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M&DC Purchasing & Supply Chain: Material Management

 

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Order Quantities

 

Contents

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

    1. How much should be ordered at one time?

    2. When should an order be placed?

  • Management must establish decision rules to answer these questions so inventory management personnel know when to order and how much. Lacking any better knowledge, decision rules are often made based on what seems reasonable. Unfortunately, such rules do not always produce the best results.

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

a. Stock-Keeping Unit (SKU)

  • Control is exercised through individual items in a particular inventory. These are called a stock-keeping unit (SKU). Two white shirts in the same inventory but of different sizes or styles would be two different SKUs. The same shirt in two different inventories would be two different SKUs.

b. Lot-Size Decision Rules

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

Lot-for-lot. The lot-for-lot

  • rule says to order exactly what is needed—no more—no less. The order quantity changes whenever requirements change. This technique requires time-phased information such as provided by a material requirements plan or a master production schedule. Since items are ordered only when needed, this system creates no unused lot-size inventory. Because of this, it is the best method for planning “A’ items and is also used in a just-in-time environment.

Fixed-order quantity

  • Fixed-order quantity rules specify the number of units to be ordered each time an order is placed for an individual item or SKU. The quantity is usually arbitrary, such as 200 units at a time. The advantage to this type of rule is that it is easily understood. The disadvantage is that it does not minimize the costs involved.

  • A variation on the fixed-order quantity system is the mm-max system. In this system, an order is placed when the quantity available falls below the order point (discussed in the next chapter). The quantity ordered is the difference between the actual quantity available at the time of order and the maximum. For example, if the order point is 100 units, the maximum is 300 units, and the quantity actually available when the order is placed is 75, the order quantity is 225 units. If the quantity actually available is 80 units, an order for 220 units is placed.
    One commonly used method of calculating the quantity to order is the economic-order quantity, which is discussed in the next section.

Order “n” periods supply.

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

c. Costs

  • As shown in the last chapter, the cost of ordering and the cost of carrying inventory both depend on the quantity ordered. Ideally, the ordering decision rules used will minimize the sum of these two costs. The best known system is the economic-order quantity.

a. Assumptions

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

    1. Demand is relatively constant and is known.

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

    3. Order preparation costs and inventory-carrying costs are constant and known.

    4. Replacement occurs all at once.

  • These assumptions are usually valid for finished goods whose demand is independent and fairly uniform. However, there are many situations where the assumptions are not valid and the EOQ concept is of no use. For instance, there is no reason to calculate the EOQ for made-to-order 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 raw material batch size. In material requirements planning, the lot-for-lot decision rule is often used, but there are also several rules used that are variations of the economic-order quantity.

b. Development of the EOQ Formula

  • Under the assumptions given, the quantity of an item in inventory decreases at a uniform rate. Suppose for a particular item, the order quantity is 200 units, and the usage 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 reaches zero. The quantity of units in inventory then increases instantaneously by Q, the quantity ordered. This is an accurate representation of the arrival of purchased parts or manufactured parts where all parts are received at once.
    From the preceding,

                                                           order quantity            200

         Average lot size inventory =                              =                     = 100 units

                                                                      2                         2

                                                            annual demand      100 X 52

       Number of orders per year =                       =                 .

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

Answer

                                                           order quantity            200

         Average cycle inventory =                              =                     = 100 units

                                                                      2                         2

                                                            annual demand       10.075         

       Number of orders per year =                       =                 . = 15.5

                                                                                                                               order quantity           650

  • Notice in the example problem the number of orders per year is rounded neither up nor down. It is an average figure, and the actual number of orders per year will vary from year to year but will average to the 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 carrying inventory 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 of increasing the other. The trick is to find the particular order quantity in which the total cost of carrying inventory and the cost of ordering will be a minimum.

Let:

A = annual usage in units

S = ordering cost in dollars per order

i   = annual carrying cost rate as a decimal of a percentage

c = unit cost in dollars

Q = 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 one unit for one year

                                       = average inventory x unit cost x carrying cost

                             Q

                        =          x c x i

                              2

Total annual costs = annual ordering costs + annual carrying costs

                               A              Q

                        =           x s +          x c x i

                              Q                2

  • everal rules used that are variations of the economic-order quantity.

e. Example Problem

  • The annual demand is 10,000 units, the ordering cost $30 per order, the carrying cost 20%, and the unit cost $15. The order quantity is 600 units. Calculate:

  1. Annual ordering cost

  2. Annual carrying cost

  3. Total annual cost

Answer

A = 10,000 units

S = $30

 i   = 0.20

C = $15

Q = 600 units

                                                    A               10,000

  1. annual ordering cost =           x S =                x $30 = $500

                                         Q                   600     

                                        Q                   600

  1. annual carrying cost=          x c x i =            x  $15 x 0.2 = $900

                                         2                      2

  1. total annual cost                                                              = $1400

  • Ideally, the total cost will be a minimum. For any situation in which the annual demand (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. Trial-and-Error Solution

  • Consider the following example:

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

Let:

A = 1000 units

S = $20 per order

c = $5 per unit

i = 20% = 0.20

Then:

                                                     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 + annual carrying cost

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

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

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

    3. 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 determine accurately the cost of carrying inventory and the cost of ordering. Since the total cost is relatively flat around the EOQ, it is not critical to have exact values. Good approximations are sufficient. Second, parts are often ordered in convenient packages such as pallet loads, cases, or dozens, and it is adequate to pick the closest package quantity to the EOQ.

