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M&DC Purchasing & Supply Chain: Material Management

Order Quantities

Contents

1. Introduction

2. Economic­Order Quantity (Eoq)

3. Variations Of The Eoq Model

4. Quantity Discounts

5. Use Of Eoq When Costs Are Not Known

6. Period­Order Quantity (P00)

1. Introduction

The objectives of inventory management are to provide the required level of customer service and toreduce the sum of all costs involved. To achieve these objectives, two basic questions must beanswered:

a. How much should be ordered at one time?

b. When should an order be placed?

Management must establish decision rules to answer these questions so inventory managementpersonnel know when to order and how much. Lacking any better knowledge, decision rules are oftenmade based on what seems reasonable. Unfortunately, such rules do not always produce the bestresults.

This chapter will examine methods of answering the first question, and the next chapter will deal withthe second question. First, we must decide what we are ordering and controlling.

a. Stock­Keeping Unit (SKU)

Control is exercised through individual items in a particular inventory. These are calleda stock­keeping unit (SKU). Two white shirts in the same inventory but of differentsizes or styles would be two different SKUs. The same shirt in two different inventorieswould be two different SKUs.

b. Lot­Size Decision Rules

The eighth edition of the APICS Dictionary defines a lot, or batch, as a quantityproduced together and sharing the same production costs and specifications. Followingare some common decision rules for determining what lot size to order at one time.

Lot­for­lot. The lot­for­lot

rule says to order exactly what is needed—no more—no less. Theorder quantity changes whenever requirements change. Thistechnique requires time­phased information such as provided by amaterial requirements plan or a master production schedule. Sinceitems are ordered only when needed, this system creates no unusedlot­size inventory. Because of this, it is the best method for planning“A’ items and is also used in a just­in­time environment.

Fixed­order quantity

Fixed­order quantity rules specify the number of units to be orderedeach time an order is placed for an individual item or SKU. Thequantity is usually arbitrary, such as 200 units at a time. Theadvantage to this type of rule is that it is easily understood. Thedisadvantage is that it does not minimize the costs involved.

A variation on the fixed­order quantity system is the mm­max system.In this system, an order is placed when the quantity available fallsbelow the order point (discussed in the next chapter). The quantityordered is the difference between the actual quantity available at thetime of order and the maximum. For example, if the order point is 100units, the maximum is 300 units, and the quantity actually available

عربى

Introduction to Material Management

Master Scheduling

Material Requirements Planning

Capacity Management

Production Activity Control

Purchasing

Forecasting

Order Quantities

Independent Demand Ordering Systems

Physical Inventory and Warehouse Management

Physical Distribution

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when the order is placed is 75, the order quantity is 225 units. If thequantity actually available is 80 units, an order for 220 units is placed.One commonly used method of calculating the quantity to order is theeconomic­order quantity, which is discussed in the next section.

Order “n” periods supply.

Rather than ordering a fixed quantity, inventory management canorder enough to satisfy future demand for a given period of time. Thequestion is how many periods should be covered? The answer isgiven later in this chapter in the discussion on the period­orderquantity system.

c. Costs

As shown in the last chapter, the cost of ordering and the cost of carrying inventoryboth depend on the quantity ordered. Ideally, the ordering decision rules used willminimize the sum of these two costs. The best known system is the economic­orderquantity.

2. Economic­Order Quantity (Eoq)

a. Assumptions

The assumptions on which the EOQ is based are as follows:

a. Demand is relatively constant and is known.

b. The item is produced or purchased in lots or batches and not continuously.

c. Order preparation costs and inventory­carrying costs are constant and known.

d. Replacement occurs all at once.

These assumptions are usually valid for finished goods whose demand is independentand fairly uniform. However, there are many situations where the assumptions are notvalid and the EOQ concept is of no use. For instance, there is no reason to calculatethe EOQ for made­to­order items in which the customer specifies the order quantity,the shelf life of the product is short, or the length of the run is limited by tool life or rawmaterial batch size. In material requirements planning, the lot­for­lot decision rule isoften used, but there are also several rules used that are variations of the economic­order quantity.

b. Development of the EOQ Formula

Under the assumptions given, the quantity of an item in inventory decreases at auniform rate. Suppose for a particular item, the order quantity is 200 units, and theusage rate is 100 units a week. Figure 10.1 shows how inventory would behave.

