# INVENTORY PLANNING AND CONTROL

* Why are inventory planning and control important?*

One of the most common problems facing operations managers is inventory planning. This is understandable since inventory usually represents a sizable portion of a firm`s total assets and, more specifically, on the average, more than 30% of total current assets in the U.S. industry. Excessive money tied up in inventory is a drag on profitability. The purpose of inventory planning is to develop policies which will achieve an optimal investment in inventory. You can do this by determining the optimal level of inventory necessary to minimize inventory related costs. Note: The ABC method of inventory control requires management to exert greatest control over the A classification items, which usually include a relatively small percentage of total items but a high percentage of the dollar volume. This method is analogous to the 80/20 rule, which says, for instance, that 20% of the customers account for 80% of the profit.

* What kinds of costs are associated with inventory?*

Inventory costs fall into three categories. They are:

- Ordering costs. These include all costs associated with preparing a purchase order.
- Carrying (holding) costs. These include storage costs for inventory items plus the cost of money tied up in inventory.
- Shortage (stockout) costs. These are costs incurred when an item is out of stock. These include the lost contribution margin on sales plus lost customer goodwill.

When and how much should I order?

Several inventory planning models are available that try to answer these questions. Three such models are

- Economic order quantity (EOQ)
- Reorder point (ROP)
- Determination of safety stock.

How does the Economic Order Quantity (eoq) model work?

The economic order quantity (EOQ) determines the order size that minimizes the sum of carrying and ordering costs.

Assumptions: Demand is assumed to be known with certainty and to remain constant throughout the year. Order cost is also known to be fixed. Also, unit carrying costs are assumed be constant. Since demand and lead time (time interval between placing an order and receiving delivery) are assumed to be determinable, no shortage costs exist. No quantity discounts are allowed.

The EOQ is computed as:

EOQ =√2DO/C

where C = carrying cost per unit, O = ordering cost per order, D = annual demand (requirements) in units.

If the carrying cost is expressed as a percentage of average inventory value (say, 12 percent per year to hold inventory), then the denominator value in the EOQ formula would be 12 percent times the price of an item. NOTE: When an item is made instead of purchasing it, the EOQ model is used to determine the economic production run size where O = cost per setup.

EXAMPLE 8.14

Assume the Los Alamitos Store buys sets of steel at $40 per set from an outside vendor. It will sell 6,400 sets evenly throughout the year. The store`s carrying cost is $8.00 per unit per year. The ordering cost is $100 per order. Therefore,

EOQ=√2(6.400)($100)/$8.00 = 160,000 = 400 sets

Total number of orders per year = D/EOQ =6,400/400 = 16 orders

Total inventory costs = Carrying cost + Ordering cost

= C x (EOQ/2) + O (D/EOQ)

= ($8.00)(400/2) + ($100)(6,400/400)

= $1,600 + $1,600 = $3,200

Based on these calculations, the Los Alamitos Store`s inventory policy should be the following:

(1)The store should order 400 sets of steel each time it places an order and order 16 times during a year.

(2)This policy will be most economical and cost the store $3,200 per year.

*How do I determine the reorder point?*

Reorder point (ROP) tells you when to place a new order. However, this requires that you know the lead time from placing to receiving an order. Reorder point (ROP) is calculated as follows:

Reorder point = (average demand per unit of lead time x lead time) + safety stock

This tells you the level of inventory at which a new order should be placed. First, multiply average daily (or weekly) demand by the lead time in days (or weeks) yielding the lead time demand. Then add safety stock to this to provide for the variation in lead time demand to determine the reorder point. NOTE: If average demand and lead time are both certain, no safety stock is necessary and should be dropped from the formula.

EXAMPLE 8.15

Using the previous example, assume lead time is constant at one week. There are 50 working weeks in the year.

Then the reorder point is 128 sets = (6,400 sets/50 weeks) x 1 week.

CONCLUSION: When the inventory level drops to 128 sets, a new order should be placed. Suppose, however, that the store is faced with variable demand for its steel and requires a safety stock of 150 additional sets to carry. Then the reorder point will be 128 sets plus 150 sets, or 278 sets.

Figure 8.8 shows this inventory system when the order quantity is 400 sets and the reorder point is 128 sets.

**Figure 8.8BASIC INVENTORY SYSTEM WITH EOQ AND REORDER POINT**

* When are these models realistic to use?*

The EOQ model described here is appropriate for a pure inventory system; that is, for single-item, single-stage inventory decisions for which joint costs and constraints can be ignored. EOQ and ROP assume that lead time and demand rates are constant and known with certainty. CAUTION: This may be unrealistic. Still, these models have been proved useful in inventory planning for many companies. There are, for instance, many businesses for which these assumptions hold to some extent. They include:

- Subcontractors who must supply parts on a regular basis to a primary contractor.
- Automobile dealerships, in which demand varies from week to week, but tends to even out over a season.

CAVEAT: When demand is not known precisely and/or other complications arise, you should not use these models. You should instead refer to probabilistic models.

What about Quantity Discounts?

EOQ does not take quantity discounts into account, which is often unrealistic in actual practice. Usually, the more you order, the lower the unit price you pay. A typical price discount schedule follows:

Order quantity (Q) | Unit price (P) |

1 to 499 | $40.00 |

500 to 999 | 39.90 |

1000 or more | 39.80 |

WHAT TO DO: If quantity discounts are offered, you must weigh the potential benefits of reduced purchase price and fewer orders that will result from buying in large quantities against the increase in carrying costs caused by higher average inventories. Hence, the buyer`s goal in this case is to select the order quantity which will minimize total costs, where total cost is the sum of carrying cost, ordering cost, and product cost:

Total cost = Carrying cost + Ordering cost + Product cost ;

Total cost = C x (Q/2) + O x (D/Q) + PD

where P = unit price, and Q = order quantity.