Order Quantity

(Q)

Ordering Costs

(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. Economic-Order Quantity Formula

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

  • This value for the order quantity is the economic-order quantity. Using the formula to calculate the EOQ in the preceding example yields:

 

h. How to Reduce Lot Size

  • Looking at the EOQ formula, there are four variables. The EOQ will increase as the annual demand (A) and the cost of ordering (S) increase, and it will decrease as the cost 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 of manufacturing. The cost of carrying inventory (i) is determined by the product itself and the cost of money to the company. As such, it is beyond the control of manufacturing.

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

  • The cost of ordering (S) is either the cost of placing a purchase order or the cost of placing a manufacturing order. The cost of placing a manufacturing order is made up from production control costs and setup costs. Anything that can be done to reduce these costs reduces the EOQ.

  • Just-in-time manufacturing emphasizes reduction of setup time. There are several reasons why this is desirable, and the reduction of order quantities is one. Chapter 15 discusses just-in-time manufacturing further.

  • There are several modifications that can be made to the basic EOQ model to fit particular circumstances. Two that are often used are the monetary unit lot-size model and the noninstantaneous receipt model.

a. Monetary Unit Lot Size

  • The EOQ can be calculated in monetary units rather than physical units. The same EOQ formula given in the preceding can be used, but the annual usage changes from units 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 the modified 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 a carrying cost of 20%. What is the EOQ in dollars?

AD = $5000

S = $20

i = 20% = 0.20

 

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

    • Purchase cost.

    • Ordering costs.

    • Carrying costs.

a. Example Problem

  • An item has an annual demand of 25,000 units, a unit cost of $10, an order preparation cost of $10, and a carrying cost of 20%. It is ordered on the basis of an EOQ, 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 Lot-Size Inventory (Qc 2)

$2500

$4900

Number of Orders per Year

50

25

Purchase Cost

$250,000

$245,000

Inventory-Carrying Cost (20%)

500

980

Order Preparation Cost ($10 each)

500

250

Total Cost

$251,000

$246,230

  • From the preceding example problem, it can be said that taking the discount results in the following:

    • There is a saving in purchase cost.

    • Ordering costs are reduced because fewer orders are placed since larger quantities are being ordered.

    • Inventory-carrying costs rise because of the larger order quantity.

  • The buyer must weigh the first two against the last and decide what to do. What counts is the total cost. Depending on the figures, it may or may not be best to take the 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 to advantage 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 carrying costs would be virtually the same (storage, capital, and risk costs) and the cost of placing an order with 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 same for all items in the family. For convenience, let:

 

a. Example Problem

  • Suppose there were a family of items for which the decision rule was to order each item four times a year. Since the cost of ordering (S) and the cost of carrying inventory (i) are not known, ordering four times a year is not based on an EOQ. Can we 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 the order quantity, the average inventory is $2636 2 = $1318.

  • Since this is a family of items where the preparation costs are the same and the carrying costs are the same, the values for K = (2S j)h/2 should be the same for all items. The preceding calculations show that they are not. The correct value for K is not 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 of orders per year (12) remains the same. Thus, the total costs associated with inventory have been reduced.

  • The economic-order quantity attempts to minimize the total cost of ordering and carrying inventory and is based on the assumption that demand is uniform. Often demand is not uniform, particularly in material requirements planning, and using the EOQ does not produce a minimum cost.

  • The period-order quantity lot-size rule is based on the same theory as the economic-order quantity. It uses the EOQ formula to calculate an economic time between orders. This is calculated by dividing the EOQ by the demand rate. This produces a time interval for which orders are placed. Instead of ordering the same quantity (EOQ), orders are placed to satisfy requirements for the calculated time interval. The number of orders placed in a year is the same as for an economic-order quantity, but the amount ordered each time varies. Thus, the ordering cost is the same but, because the order quantities are determined by actual demand, the carrying cost is reduced.

                                                                                EOQ

                  Period-order 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 the period-order 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 order receipts using the economic-order quantity. Next, calculate the period-order quantities and the planned order receipts. In both cases, calculate the ending inventory and the total inventory carried over the ten weeks.

Week

1

2

3

4

5

6

7

8

9

10

Total

Net Requirements

100

50

150

 

75

200

55

80

150

30

890

Planned Order Receipt

 

 

 

 

 

 

 

 

 

 

 

Answer

EQQ = 250 units

Week

1

2

3

4

5

6

7

8

9

10

Total

Net Requirements

100

50

150

 

75

200

55

80

150

30

890

Planned Order Receipt

 250

 

250

 

 

250

 

 

250

 

 

Ending Inventory

150

100

200

200

125

175

120

40

140

110

1360

 

Week

1

2

3

4

5

6

7

8

9

10

Total

Net Requirements

100

50

150

 

75

200

55

80

150

30

890

Planned Order Receipt

300

 

 

 

330

 

 

260

 

 

 

Ending Inventory

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 870 units over the ten-week period.

c. Practical Considerations When Using the EOQ

Lumpy demand.

  • The EOQ assumes that demand is uniform and replenishment occurs all at once. When this is not true, the EOQ will not produce the best results. It is better to use the period-order quantity.

Anticipation inventory.

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

Minimum order.

  • Some suppliers require a minimum order. This minimum may be based on the total order rather than on individual items. Often these are 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 the amount shipped. A full load costs less per ton to ship than a part load. This is similar to the price break given by suppliers for large quantities. The same type of analysis can be used.

Multiples.

  • Sometimes, order size is constrained by package size. For example, a supplier may ship only in skid-load lots. In these cases, the unit used should be the minimum package size.