Dynamic Lot Sizing

This assignment discusses the concept of dynamic lot sizing, which is an advanced form of Equilibrium Order Quantity. It explains the importance of dynamic lot sizing in inventory control and cost minimization. The assignment also provides solutions to problems related to dynamic lot sizing using different methods such as EOQ, period balancing, Silver Meal Heuristic, and Wagner-Whitin Algorithm.

Dynamic Lot Sizing
Introduction
The assignment deals with dynamic lot sizing which is an advanced form of Equilibrium Order
Quantity under which it is understood that demand varies over time and does not remain constant as in
Equilibrium Order Quantity wherein it is assumed that demand remains same throughout the year and
accordingly the order size is determined. The concept of dynamic lot sizing has been introduced by
Harvey M. Wagner and Thomson M.Whitin in 1958 and plays a very vital role in inventory control,
inventory management and cost minimization.
The dynamic lot sizing encompasses season effect of business and plans the business inventory
management strategy accordingly.
Problems
In the given case, the demand and month data has been provided here-in-below:
Month Demand
1 300
2 700
3 800
4 900
5 3300
6 200
7 600
8 900
9 200
10 300
11 1000
12 800
Total 10000
Answer 1
Ordering Cost 504
Holding Cost 3
Annual Demand' 10000
The Equilibrium Order Quantity formula has been presented here-in-below:

On the basis of said formula, the computation of Eoq has been done where in D= 10000, S= 504 and
H = 3 Per unit per year
https://corporatefinanceinstitute.com/resources/knowledge/finance/what-is-eoq-formula/
The computation of EOQ results in 1833 units per lot. Accordingly, 6 lots has to be ordered and the
frequency of order shall be 2 months. Also, holding cost shall be computed by dividing the aforesaid
quantity by 2 and multiplying the same with rate of holding.
The holding cost is = 1833*3/2= 2749.5
Ordering Cost= 6*504= 3024
Total Cost under EOQ= Holding Cost + Ordering Cost = 5773.5
Since, the price of the product has not been provides, the said detail has not been considered for
analysis.

Answer 2
The second question deals with dynamic lot sizing where in the quantity has to be determined for
ordering on the basis of demand with the objective of minimisation of cost.
The function of the said equation has been detailed as under:
Let Yi be the order placed for Month i
Xi represents the quantity ordered
N= Number of orders Or Summation Yi
Objective Function
Minimisation of Cost : (3/12) * (I1+ I2+I3……..+I12)+ 504*N
Further, X1≥ D1 …………………………… and I1= X1-D1
X1+X2≥ D1+D2……………………………..and I2=X1+X2-D1-D2
……. (12 CONSTRAINT)
Summation of Xj must be greater than equal to Dj where i=1,2,3……12
Xi ≥ MYi
Yi=0,1 Xi Integer
Thus, Minimisation of Cost : (3/12) * [(12X1+ 11X2+10X3……..+X12)- (12D1+ 11D2+ 10D3+
……..+ D12)]+ 504*N
By using solver we get we get, Y1= Y3= Y5=Y6=Y8= Y9=Y11
Accordingly, there shall be 7 order placed. Thus the cost of ordering shall be 504*7= 3528
Further, the quantity ordered shall be
X1= 1000
X3= 1700
X5=3300
X6= 800
X8=900
X9= 500
X11= 1800

Order 1000 700 1700 900 3300 800 600 900 500 300 1800 800
Demand 300 700 800 900 3300 200 600 900 200 300 1000 800
Holding Cost of
Inventory per unit 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25
Holding Cost 175 0 225 0 0 150 0 0 75 0 200 0 825
Ordering Cost 1 1 1 1 1 1 1 3528
4353
Cost under average inventory
Order 1000 1700 3300 800 900 500 1800
Demand 300 700 800 900 3300 200 600 900 200 300 1000 800
Holding Cost of Inventory per unit 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25
Holding Cost 212.5 87.5 325 112.5 412.5 175 75 112.5 100 37.5 325 100 2075
Ordering Cost 1 1 1 1 1 1 1 3528
Total Cost 5603

Answer 3
Under, the third question a lot for lot heuristic method has been used whereby the order is made every month and cost is calculated on the basis of average
inventory. The computation of the same involves ordering the quantity required for every month. Thus, this method involves 12 ordering and correspondingly
the computation has been detailed here-in-below:
K =1
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
Inventory Holding
Cost 450 1050 1200 1350 4950 300 900 1350 300 450 1500 1200 1250
Ordering Cost 504 504 504 504 504 504 504 504 504 504 504 504 6048
7298
Note : Holding cost has been initially calculated yearly and then divided by 12 at the total.
Answer 4
Under part Period balancing method, order made is generally for a period greater than zero. Under, this method heuristic method chooses a size which is
generally equal to demand of some K period wherein K>0, Further, holding and ordering cost under this method is computed for K=1,2,3,… until it reaches a
situation whereby the holding cost exceeds the ordering cost. The best K shall be a situation whereby the absolute difference between the holding cost and
ordering cost is minimum. Thus, under this method absolute difference shall be minimum. The computation considering holding cost on average inventory
has been provided as under:
K =1
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
Inventory Holding
Cost 450 1050 1200 1350 4950 300 900 1350 300 450 1500 1200 1250
Ordering Cost 504 504 504 504 504 504 504 504 504 504 504 504 6048
7298
Note : Holding cost has been initially calculated yearly and then divided by 12 at the total.