The vertical lines represent stock arriving all at once as the stock on hand reacheszero. The quantity of units in inventory then increases instantaneously by Q, thequantity ordered. This is an accurate representation of the arrival of purchased parts ormanufactured parts where all parts are received at once.From the preceding,

orderquantity 200 Average lot size inventory = = = 100 units 2 2

annualdemand 100 X 52 Number of orders per year = = . orderquantity 200

= 26 times per year

c. Example Problem

The annual demand for an SKU is 10,075 units, and it is ordered in quantities of 650

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units. Calculate the average inventory and the number of orders placed per year.

Answer

orderquantity 200 Average cycle inventory = = = 100 units 2 2

annualdemand 10.075 Number of orders per year = = . = 15.5 orderquantity 650

Notice in the example problem the number of orders per year is rounded neither up nor down. It is anaverage figure, and the actual number of orders per year will vary from year to year but will average tothe calculated figure. In the example, 16 orders will be placed in one year and 15 in the second.

d. Relevant costs

The relevant costs are as follows:

Annual cost of placing orders.

Annual cost of carrying inventory.

As the order quantity increases, the average inventory and the annual cost of carryinginventory increase, but the number of orders per year and the ordering cost decrease.It is a bit like a seesaw where one cost can be reduced only at the expense ofincreasing the other. The trick is to find the particular order quantity in which the totalcost of carrying inventory and the cost of ordering will be a minimum.

Let:

A = annual usage in unitsS = ordering cost in dollars per orderi = annual carrying cost rate as a decimal of a percentagec = unit cost in dollarsQ = order quantity in units

Then:

Annual ordering cost = number of orders x costs per order

A = x S Q

Annual carrying cost = average inventory x cost of carrying oneunit for one year

= average inventory x unit cost x carryingcost

Q = x c x i 2

Total annual costs = annual ordering costs + annual carryingcosts

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A Q = x s + x c x i Q 2

everal rules used that are variations of the economic­order quantity.

e. Example Problem

The annual demand is 10,000 units, the ordering cost $30 per order, the carrying cost20%, and the unit cost $15. The order quantity is 600 units. Calculate:

a. Annual ordering cost

b. Annual carrying cost

c. Total annual cost

Answer

A = 10,000 unitsS = $30 i = 0.20C = $15Q = 600 units A 10,000

a. annual ordering cost = x S = x $30 = $500 Q 600

Q 600b. annual carrying cost= x c x i = x $15 x 0.2 = $900 2 2

c. total annualcost = $1400

Ideally, the total cost will be a minimum. For any situation in which the annualdemand (A), the cost of ordering (S), and the cost of carrying inventory (i) are given,the total cost will depend upon the order quantity (Q).

f. Trial­and­Error Solution

Consider the following example:

a. A hardware supply distributor cSarries boxes of 3­inch bolts in stock. Theannual usage is 1000 boxes, and demand is relatively constant throughout theyear. Ordering costs are $20 per order, and the cost of carrying inventory isestimated to be 20%. The cost per unit is $5.

Let:A = 1000 unitsS = $20 per orderc = $5 per uniti = 20% = 0.20Then:

A 1000 Annual ordering cost = x S = x 20 Q Q

Q Q Annual carrying cost = x c x i = x 5 x 0.20 2 2

Total annual cost = annual ordering cost + annualcarrying cost

Figure 10.2 is a tabulation of the costs for different order quantities. The results from

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the table in Figure 10.2 are represented on the graph of Figure 10.3.

Figures 10.2 and 10.3 show the following important facts:

a. There is an order quantity in which the sum of the ordering costs and carryingcosts is a minimum.

b. This EOQ occurs when the cost of ordering equals the cost of carrying.

c. The total cost varies little for a wide range of lot sizes about EOQ.

The last point is important for two reasons. First, it is usually difficult to determineaccurately the cost of carrying inventory and the cost of ordering. Since the total costis relatively flat around the EOQ, it is not critical to have exact values. Goodapproximations are sufficient. Second, parts are often ordered in convenient packagessuch as pallet loads, cases, or dozens, and it is adequate to pick the closest packagequantity to the EOQ.

