Economic Order Quantity

Term Paper: EOQ & EPQ

Economic Order Quantity (EOQ) & Economic Production Quantity (EPQ) Inventories are, e.g., idle goods in storage, raw materials waiting to be used, in-process materials, finished goods, individuals. A good inventory model allows us to: Smooth out time gap between supply and demand; e.g., supply of corn. Contribute to lower production costs; e.g., produce in bulk Provide a way of "storing" labour; e.g., make more now, free up labour later Provide quick customer service; e.g., convenience.

For every type of inventory models, the decision maker is concerned with the main question: When should a replenishment order be placed? One may review stock levels at a fixed interval or re-order when the stock falls to a predetermined level; e. g., a fixed safety stock level.

Keywords, Notations Often Used for the Modeling and Analysis Tools for Inventory Control Manish Kumar, Department of Business Administration (LUMBA), University of Lucknow.

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Term Paper: EOQ & EPQ

Demand rate: A constant rate at which the product is withdrawn from x inventory Ordering cost: It is a fixed cost of placing an order independent of the C1 amount ordered. Set-up cost Holding cost: C2

This cost usually includes the lost investment income caused by having the asset tied up in inventory. This is not a real cash flow, but it is an important component of the cost of inventory. If P is the unit price of the product, this component of the cost is often computed by iP, where i a percentage that includes opportunity cost, allocation cost, insurance, etc. It is a discount rate or interest rate used to compute the inventory holding cost.

Shortage cost: There might be an expense for which a shortage occurs. C3 Backorder cost: C4

This cost includes the expense for each backordered item. It might be also an expense for each item proportional to the time the customer must wait.

Lead time: L

It is the time interval between when an order is placed and when the inventory is replenished.

The widely used deterministic and probabilistic models are presented in the following sections. The Classical EOQ Model: This is the simplest model constructed based on the conditions that goods arrive the same day they are ordered and no shortages allowed. Clearly, one must reorder when inventory reaches 0, or considering lead time L The following figure shows the change of the inventory level with time:

Manish Kumar, Department of Business Administration (LUMBA), University of Lucknow.

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Term Paper: EOQ & EPQ

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The figure shows time on the horizontal axis and inventory level on the vertical axis. We begin at time 0 with an order arriving. The amount of the order is the lot size, Q. The lot is delivered all at one time causing the inventory to shoot from 0 to Q instantaneously. Material is withdrawn from inventory at a constant demand rate, x, measured in units per time. After the inventory is depleted, the time for another order of size Q arrives, and the cycle repeats. The inventory pattern shown in the figure is obviously an abstraction of reality in that we expect no real system to operate exactly as shown. The abstraction does provide an estimate of the optimum lot size, called the economic order quantity (EOQ), and related quantities. We consider alternatives to those assumptions later on these pages. Ordering Total Cost =

C1x/Q

Holding +

C2/(2Q)

The Optimal Ordering Quantity = Q* = (2xC1/C2) 1/2, therefore, The Optimal Reordering Cycle = T* = [2C1/(xC2)]1/2 Numerical Example 1: Suppose your office uses 1200 boxes of typing paper each year. You are to determine the quantity to be ordered, and how often to order it. The data to consider are the demand rate x = 1200 boxes per year; the ordering cost C1 = $5 per order; holding cost C2 = $1.20 per box, per year. The optimal ordering quantity is Q* = 100 boxes, this gives number of orders = 1200/100 = 12, i.e., 12 orders per year, or once a month. Notice that one may incorporate the Lead Time (L), that is the time interval between when an order is placed and when the inventory is replenished. Manish Kumar, Department of Business Administration (LUMBA), University of Lucknow.

Term Paper: EOQ & EPQ

Models with Shortages: When a customer seeks the product and finds the inventory empty, the demand can be satisfied later when the product becomes available. Often the customer receives some discount which is included in the backorder cost. A model with backorders is illustrated in the following figure:

In this model, shortages are allowed some time before replenishment. Regarding the response of a customer to the unavailable item, the customer will accept later delivery which is called a backorder. There are two additional costs in this model; namely, the shortage cost (C3), and the backorder cost (C4). Since replenishments are instantaneous, backordered items are delivered at the time of replenishment and these items do not remain in inventory. Backorders are as a negative inventory; so the minimum inventory is a negative number; therefore the difference between the minimum and maximum inventory is the lot size. Ordering Total Cost =

xC1/Q

Holding +

(Q-S)2C2/(2Q) +

Shortage + Backorder xSC3/Q + S2C4/(2Q)

If xC3 2 < 2C1C2, then Q* = M/(C2C4), and S* = M/(C2C4 +C42) - (xC3)/(C2 + C4), where, M = {xC2C4[2C1(C2 + C4) - C32]}1/2 Otherwise, Manish Kumar, Department of Business Administration (LUMBA), University of Lucknow.

