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6.5 EOQ Without the Instantaneous Receipt Assumption

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.

Annual Carrying Cost for 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:

\begin{array}{l} Q & = & number of units per order, or production run \\ C_{s} & = & setup cost \\ C_{h} & = & holding or carrying cost per unit per year \\ p & = & daily production rate \\ d & = & daily demand rate \\ t & = & length of production run in days \end{array}

$\begin{array}{l} Q & = & number of units per order, or production run \\ C_{s} & = & setup cost \\ C_{h} & = & holding or carrying cost per unit per year \\ p & = & daily production rate \\ d & = & daily demand rate \\ t & = & length of production run in days \end{array}$

A line graph showing when production is and is not taking place during the inventory cycle. — Figure 6.5 Inventory Control and the Production Process

Figure 6.5 Full Alternative Text

The maximum inventory level is as follows:

\begin{array}{l} (Total produced during the production run) - (Total used during production run) \\ = (Daily production rate) (Number of days of production) \\ - (Daily demand) (Number of days of production) \\ = (p t) - (d t) \end{array}

$\begin{array}{l} (Total produced during the production run) - (Total used during production run) \\ = (Daily production rate) (Number of days of production) \\ - (Daily demand) (Number of days of production) \\ = (p t) - (d t) \end{array}$

Since

Total produced = Q = p t,

$Total produced = Q = p t,$

we know that

\begin{array}{l} t & = & \frac{Q}{p} \\ Maximum inventory level & = & p t - d t = p \frac{Q}{p} - d \frac{Q}{p} = Q (1 - \frac{d}{p}) \end{array}

$\begin{array}{l} t & = & \frac{Q}{p} \\ Maximum inventory level & = & p t - d t = p \frac{Q}{p} - d \frac{Q}{p} = Q (1 - \frac{d}{p}) \end{array}$

Since the average inventory is one-half of the maximum, we have

Average inventory = \frac{Q}{2} (1 - \frac{d}{p})

$Average inventory = \frac{Q}{2} (1 - \frac{d}{p})$ (6-9)

and

Annual holding cost = \frac{Q}{2} (1 - \frac{d}{p}) C_{h}

$Annual holding cost = \frac{Q}{2} (1 - \frac{d}{p}) C_{h}$ (6-10)

Annual Setup Cost or Annual Ordering Cost

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,

Annual setup cost = \frac{D}{Q} C_{s}

$Annual setup cost = \frac{D}{Q} C_{s}$ (6-11)

and

Annual ordering cost = \frac{D}{Q} C_{o}

$Annual ordering cost = \frac{D}{Q} C_{o}$ (6-12)

Determining the Optimal Production Quantity

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,

\begin{array}{l} Annual holding cost & = & Annual setup cost \\ \frac{Q}{2} (- \frac{d}{p}) C_{h} & = & \frac{D}{Q} C_{s} \end{array}

$\begin{array}{l} Annual holding cost & = & Annual setup cost \\ \frac{Q}{2} (- \frac{d}{p}) C_{h} & = & \frac{D}{Q} C_{s} \end{array}$

Solving this for Q, we get the optimal production quantity $(Q^{*}) :$ $(Q^{*}) :$

Q^{*} = \sqrt{\frac{2 D C_{s}}{C_{h} (1 - \frac{d}{p})}}

$Q^{*} = \sqrt{\frac{2 D C_{s}}{C_{h} (1 - \frac{d}{p})}}$ (6-13)

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 $C_{o}$ $C_{o}$ replaces $C_{s}$ $C_{s}$ in the formula.

