A LotSizing Case Study of a Socket Manufacturer
One of my customers is meeting with the following problem: how to optimize production batch size during an upcoming national promotion that will greatly increase demand?
The end products are sets of sockets, each of which has a different diameter.
An important challenge is that demand is in “set” that include many different diameters but actual production is grouped by “sockets of the same diameter”.
This is because significant changeover time is required to setup the bottleneck forming machine for production of a given diameter. Such changeover time varies from 24 hours. In some cases it can take an entire shift. Therefore until now, they have grouped sockets of the same diameter by product batches. Such “lotforlot” production is simple to execute and minimizes the number of changeover required.
However, their business model has recently changed. Marketing is trying to conduct a national promotion on certain sets. Therefore the demand of a number of sets will be greatly increased. This creates a new problem for using “lotforlot” rules in production. If you focus on making 1 diameter before making the next one, then you will accumulate a lot of inventory before you can ship sets. The cost of holding inventory and the responsiveness to demand fluctuation will become a much significant problem than when demand is in smaller batches. Given that, should you divide the demand quantity into smaller batches and if so, how should you determine the size of which?
Assuming that each set has 5 pieces and each piece cost $10, we can calculate the following 2 cases with setup cost estimated at $100 and inventory holding cost for each piece estimated at $0.04 per day.
Case 1 (Divide 10,000 pieces of demand into 2 batches, each of which 5000 pieces)
Day 1 
Day 2 
Day 3 
Day 4 
Day 5 
Day 6 
Day 7 
Day 8 
Day 9 
Day 10 

Demand 
0 
0 
0 
0 
0 
5000 
0 
0 
0 
0 
5000 

Production Qty 
1000 
1000 
1000 
1000 
1000 
1000 
1000 
1000 
1000 
1000 

Inventory 
1000 
2000 
3000 
4000 
5000 
1000 
2000 
3000 
4000 
5000 

Setup cost 
100 
100 
100 
100 
100 
100 
100 
100 
100 
100 
1000 

Holding cost 
40 
80 
120 
160 
200 
40 
80 
120 
160 
200 
1200 

Total Cost 
140 
180 
220 
260 
300 
140 
180 
220 
260 
300 
2200 
Case 2 (Produce 10,000 as 1 batch)
Day 1 
Day 2 
Day 3 
Day 4 
Day 5 
Day 6 
Day 7 
Day 8 
Day 9 
Day 10 

Demand 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
10000 

Production Qty 
1000 
1000 
1000 
1000 
1000 
1000 
1000 
1000 
1000 
1000 

Inventory 
1000 
2000 
3000 
4000 
5000 
6000 
7000 
8000 
9000 
10000 

Setup cost 
100 
100 
100 
100 
100 
500 

Holding cost 
40 
80 
120 
160 
200 
240 
280 
320 
360 
400 
2200 

Total Cost 
40 
180 
120 
260 
200 
340 
280 
420 
360 
500 
2700 
Notice that when taking both setup cost and inventory holding cost into considerations, it is actually cheaper to divide the demand into 2 batches (Case 1).
Note that there is 5 changeover required for gather 5 diameters of sockets for each set. Therefore the smaller batches of demand, the higher setup cost but at the same time inventory holding cost is reduced.
What happen if we further divide demand into smaller batches?
Case 3 (Divide 10,000 pieces of demand into 5 batches, each of which 2000 pieces)
Day 1 
Day 2 
Day 3 
Day 4 
Day 5 
Day 6 
Day 7 
Day 8 
Day 9 
Day 10 

Demand 
0 
0 
2000 
0 
2000 
0 
2000 
0 
2000 
0 
2000 

Production Qty 
1000 
1000 
1000 
1000 
1000 
1000 
1000 
1000 
1000 
1000 

Inventory 
1000 
2000 
1000 
2000 
1000 
2000 
1000 
2000 
1000 
2000 

Setup Cost 
200 
300 
200 
300 
200 
300 
200 
300 
200 
300 
2500 

Holding Cost 
40 
80 
40 
80 
40 
80 
40 
80 
40 
80 
600 

Total Cost 
240 
380 
240 
380 
240 
380 
240 
380 
240 
380 
3100 
Our analysis is showing that further reduction of batch size from Case 1 to Case 3 has increased the total cost.
Therefore, under the given conditions, the optimal demand batch size exist somewhere between 2000 and 10,000.
How to determine the optimal batch size? At first glance, this may seem like a case that requires dynamic lotsizing procedure of WagnerWhitin or the like because demand is not constant. However, this problem actually can be simplified and hence solved in a similar way to EOQ calculation.
Without gong into the detail of the derivation of the formula, I will just give the results:
Therefore the optimal demand batch size is 10000/2=5000 which is case 1 and the total cost is about 18% lower than “lotforlot” production of case 2.
On the other hand, it is important to further reduce setup time in order to reduce overall cost and improve demand responsiveness. For example, if S is reduced from 500 to 100, N can be calculated to 4.47≈5 and hence a demand batch of 2000 with a total cost 30% of the original total cost of case 3.
From this calculation, we learn that :
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