A manufacturing plant needs to make 100,000 lamps annually. Each lamp costs $25 to make and it costs $500 to set up the factory to produce lamps. It costs the plant $1 to store a lamp for 1 year. How many lamps should the plant produce in each batch of lamps to minimize their total costs? The quantity of lamps produced in each batch or lot that results in the lowest costs possible is called the economic lot size. To find the economic lot size, we need to analyze the costs for the plant and to model these costs. What kind of costs will be incurred? Three different types of costs are described in this problem. A fixed cost of $500 is incurred to set up the factory, even if no lamps are made. A variable cost of $25 per lamp is incurred for labor, materials, and transportation. Since the demand for lamps occurs throughout the year, we’ll need to store some of them in a warehouse at a cost of $1 per lamp for a year. Let’s try to solve this naively and simply produce all 100,000 lamps in one batch. If all of the lamps are produced in one batch, the production line will need to be set up once for a cost of $500. To produce 100,000 lamps at a cost of $25 per lamp will cost 100,000 $25 or $2,500,000. If the demand throughout the year is uniform, we can expect to have an average inventory of average inventory 
100, 000  0
 50, 000 2
It will cost $1 to store each of those lamps for storage costs of 50,000 $1 or $50,000. The total cost to produce one batch of 100,000 lamps is Total Cost  500  100, 000  25   50, 000 1  2,550,500  

 
fixed
variable
storage
The largest cost in this sum is the variable costs. If the cost to make a lamp is fixed and we can’t change the number of lamps produced each year, changing the batch size won’t affect this term. We can lower the storage costs by producing fewer lamps in each batch. But this increases the fixed costs since the production line will need to be set up more often. Suppose we produce 50,000 lamps in two lots. The production line will need to be set up twice at a cost of 2 $500 or $1000. We will still produce 100,000 lamps at a cost of $25 per lamp. This will cost 100,000 $25 or $2,500,000. The average inventory is reduced to average inventory 
50, 000  0
 25, 000 2
for storage costs of 25,000 $1 or $25,000. The total cost is now Total Cost  2  500   100, 000  25   25, 000 1  2,526, 000 

 

 
fixed
variable
storage
In this case, the storage costs dropped by $25,000, but the fixed costs increased by $500. This results in a lowering of the total costs. Continuing with this strategy, we can fill out the following table: Size of Batch Fixed Costs 100,000 500 50,000 2  500  10,000 10  500  1000 100  500 
Variable Costs 100, 000  25 
100, 000  25 
100, 000  25 
100, 000  25 
Storage Costs Total Cost 50, 000 1
500  100, 000  25   50, 000 1  2,550,500 25, 000 1
2  500   100, 000  25   25, 000 1  2,526, 000 5, 000 1 10  500   100, 000  25   5000 1  2,510, 000 500 1 100  500   100, 000  25   500 1  2,550,500 As the lot size decreases, the fixed cost increase and the storage costs decrease. Initially, the storage costs are much higher. But for smaller batch sizes, the fixed costs are much higher. Somewhere in the middle is a batch size whose total cost is as small as possible. To find the batch size q that minimizes the total cost, we need to find a function that models the total cost as a function of q. Examine the patterns in the table. Each fixed cost in the second column is a product of $500 and the number of batches that will be produced. We get the number of batches by . This means a batch size of q will have fixed costs of dividing 100,000 by the batch size, 100,000
q
Fixed Costs 
100, 000
 500  q
In every row of the second column, the variable costs are the same. So changing the size of the batch has no effect on the variable costs. The storage costs are the product of the average inventory and the unit storage cost. If the batch size is q, the storage costs are Storage Costs 
q
1 2
Let’s add these terms to the table. Size of Batch Fixed Costs Variable Costs Storage Costs Total Cost 100,000 500 100, 000  25  50, 000 1
500  100, 000  25   50, 000 1  2,550,500 50,000 2  500  100, 000  25  25, 000 1
2  500   100, 000  25   25, 000 1  2,526, 000 10,000 10  500  100, 000  25  5, 000 1 10  500   100, 000  25   5000 1  2,510, 000 500 1 100  500   100, 000  25   500 1  2,550,500 1000 q 100  500  100, 000  25  100,000
q
 500  100, 000  25  q
2
1 100, 000
q
 500   100, 000  25  1 q
2
Notice that the expression in the last row for the total cost preserves the pattern for all values of q above it. Thus the total cost as a function of the batch size q is TC (q) 
100, 000
q
 500   100, 000  25  1 q
2
Before we take the derivative to find the critical points, let’s simplify this function, TC (q ) 
q
50, 000, 000
 2,500, 000 
q
2
1
 50, 000, 000q 1  2,500, 000  q
2 Using the power rule, the derivative is computed as TC (q )  50, 000, 000q 2 
50, 000, 000 1


q2
2
1
2
Notice that the variable costs drop out of the critical point calculation meaning that the variable costs have nothing to do with the economic lot size. This function is undefined at q  0 . But a batch size of 0 is not a reasonable answer since a total of 100,000 lamps must be made. More critical points can be found by setting the derivative equal to 0: 
50, 000, 000 1
  0 q2
2
Clear the fractions by multiplying each term by 2q 2 to yield 100, 000, 000  q 2  0 Add ‐100,000,000 to both sides, q 2  100, 000, 000 Square root both sides of the equation to give quantities q equal to 10,000. Only the quantity 10,000 lamps makes sense. But is this critical point a relative minimum or a relative maximum? To check, we’ll substitute the critical point into the second derivative and determine the concavity at the critical point. The second derivative is TC (q ) 
100, 000, 000
q3
At the critical point, TC (10, 000) 
100, 000, 000
0 10, 0003
so the function is concave up. This tells us that q  10, 000 is a relative minimum. The total cost at that point is TC (10, 000) 
50, 000, 000
10, 000
 2,500, 000 
 2,510, 000 10, 000
2
meaning that a lot size of 10,000 lamps leads to the lowest total cost possible of $2,510,000.