Order Quantity(Q)

OrderingCosts(AS/Q)

Carrying Costs(Qci/2)

Total Costs

50 $400 $25 $425

100 200 50 250

150 133 75 208

200 100 100 200

250 80 125 205

300 67 150 217

350 57 175 232

400 50 200 250

g. Economic­Order Quantity Formula

The previous section showed that the EOQ occurred at an order quantity in which theordering costs equal the carrying costs. If these two costs are equal, the followingformula can be derived:

This value for the order quantity is the economic­order quantity. Using the formula tocalculate the EOQ in the preceding example yields:

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h. How to Reduce Lot Size

Looking at the EOQ formula, there are four variables. The EOQ will increase as theannual demand (A) and the cost of ordering (S) increase, and it will decrease as thecost of carrying inventory (i) and the unit cost (c) increase.

The annual demand (A) is a condition of the marketplace and is beyond the control ofmanufacturing. The cost of carrying inventory (i) is determined by the product itselfand the cost of money to the company. As such, it is beyond the control ofmanufacturing.

The unit cost (c) is either the purchase cost of the SKU or the cost of manufacturingthe item. Ideally, both costs should be as low as possible. In any event, as the unitcost decreases, the EOQ increases.

The cost of ordering (S) is either the cost of placing a purchase order or the cost ofplacing a manufacturing order. The cost of placing a manufacturing order is made upfrom production control costs and setup costs. Anything that can be done to reducethese costs reduces the EOQ.

Just­in­time manufacturing emphasizes reduction of setup time. There are severalreasons why this is desirable, and the reduction of order quantities is one. Chapter 15discusses just­in­time manufacturing further.

3. Variations Of The Eoq Model

There are several modifications that can be made to the basic EOQ model to fit particularcircumstances. Two that are often used are the monetary unit lot­size model and thenoninstantaneous receipt model.

a. Monetary Unit Lot Size

The EOQ can be calculated in monetary units rather than physical units. The sameEOQ formula given in the preceding can be used, but the annual usage changes fromunits to dollars.

AD = annual usage in dollars

S = ordering costs in dollars

i = carrying cost rate as a decimal of a percent

Because the annual usage is expressed in dollars, the unit cost is not needed in themodified EOQ equation.

The EOQ in dollars is:

b. Example Problem

An item has an annual demand of $5000, preparation costs of $20 per order, and acarrying cost of 20%. What is the EOQ in dollars?

AD = $5000

S = $20

i = 20% = 0.20

4. Quantity Discounts

When material is purchased, suppliers often give a discount on orders over a certain size. This can bedone because larger orders reduce the supplier’s costs; to get larger orders, they are willing to offervolume discounts. The buyer must decide whether to accept the discount, and in doing so, mustconsider the relevant costs:

Purchase cost.

Ordering costs.

Carrying costs.

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a. Example Problem

An item has an annual demand of 25,000 units, a unit cost of $10, an orderpreparation cost of $10, and a carrying cost of 20%. It is ordered on the basis of anEOQ, but the supplier has offered a discount of 2% on orders of $10,000 or more.Should the offer be accepted?

Answer

AD = 25,000 x $10 = $250,000

S = $10

i = 20%

Discounted order quantity = $l0,000 X 0.98 = $9,800 No discount Discount

lot size

Unit Price $10 $9.80

Lot Size $5000 $9800

Average Lot­Size Inventory (Qc ±2)

$2500 $4900

Number of Orders per Year 50 25

Purchase Cost $250,000 $245,000

Inventory­Carrying Cost (20%) 500 980

Order Preparation Cost ($10each)

500 250

Total Cost $251,000 $246,230

From the preceding example problem, it can be said that taking the discount resultsin the following:

There is a saving in purchase cost.

Ordering costs are reduced because fewer orders are placed since largerquantities are being ordered.

Inventory­carrying costs rise because of the larger order quantity.

The buyer must weigh the first two against the last and decide what to do. Whatcounts is the total cost. Depending on the figures, it may or may not be best to takethe discount.

5. Use Of Eoq When Costs Are Not Known

The EOQ formula depends upon the cost of ordering and the cost of carrying inventory. In practice,these costs are not necessarily known or easy to determine. However, the formula can still be used toadvantage when applied to a family of items.

