When a firm receives its inventory over a period of time, a new model is needed that does not require the instantaneous inventory receipt assumption. This new model is applicable when inventory continuously flows or builds up over a period of time after an order has been placed or when units are produced and sold simultaneously. Under these circumstances, the daily demand rate must be taken into account. Figure 6.5 shows inventory levels as a function of time. Because this model is especially suited to the production environment, it is commonly called the production run model.
In this model, instead of having an ordering cost, there will be a setup cost. This is the cost of setting up the production facility to manufacture the desired product. It normally includes the salaries and wages of employees who are responsible for setting up the equipment, engineering and design costs of making the setup, paperwork, supplies, utilities, and so on. The carrying cost per unit is composed of the same factors as in the traditional EOQ model, although the annual carrying cost equation changes due to a change in average inventory.
The optimal production quantity can be derived by setting the setup cost equal to the holding or carrying cost and solving for the order quantity. Let’s start by developing the expression for carrying cost. You should note, however, that making setup cost equal to carrying cost does not always guarantee optimal solutions for models more complex than the production run model.
As with the EOQ model, the carrying cost of the production run model is based on the average inventory, and the average inventory is one-half the maximum inventory level. However, since the replenishment of inventory occurs over a period of time and demand continues during this time, the maximum inventory will be less than the order quantity Q. We can develop the annual carrying, or holding, cost expression using the following variables:
The maximum inventory level is as follows:
Since
we know that
Since the average inventory is one-half of the maximum, we have
and
When a product is produced over time, setup cost replaces ordering cost. Both of these are independent of the size of the order and the size of the production run. This cost is simply the number of orders (or production runs) times the ordering cost (setup cost). Thus,
and
When the assumptions of the production run model are met, costs are minimized if the setup cost equals the holding cost. We can find the optimal quantity by setting these costs to be equal and solving for Q. Thus,
Solving this for Q, we get the optimal production quantity
It should be noted that if the situation does not involve production but rather involves the receipt of inventory gradually over a period of time, this same model is appropriate, but replaces in the formula.
Brown Manufacturing produces commercial refrigeration units in batches. The firm’s estimated demand for the year is 10,000 units. It costs about $100 to set up the manufacturing process, and the carrying cost is about 50 cents per unit per year. When the production process has been set up, 80 refrigeration units can be manufactured daily. The demand during the production period has traditionally been 60 units each day. Brown operates its refrigeration unit production area 167 days per year. How many refrigeration units should Brown Manufacturing produce in each batch? How long should the production part of the cycle shown in Figure 6.5 last? Here is the solution:
If units and we know that 80 units can be produced daily, the length of each production cycle will be days. Thus, when Brown decides to produce refrigeration units, the equipment will be set up to manufacture the units for a 50-day time span. The number of production runs per year will be This means that the average number of production runs per year is 2.5. There will be three production runs in one year with some inventory carried to the next year, so only two production runs are needed in the second year.
The Brown Manufacturing production run model can also be solved using Excel QM. Program 6.2A contains the input data and the Excel formulas for this problem. Program 6.2B provides the solution results, including the optimal production quantity, maximum inventory level, average inventory level, and number of setups.