Use these steps to find economic order size with price discounts:

Compute the common EOQ when price discounts are ignored and the corresponding costs using the new cost after discount.

Compute the costs for those quantities greater than EOQ at which price reductions occur.

Select the value of Q which will result in the lowest total cost.

EXAMPLE 8.16

Using the information from the previous examples and the discount schedule shown previously, try to determine the EOQ. Recall that EOQ = 400 sets without discount. The total cost for this is:

Total cost = $8.00(400/2) + $100(6,400/400) + $40.00(6,400)

Total cost =$1,600 +1,600 + 256,000 = $259,200

The further we move from the point 400, the greater will be the sum of the carrying and ordering costs. Thus, 400 is obviously the only candidate for the minimum total cost value within the first price range. Q=500 is the only candidate within the $39.90 price range and Q=1,000 is the only candidate within the $39.80 price bracket. These three quantities are evaluated below and pictured in Figure 8.9.

ANNUAL COSTS WITH VARYING ORDER QUANTITIES

Order Quantity (Q) | 400 | 500 | 1,000 |

Purchase price (P) | $40 | $39.90 | $39.80 |

Carrying cost (C x Q/2) $8 x (order quantity/2) | $1,600 | $2,000 | $4,000 |

Ordering cost (O x D/Q) $100 x (6,400/order quantity) | 1,600 | 1,280 | 640 |

Product cost (PD) Unit price x 6,400 | 256,000 | 255,360 | 254,720 |

Total cost | $259,200 | $258,640 | $259,360 |

**Figure 8.9 **

**COST WITH QUANTITY DISCOUNT PROBLEM**

CONCLUSION: The EOQ with price discounts is 500 sets. The manufacturer is justified in going to the first price break but the extra carrying cost of going to the second price break more than outweighs the savings in ordering and in the cost of the product itself.

What can I do when lead time and demand are uncertain?

When lead time and demand are not certain, you must carry extra units of inventory, called safety stock, as protection against possible stockouts. To determine the appropriate level of safety stock size, you must consider the service level or stockout costs.

Service level can be defined as the probability that demand will not exceed supply during the lead time. Thus, a service level of 90 percent implies a probability of 90 percent that demand will not exceed supply during lead time. Figure 8.10 shows a service level of 90%.

Here are three cases for computing the safety stock. The first two do not recognize stockout costs; the third does.

Case 1: Variable demand rate, constant lead time

ROP = Expected demand during lead time + safety stock

u = LT + z √ LT (σu)

where

u = average demand

LT=lead time

σu =standard deviation of demand rate

z = standard normal variate as defined in Table 8.1 - Normal Distribution table.

For a normal distribution, a given service level amounts to the shaded area under the curve to the left of ROP in Figure 8.10.

**Figure 8.10SERVICE LEVEL OF 90 PERCENT**

EXAMPLE 8.17

Norman`s Pizza uses large cases of tomatoes at an average rate of 50 cans per day. The demand can be approximated by a normal distribution with a standard deviation of 5 cans per day. Lead time is 4 days. Thus,

u = 50 cans per day.

LT= 4 days

σu = 5 cans

How much safety stock is necessary for a service level of 99%? And what is the ROP?

For a service level of 99%, z = 2.33 (from Table 8.1). Thus,

Safety stock = 2.33 √4 (5) = 23.3 cans

ROP = 50(4) + 23.3 = 223.3 cans

**Figure 8.11**

**SERVICE LEVEL OF 99 PERCENT**

ROP = Expected demand during lead time + safety stock

= uLT + z u (σLT)

where

u = constant demand

LT = average lead time

σLT =standard deviation of lead time

EXAMPLE 8.18

SVL`s Hamburger Shop uses 10 gallons of cola per day. The lead time is normally distributed with a mean of 6 days and a standard deviation of 2 days. Thus,

u = 10 gallons per day.

LT = 6 days

σLT = 2 days

How much safety stock is necessary for a service level of 99%? And what is the ROP?

For a service level of 99%, z = 2.33. Thus,

Safety stock = 2.33 (10)(2) = 46.6 gallons

ROP = 10(6) + 46.6 = 106.6 gallons

(note: z = 2.33 at 99% service kevel)

Case 3: Incorporation of stockout costs

This case specifically recognizes the cost of stockouts or shortages, which can be quite expensive. Lost sales and disgruntled customers are examples of external costs. Idle machine and disrupted production scheduling are examples of internal costs. WHAT TO DO: You can use the probability approach to determine the optimal safety stock size in the presence of stockout costs. Here is an example.

EXAMPLE 8.19

Refer to Example 8.15. The total demand over a one-week period is estimated as follows:

Total demand | Probability |

78 | 0.2 |

128 | 0.4 |

178 | 0.2 |

228 | 0.1 |

278 | 0.1 |

1.00 |

A stockout cost is estimated at $12.00 per set. Recall that the carrying cost is $8.00 per set.

Figure 8.12 shows the computation of safety stock. CONCLUSION: The computation shows that the total costs are minimized at $1,200, when a safety stock of 150 sets is maintained. Thus, the reorder point is: 128 sets + 150 sets = 278 sets.

**Figure 8.12COMPUTATION OF SAFETY STOCK**

** **

** **

**
**

**Table 8.1**

**VALUES OF ZP FOR SPECIFIED PROBABILITIES p**