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Term Paper: EOQ & EPQ Q* = (2xC1/C2)1/2, with S* = 0. However, if shortage cost C3 = 0, the above optimal decision values will reduce to: Q* = [2xC1(C2 + C4)/(C2C4)]1/2, and , S* = [2xC1C2/(C2C4 + C42)]1/2 Numerical Example 2: Given C3 = 0, and C4 = 2 C2, would you choose this model? Since S* = Q*/3 under this condition, the answer is, a surprising "Yes". One third of orders must be back-ordered. Numerical Example 3: Consider the numerical example no. 1 with shortage cost of C4 = $2.40 per unit per year. The optimal decision is to order Q* = 122 units, allowing shortage of level S = 81.5 units. Production and Consumption Model: The model with finite replenishments is illustrated in the following figure:

Rather than the lot arrives instantaneously, the lot is assumed to arrive continuously at a production rate K. This situation arises when a production process feeds the inventory and the process operates at the rate K greater than the demand rate x. The maximum inventory level never reaches Q because material is withdrawn at the same time it is being produced. Production takes place at the beginning of the cycle. At the end of production period, the inventory is drawn down at the demand rate x until it reaches 0 at the end of the cycle. Ordering

Holding

Manish Kumar, Department of Business Administration (LUMBA), University of Lucknow.

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Term Paper: EOQ & EPQ

Total Cost =

xC1/Q

+

(K-x)QC2/ (2K)

Optimal Run Size Q* = {(2C1xK)/[C2(K - x)] }1/2 Run Length = Q*/K Depletion Length = Q*(K-x)/(xK) Optimal Cycle T* = {(2C1)/[C2x(1 - x/K)] }1/2 Numerical Example 3: Suppose the demand for a certain energy saving device is x = 1800 units per year (or 6 units each day, assuming 300 working days in a year). The company can produce at an annual rate of K = 7200 units (or 24 per day). Set up cost C1 = $300. There is an inventory holding cost C2 = $36 per unit, per year. The problem is to find the optimal run size, Q. Q* = 200 units per production run. The optimal production cycle is 200/7200 = 0.0278 years, that is 8 and 1/3 of a day. Number of cycle per year is 1800/200 = 9 cycles. Production and Consumption with Shortages: Suppose shortages are permitted at a backorder cost C4 per unit, per time period. A cycle will now look like the following figure:

If we permit shortages, the peak shortage occurs when production commences at the beginning of a cycle. It can be shown that: Optimal Production = q* = {[(2C1x)/C2][K/(K- x)][(C2+C4)/C4]}1/2 Period per Cycle Is: T = q/x Optimal Inventory Is: Q* = t2(K-x) Manish Kumar, Department of Business Administration (LUMBA), University of Lucknow.

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Term Paper: EOQ & EPQ Optimal Shortage Is: P* = t1(K-x); Total Cost Is: TC = {[(C2t22 + C4t12)(K-x)] + [(2C1x)/K] }/ {2(t1+t2)}, where, t1 = {[2xC1C2]/[C4K(K-x)(C2+C4)]}1/2, and t2 = {[2xC1C4]/[C2K(K-x)(C2+C4)]}1/2 You may like using Inventory Control Models JavaScript for checking your computation. You may also perform sensitivity analysis by means of some numerical experimentation for a deeper understanding of the managerial implications in dealing with uncertainties of the parameters of each model Further Reading: Zipkin P., Foundations of Inventory Management, McGraw-Hill, 2000. Optimal Order Quantity Discounts The solution procedure for determination of the optimal order quantity under discounts is as follows: • Step 1: Compute Q for the unit cost associated with each discount category. • Step 2: For those Q that are too small to receive the discount price, adjust the order quantity upward to the nearest quantity that will receive the discount price. • Step 3: For each order quantity determined from Steps 1 and 2, compute the total annual inventory price using the unit price associated with that quantity. The quantity that yields the lowest total annual inventory cost is the optimal order quantity. Quantity Discount Application: Suppose the total demand for an expensive electronic machine is 200 units, with ordering cost of $2500, and holding cost of $190, together with the following discount price offering: Order Size 1-49 50-89 90+

Price $1400 $1100 $900

Manish Kumar, Department of Business Administration (LUMBA), University of Lucknow.

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Term Paper: EOQ & EPQ

The Optimal Ordering Quantity: Q* = (2xC1/C2) 1/2 = [ 2(2500)(200)/190] 1/2 = 72.5 units. The total cost is = [(2500)(200)/72.5] + [(190)(72.5)/2] + [(1100)(200)] = $233784 The total cost for ordering quantity Q = 90 units is: TC(90) = [(2500)(200)/90] + [(190)(90)/2] + [(900)(200)] = $233784, this is the lowest total cost order quantity. Therefore, should order Q = 90 units

Manish Kumar, Department of Business Administration (LUMBA), University of Lucknow.

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