Optimal Order Quantity and Total Cost in Inventory Control

Learn about the optimal order quantity and total cost in inventory control. Understand the frequency of orders and time between orders. Find out the minimum total cost and the lot-for-lot production policy. Explore the variable holding cost and the silver-meal heuristic. Discover the summary of the results and explanations.

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
---
---
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)

=$4080
Question 5
Silver- meal Heuristic
S=1
j=1, ps=300;TVCUT(1,1) =[504+3*0]/=504
j=2, ps=1000;TVCUT(1,2) =[504+3*(700+0)]/2= 1302 (>504 reset)
thus t=1 and p1= 300
Variable cost= (2*504+0.3(700+0+900+0+200+0+900+0+300+0+800+0))
$2148
t=1
F(1) = 504, ands1∗ =1
t=2
F(2) = Min {[F(0)+ A+ HC(1,2)], [F(1)+ A+HC(2,2)}
= Min {[0+504+3×700)], [504 +504+0]} = Min {2604, 1008}=1008;
s2*=2
t=3
F(3) = Min { [F(0)+ A+ HC(1,3)], [F(1)+ A+HC(2,3)], [F(2)+ A+HC(3,3)] }
= Min { [0+504+3×800+3×1500)], [504+504+3 ×1500], [1008+504+0] }
= Min {7404, 5508, 1,512} = 1512
S4*=3
t=4
F(4) = Min { [F(0)+ A+HC(1,4)], [F(1)+ A+HC(2,4)], [F(2)+ A+HC(3,4)],
[F(3)+ A+HC(4,4)] }
= Min { [0+504+3 ×2400+3×1700+3×900], [504+504+3×1700+3×900],
[1008+504+3×900], [1512+504+0] }
= Min {16512, 8808, 4212, 2016} = 2016
S4*=4
t=5
S4*=4
P4=D4+D5+D6+D7++ D8+ D9+ D10+ D11+ D12= 8200 and P1=D1+D2+D3=1800.

Table showing the summary of the results plus an explanation
Summary
1.
Holdin
g cost
=3
Holding
cost=3
Ordering
cost=504
Optimal order
quantity=1833.03
Total
Cost=5499.09
Time
between
orders=2
months
Number of
orders
place=5.45
Frequency
of orders
is every 2
months or
once every
60 days
2. Minimum
cost=6,048
So,
essentially,
the
optimal
solution
yields a
lot-for-lot
production
policy
3. Total cost
=6,048
Total cost
= Annual
holding
cost +
Annual
ordering
cost
4. Variable
cost=$4080
Holding
cost=4080
Average
inventory
is used to
get the
holding
cost
5. P4=1800 The result
is obtained
from the
total
demands
for the
months

References
Axsater, S.(2006). Inventory control. Second Edition, MA: Springer US International Series in
operations research and management science).
Gorham, T.(1998) ‘Dynamic order quantities, ’Production and Inventory Management,9(1),
pp.75-81