K =2
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
Inventory Holding
Cost 2550 1050 3900 1350 5550 300 3600 1950 1200 3450 3900 1200 2500
Ordering Cost 504 504 504 504 504 504 3024
5524
Note : Holding cost has been initially calculated yearly and then divided by 12 at the total.
K =3
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
Inventory Holding
Cost 4950 3450 3900 11850 5550 2100 4200 1950 1200 5850 3900 1200 4175
Ordering Cost 504 504 504 504 2016
6191
Note : Holding cost has been initially calculated yearly and then divided by 12 at the total.
On perusal, of the above table it can be seen that at K=3, holding cost exceeds the ordered cost. Further, the absolute difference of holding cost and inventory
cost is minimum under K=2. Thus, the said ordering shall be considered under period balancing method. Accordingly, the cost of holding shall be 2500 and
cost of ordering shall be 3024. Total cost shall be 5524.
Answer 5

Under Silver Meal Heuristic method, order made is generally for a period greater than zero. Under, this method heuristic method chooses a size which is
generally equal to demand of some K period wherein K>0, Further, for K =1 , the demand is for period 1 for K=2 demand is for 2 periods, for K=3 demand is
for 3 period and so on.
Further, under this method cost is computed for each period ie. K=1,2,3,,…. and so on until the average cost of the period starts increasing. The best K under
this method shall be the last period uptill which K decreases. The computation considering holding cost on average inventory has been provided as under:
K =1
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
Inventory Holding
Cost 450 1050 1200 1350 4950 300 900 1350 300 450 1500 1200 1250
Ordering Cost 504 504 504 504 504 504 504 504 504 504 504 504 6048
7298
Note : Holding cost has been initially calculated yearly and then divided by 12 at the total.
K =2
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
Inventory Holding
Cost 2550 1050 3900 1350 5550 300 3600 1950 1200 3450 3900 1200 2500
Ordering Cost 504 504 504 504 504 504 3024
5524
Note : Holding cost has been initially calculated yearly and then divided by 12 at the total.

K =3
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
Inventory Holding
Cost 4950 3450 3900 11850 5550 2100 4200 1950 1200 5850 3900 1200 4175
Ordering Cost 504 504 504 504 2016
6191
Note : Holding cost has been initially calculated yearly and then divided by 12 at the total.
K =4
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
Inventory Holding
Cost 7650 6150 3900 450 10050 4800 3600 450 6600 5850 3900 400 4483.333
Ordering Cost 504 504 504 1512
5995.333
Note : Holding cost has been initially calculated yearly and then divided by 12 at the total.
K = 6
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
Inventory Holding
Cost 18150 16650 14400 10950 5550 300 10500 8250 6600 5550 3900 1200 8500

Ordering Cost 504 504 1008
9508
Note : Holding cost has been initially calculated yearly and then divided by 12 at the total.
On perusal, of the above table it can be seen that at K=2 wherein the average cost of the period starts increasing.. Thus, the said ordering shall be considered
under silver meal heuristic method. Accordingly, the cost of holding shall be 2500 and cost of ordering shall be 3024. Total cost shall be 5524.
Answer 6
Under Wagner- Whitin Algorithm, an optimal approach is defined using F(t) where in F(t) shall mean minimum cost that shall be incurred to satisfy demand
in period 1 through t;
St* shall represent the best period to order which shall satisfy the demand for period t
On the basis of above the following computation has been made:
F(t)= Min s greater than equal to t {F(s-1)+ A+ HC(s,t)} Assuming A=10
F(1)=A.S1*=1
F(0)=0
t=1
F(1)= 10, and S1*= 1
t=2
F(2)= Min {[F(0)+ A+ HC(1,2)], [F(1)+ A+HC (2,2)}
'= Min{[0+10+700*.25],[10+ 10+0]}
'=Min{185,20}
=20
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+10+1500*.25+ 800*.25 ], [10+10+800*.25], [20+ 10+0]}
'=Min{585,220, 30}
=30
S3=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+10+2400*.25+ 1700*.25 + 900*.25 ], [10+10+1700*.25+ 900*.25],[20+10+ 900*.25] [30+ 10+0]}
'=Min{1260,670, 255,40}
40
S4=4
Similarly
F(5)=50 S5=5
F(6)=60 S6=6
F(7)=70 S7=7
F(8)=80 S8=8
F(9)=90 S9=9
F(10)=100 S10=10
F(11)=110 S11=11
F(12)=120 S12=12
Thus, there shall be 12 orders each every month

K =1
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
Inventory Holding
Cost 450 1050 1200 1350 4950 300 900 1350 300 450 1500 1200 1250
Ordering Cost 504 504 504 504 504 504 504 504 504 504 504 504 6048
7298
Summary of Results
Sl. No Answer Holding Cost Ordering Cost Total Cost
1 1 2749.5 3024 5773.5
2 2 2075 3528 5603
3 3 1250 6048 7298
4 4 2500 3024 5524
5 5 2500 3024 5524
6 6 1250 6048 7298
On perusal of the above table, it can be seen that different method give different results. However, the best method under this scenario shall be silver meal and
period balancing method where in the total cost of inventory holding is minimum. Hence, it is advised to adopt these methods.