Production Run Model

\begin{array}{l} Annual holding cost & = & \frac{Q}{2} (1 - \frac{d}{p}) C_{h} \\ Annual setup cost & = & \frac{D}{Q} C_{s} \\ Optimal production quantity Q^{*} & = & \sqrt{\frac{2 D C_{s}}{C_{h} (1 - \frac{d}{p})}} \end{array}

$\begin{array}{l} Annual holding cost & = & \frac{Q}{2} (1 - \frac{d}{p}) C_{h} \\ Annual setup cost & = & \frac{D}{Q} C_{s} \\ Optimal production quantity Q^{*} & = & \sqrt{\frac{2 D C_{s}}{C_{h} (1 - \frac{d}{p})}} \end{array}$

Brown Manufacturing Example

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:

\begin{array}{l} Annual demand & = & D = 10, 000 units \\ Setup cost & = & C_{s} = $ 100 \\ Carrying cost & = & C_{h} = $ 0.50 per units per year \\ Daily production rate & = & p = 80 units daily \\ Daily demand rate & = & d = 60 units daily \end{array}

$\begin{array}{l} Annual demand & = & D = 10, 000 units \\ Setup cost & = & C_{s} = $ 100 \\ Carrying cost & = & C_{h} = $ 0.50 per units per year \\ Daily production rate & = & p = 80 units daily \\ Daily demand rate & = & d = 60 units daily \end{array}$

\begin{array}{l} Q^{*} & = & \sqrt{\frac{2 D C_{s}}{C_{h} (1 - \frac{d}{p})}} \\ = & \sqrt{\frac{2 \times 10, 000 \times 100}{0.5 (1 - \frac{60}{80})}} \\ = & \sqrt{\frac{2, 000, 000}{0.5 (\frac{1}{4})}} = \sqrt{16, 000, 000} \\ = & 4, 000 units \end{array}

$\begin{array}{l} Q^{*} & = & \sqrt{\frac{2 D C_{s}}{C_{h} (1 - \frac{d}{p})}} \\ = & \sqrt{\frac{2 \times 10, 000 \times 100}{0.5 (1 - \frac{60}{80})}} \\ = & \sqrt{\frac{2, 000, 000}{0.5 (\frac{1}{4})}} = \sqrt{16, 000, 000} \\ = & 4, 000 units \end{array}$

If $Q^{*} = 4, 000$ $Q^{*} = 4, 000$ units and we know that 80 units can be produced daily, the length of each production cycle will be $Q / p = 4, 000 / 80 = 50$ $Q / p = 4, 000 / 80 = 50$ 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 $D / Q = 10, 000 / 4, 000 = 2.5.$ $D / Q = 10, 000 / 4, 000 = 2.5.$ 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.

Using Excel QM for Production Run Models

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.

Screenshot showing data input and formulas for Brown Manufacturing. — Program 6.2A Excel QM Formulas and Input Data for the Brown Manufacturing Problem

Figure 6.2A Full Alternative Text

A screenshot of the solutions for Brown Manufacturing data. The data table from the previous figure is replicated. The results table is replicated as well, with data included instead of formulas. — Program 6.2B Excel QM Solutions for the Brown Manufacturing Problem

Figure 6.2B Full Alternative Text

In Action U.S. Defense Logistics Agency Saves on Inventory Costs Through Risk Management System

The U.S. Defense Logistics Agency (DLA; www.dla.mil) had cost overruns in its inventory supply chain. Some items had stable but infrequent demand, while others had highly variable demand patterns. The Logistics Management Institute (LMI; www.lmi.org) developed an inventory control solution named PNG that contained two modules: Peak Policy and Next Gen. The system utilized a risk management approach to control inventory.

Since the DLA implemented the two modules in January 2013, customer service metrics have improved, buyer workload has decreased, and inventory cost savings have totaled nearly $400 million per year.

Source: Based on Tovey C. Bachman, Pamela J. Williams, Kristen M. Cheman, Jeffrey Curtis, and Robert Carroll, “PNG: Effective Inventory Control for Items with Highly Variable Demand,” Interfaces 46, 1 (January–February 2016): 18–32, © Trevor S. Hale.

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Table of Contents for 6.5 EOQ Without the Instantaneous Receipt Assumption

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6.5 EOQ Without the Instantaneous Receipt Assumption