (D) = 10000
Holding cost/unit (H) = 3
Ordering cost (S) = 504
Optimal order quantity (Q*) = sqrt (2SD/H) = sqrt (2*10000*504/3) = 1833.03 or 1833 units
Total cost = Annual holding cost + Annual ordering cost = HQ/2 + DS/Q =3*1833.03/2 +
10000*504/1833.03 = 5499.09
Number of orders place = (D/Q) = (10000/1833.03) = 5.45 or 6 orders. This means frequency of
orders is every 2 months or once every 60 days
Time between orders is 2 month.
Question 2
Let Pj, and Ej be the production and ending inventory respectively in Month-j. Also, let Yj be a
binary integer such that Yj=1 when Pj > 0 and Yj=0 otherwise. for j=1,2,...,12
Minimize Z = total cost = 3*(E1 + E2 + ... + E12) + 504*(Y1 + Y2 + ... + Y12)
Subject to,
Production balance equations:
0 + P1 - E1 = 300
E1 + P2 - E2 = 700
E2 + P3 - E3 = 800
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E11 + P12 - E12 = 800

Linking of Pj and Yj:
Pj - 99999 Yj <= 0
Pj, Ej >= 0
Yj = {0,1}
Excel solution
2Month-j 3 Demand Total 10 300 300 700 800 900 3300 200 600 900 1000 800 5 Production Pj
UM(C5:N5) UM(C6 N6) 6 Ending inventory
Solver Parameters Set Objective: O Max O Min O Value Of: By Changing Variable Cells:
Subject to the Constraints: $4 : $N$4-bi
Solution
Month-j Demand 12345 6789101112 Total 300 700 800 900 3300 200 600 900 200 300 1000
800 10000 Production, Pj Ending inventory
So, essentially, the optimal solution yields a lot-for-lot production policy. The minimum total
cost is $6,048.
With holding cost levied on the average inventory, the formulation will be as follows:
0 2Month- 3 Demand 12 800 Total 300 700 800 900 3300 200 600 900 300 1000 –
SUM(C3 N3) Production, P 800 UM(C5 N5) UM(C6 N6) 3

Optimal solution
Month-j Demand 1,2,3,4,5,6,7,8,9,10,11,12 Total 300 700 800 900 3300 200 600 900 200 300
1000 800 10000 Production, Pj Ending inventory
There will be no change because this is a lot-for-lot policy. The inventory is zero (ending and
average both).
Question 3
Total cost =12*504
= $6,048
Question 4
Part period balancing period
S=1
j=1, ps=300; HC (1,1)=3*0=0<504
j=2, ps=1000; HC (1,2) =3*(700+0)=2100> 504
t=2
p1=1000,p2=0
reset: s=3
j=3, ps=800; HC (3,3)=3*0=0<504
j=4, ps=3300; HC (3,4) =3*(800+0)=2700> 504
p3=3300,
variable cost= 2*504+( 700+0+900+0+200+0+900+0+300+0)