Lot Sizing

This model will perform lot-sizing for determining total holding, setup and stockout costs when demands are not equal in each period. Standard methods include the economic order quantity( EOQ ), period order quantity ( POQ ), lot for lot, part-period balancing method, or Wagner- Whitin which finds the optimal schedule. Lot sizing is almost invariably discussed in association with MRP systems.

The data

Consider the following example:

Week	Demand
July 11
July 18
July 25
August 1
August 8
August 15	5
2
4
8
9
3
Holding costs $2 per unit per week and the cost to set up a production run is $21. There is no initial inventory, nor is there a lead time. A data screen for our problem appears next. The data to be given includes demands on the left and costs and other information on the right of the table.

Six methods are available in the method box above the data.

1.	Wagner- Whitin finds the production schedule which minimizes the total costs (holding + setup).

2.	Lot-for-lot is the traditional MRP way of ordering exactly what is needed in every period. (This is optimal if setup costs are 0.)

3.	The EOQ method computes the EOQ based on the average demand over the period and orders in lots of this size. Enough lots are ordered to cover the demand.

4.	The period order quantity ( POQ ) translates the EOQ into time units (number of periods) rather than an order quantity. The POQ is the length of time an EOQ order will cover, rounded off to an integer. For example, if the demand rate averages 100 units per period and the EOQ is 20 units per order, the POQ is 100/20 = 5 periods.

5.	Part period balancing. This is a well-known, widely-used heuristic that is covered in many books.

6.	User defined. The user may define the production quantities.

Demands. The demands in each period are to be given. The demands are integers.

Produce. This column is used only for the user-defined option. Enter the number of units to be produced. If an option other than user-defined is chosen, the program will revise this column and display it as output.

The information on the right includes:

Holding cost. The cost of holding one unit for one period is to be entered here. The holding cost is charged against the inventory at the end of the period.

Stockout cost. The cost of being short one unit for one period is entered here. The stockout cost is charged against units short at the end of the period. In general, stockouts only occur if the lead time prevents beginning demands from being met.

Shortage cost. The cost of being short one unit for one period is to be entered here. The shortage cost is charged against the inventory at the end of the period if the inventory is negative. Due to lead time or under the user- defined option it is possible for the inventory to be negative. (For example, the user could define production to be 0 in every period).

NOTE: We generally assume that the holding and shortage costs are charged against the inventory that is on hand at the end of the period.

Setup cost. This is the cost of each production run. It is charged only in the periods that have positive production.

Initial inventory. It is possible to allow for a situation where there is beginning inventory.

Lead time. This will offset the requirements and produce n periods earlier. (See example 3)

Example 1 - A six period lot sizing problem

The solution for our example is displayed in the preceding screen. The production column has been derived by the program. The extra columns that are derived contain the following information.

Inventory. This is the amount of inventory on hand at the end of the period. In the example, there are six units left after period 1, four units left after period 2, and three units on hand after period 5. The holding cost is charged against these amounts.

Holding cost. This is the cost of holding inventory at the end of this period. It is simply the number of units on hand multiplied by the holding cost per unit, which in this example is $2.

Setup cost. This is $0 if no production occurs or the setup cost if production occurs during this period. In the example setups occur in periods 1, 4 and 5, so the setup cost of $21 is listed in these three periods but not in the other three periods.

Totals. The total inventory, holding costs and setup costs are listed at the bottom of each column. Thirteen units were held for one month at a cost of $26.00. Three setups occurred at a total cost of $63.

Total cost. The sum of the setup and holding costs are displayed in the bottom left hand corner. The total cost in this example is $89. Since we used Wagner- Whitin this solution is optimal.

Example 2 - Using the EOQ

One of the options for placing orders is to use the economic order quantity. The EOQ is computed based on the average demand over the periods. In the example, the EOQ is based on the demand rate of 31 units per 6 periods (31/6 = 5.167). Using the holding cost and setup cost with this demand generates an EOQ of 10 (after rounding), as shown near the bottom of the screen. The program will place an order for 10 units every time that the inventory is insufficient to cover the demand. For example, the first order for 10 units is placed in period 1. This covers the demand in period 1 and the demand in period 2. In period 3 we need another order of 10 units. Using this method in this example generates four orders (which total 40 units - not 31 units) and a total cost of $142.

Note that the EOQ method will likely order more units than needed and therefore have higher holding costs than necessary.

Example 3- Using the POQ

We have modified our previous two examples by adding an initial inventory of 6 units and a lead time of 1 week. We also have changed the method to the POQ .

One of the options for placing orders is to use the period order quantity. The POQ is the EOQ but expressed in time rather than units. In our example the POQ is the 10 units divided by the average demand rate and rounded off, which is two periods, as seen in the following screen. The program will place an order to cover every two periods.

Because there is a lead time, the results screen includes an extra column for the order release. For example, the order due on July 18 must be released on July 11 due to this lead time. The order quantities are the same as without the lead time but the orders are released earlier due to the lead time. Notice that if we had used a one week lead time but not added the initial inventory to cover the first period then there would have been an unavoidable shortage in period 1.

Example 4 - Lot-for-lot ordering

Lot-for-lot (not shown) ordering is very straightforward and a common way for MRP systems to operate. The exact amount demanded is always ordered. This is optimal if there is no setup cost.