For a family of items, the ordering costs and the carrying costs are generally the same for each item.For instance, if we were ordering hardware items—nuts, bolts, screws, nails, and so on—the carryingcosts would be virtually the same (storage, capital, and risk costs) and the cost of placing an orderwith the supplier would be the same for each item. In cases such as this, the cost of placing an order(S) is the same for all items in the family as is cost of carrying inventory (i).Now

where A (annual demand) is in dollars.

Since S is the same for all the items and i is the same for all items, the ratio 2S ± i must be the samefor all items in the family. For convenience, let:

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a. Example Problem

Suppose there were a family of items for which the decision rule was to order eachitem four times a year. Since the cost of ordering (S) and the cost of carryinginventory (i) are not known, ordering four times a year is not based on an EOQ. Canwe come up with a better decision rule even if the EOQ cannot be calculated?

The sum of all the lots is $2636. Since the average inventory is equal to half theorder quantity, the average inventory is $2636 2 = $1318.

Since this is a family of items where the preparation costs are the same and thecarrying costs are the same, the values for K = (2S j)h/2 should be the same for allitems. The preceding calculations show that they are not. The correct value for K isnot known, but a better value would be the average of all the values:

This value of K can be used to recalculate the order quantities for each item.

The average inventory has been reduced from $1318 to $726 while the number oforders per year (12) remains the same. Thus, the total costs associated withinventory have been reduced.

6. Period­Order Quantity (P00)

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The economic­order quantity attempts to minimize the total cost of ordering and carrying inventory andis based on the assumption that demand is uniform. Often demand is not uniform, particularly inmaterial requirements planning, and using the EOQ does not produce a minimum cost.

The period­order quantity lot­size rule is based on the same theory as the economic­order quantity. Ituses the EOQ formula to calculate an economic time between orders. This is calculated by dividingthe EOQ by the demand rate. This produces a time interval for which orders are placed. Instead ofordering the same quantity (EOQ), orders are placed to satisfy requirements for the calculated timeinterval. The number of orders placed in a year is the same as for an economic­order quantity, but theamount ordered each time varies. Thus, the ordering cost is the same but, because the orderquantities are determined by actual demand, the carrying cost is reduced.

EOQ Period­order quantity = . average weekly usage

a. Example Problem

The EOQ for an item is 2800 units, and the annual usage is 52,000 units. What is theperiod­order quantity?

Answer

When an order is placed it will cover the requirements for the next three weeks.

b. Example Problem

Given the following MRP record and an EOQ of 250 units, calculate the planned orderreceipts using the economic­order quantity. Next, calculate the period­order quantitiesand the planned order receipts. In both cases, calculate the ending inventory and thetotal inventory carried over the ten weeks.

Week 1 2 3 4 5 6 7 8 9 10 Total

NetRequirements

100 50 150 75 200 55 80 150 30 890

Planned OrderReceipt

AnswerEQQ = 250 units

Week 1 2 3 4 5 6 7 8 9 10 Total

NetRequirements

100 50 150 75 200 55 80 150 30 890

Planned OrderReceipt

250 250 250 250

EndingInventory

150 100 200 200 125 175 120 40 140 110 1360

Week 1 2 3 4 5 6 7 8 9 10 Total

NetRequirements

100 50 150 75 200 55 80 150 30 890

Planned OrderReceipt

300 330 260

EndingInventory

200 150 0 0 255 55 0 180 30 0 870

Notice in the example problem the total inventory is reduced from 1360 units to 870units over the ten­week period.

c. Practical Considerations When Using the EOQ

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Lumpy demand.

The EOQ assumes that demand is uniform and replenishmentoccurs all at once. When this is not true, the EOQ will not producethe best results. It is better to use the period­order quantity.

Anticipation inventory.

Demand is not uniform, and stock must be built ahead. It is better toplan a buildup of inventory based on capacity and future demand.

Minimum order.

Some suppliers require a minimum order. This minimum may bebased on the total order rather than on individual items. Often theseare C items where the rule is to order plenty, not an EOQ.

Transportation inventory.

As will be discussed in Chapter 13, carriers give rates based on theamount shipped. A full load costs less per ton to ship than a partload. This is similar to the price break given by suppliers for largequantities. The same type of analysis can be used.

Multiples.

Sometimes, order size is constrained by package size. Forexample, a supplier may ship only in skid­load lots. In these cases,the unit used should be the minimum package size.