MANAGING INVENTORIES CHAPTER 12 DAVID A. COLLIER AND JAMES R. EVANS

LO1 Explain the importance of inventory, types of inventories, and key decisions and costs. LO2 Describe the major characteristics that impact inventory decisions. LO3 Describe how to conduct an ABC inventory analysis. LO4 Explain how a fixed order quantity inventory system operates, and how to use the EOQ and safety stock models. LO5 Explain how a fixed period inventory system operates. LO6 Describe how to apply the single period inventory model.

Banana Republic is a unit of San Francisco’s Gap, Inc Banana Republic is a unit of San Francisco’s Gap, Inc. and accounts for about 13 percent of Gap’s sales. As Gap shifted its product line to basics such as cropped pants, jeans, and khakis, Banana Republic had to move away from such staples and toward trends, trying to build a name for itself in fashion circles. But fashion items, which have a much shorter product life cycle and are riskier because their demand is more variable and uncertain, bring up a host of operations management issues. In one recent holiday season, the company had bet that blue would be the top-selling color in stretch merino wool sweaters. They were wrong. Marka Hansen, company president noted, “The No. 1 seller was moss green. We didn’t have enough.”

What do you think? Can you cite any experiences in which the lack of appropriate inventory at a retail store has caused you as the customer to be dissatisfied?

Inventory is any asset held for future use or sale. Objectives: Inventory Management involves planning, coordinating, and controlling the acquisition, storage, handling, movement, distribution, and possible sale of raw materials, component parts and subassemblies, supplies and tools, replacement parts, and other assets that are needed to meet customer wants and needs. Maintain sufficient inventory Incur lowest possible cost

Understanding Inventory Raw materials, component parts, subassemblies, and supplies are inputs to manufacturing and service-delivery processes. Work-in-process (WIP) inventory consists of partially finished products in various stages of completion that are awaiting further processing. Finished goods inventory is completed products ready for distribution or sale to customers. Safety stock inventory is an additional amount of inventory that is kept over and above the average amount required to meet demand.

Exhibit 12.1 Role of Inventory in the Value Chain

Inventory Management Decisions and Costs Inventory managers deal with two fundamental decisions: When to order items from a supplier or when to initiate production runs if the firm makes its own items. How much to order or produce each time a supplier or production order is placed.

Inventory Management Decisions and Costs Four categories of inventory costs: Ordering or setup costs Inventory-holding costs Shortage costs Unit cost of the stock-keeping units (SKUs)

Inventory Management Decisions & Costs Ordering costs or setup costs are incurred as a result of the work involved in placing purchase orders with suppliers or configuring tools, equipment, and machines within a factory to produce an item. Inventory-holding costs or inventory-carrying costs are the expenses associated with carrying inventory.

Inventory Management Decisions & Costs Shortage costs or stockout costs are the costs associated with a SKU being unavailable when needed to meet demand. Unit cost is the price paid for purchased goods or the internal cost of producing them.

Inventory Characteristics Number of items: each item is identified by a unique identifier, called a stock-keeping unit (SKU). A stock-keeping unit (SKU) is a single item or asset stored at a particular location.

Inventory Characteristics Nature of Demand: Independent demand is demand for an SKU that is unrelated to the demand for other SKUs and needs to be forecast. Dependent demand is demand directly related to the demand for other SKUs and can be calculated without needing to be forecast. Demand can either be constant (deterministic) or uncertain (stochastic). Static demand is stable demand. Dynamic demand varies over time.

Inventory Characteristics Number and Duration of Time Periods: Lead Time: Single period Multiple time periods The lead time is the time between placement of an order and its receipt.

Inventory Characteristics Stockouts: A stockout is the inability to satisfy demand for an item. A backorder occurs when a customer is willing to wait for an item. A lost sale occurs when the customer is unwilling to wait and purchases the item elsewhere.

ABC Inventory Analysis ABC inventory analysis categorizes SKUs into three groups according to their total annual dollar usage. “A” items account for a large dollar value but a relatively small percentage of total items. “C” items account for a small dollar value but a large percentage of total items. “B” items are between A and C.

ABC Inventory (Pareto) Analysis “A” items account for a large dollar value but relatively small percentage of total items (e.g., 10% to 30 % of items, yet 60% to 80% of total dollar value). “C” items account for a small dollar value but a large percentage of total items (e.g., 50% to 60% of items, yet about 5% to 15% of total dollar value). These can be managed using automated computer systems. “B” items are between A and C.

Solved Problem The data shows projected annual dollar usage for 20 items. Exhibit 12.3 shows the data sorted, and indicates that about 70% of total dollar usage is accounted for by the first 5 items. Exhibit 12.2 Usage-Cost Data for 20 Inventoried Items

Exhibit 12.3 ABC Analysis Calculations

Exhibit 12.4 ABC Histogram for the Results from Exhibit 12.3

Managing Fixed Quantity Inventory Systems In a fixed quantity system (FQS), the order quantity or lot size is fixed; the same amount, Q, is ordered every time. The fixed order (lot) size, Q, can be a box, pallet, container, or truck load. Q does not have to be economically determined, as we will do for the EOQ model later.

Managing Fixed Quantity Inventory Systems The process of triggering an order is based on the inventory position. Inventory position (IP) is the on-hand quantity (OH) plus any orders placed but which have not arrived (scheduled receipts, or SR), minus any backorders (BO). IP = OH + SR – BO [12.1]

Managing Fixed Quantity Inventory Systems When inventory falls at or below a certain value, r, called the reorder point, a new order is placed. The reorder point is the value of the inventory position that triggers a new order.

Exhibit 12.5 Summary of Fixed Quantity System (FQS)

Exhibit 12.6 Fixed Quantity System (FQS) under Stable Demand

Exhibit 12.7 Fixed Quantity System (FQS) with Highly Variable Demand

The EOQ Model The Economic Order Quantity (EOQ) model is a classic economic model developed in the early 1900s that minimizes total cost, which is the sum of the inventory-holding cost and the ordering cost.

The EOQ Model Assumptions: Only a single item (SKU) is considered. The entire order quantity (Q) arrives in the inventory at one time. Only two types of costs are relevant—order/setup and inventory holding costs. No stockouts are allowed. The demand for the item is deterministic and continuous over time. Lead time is constant.

The EOQ Model Cycle inventory (also called order or lot size inventory) is inventory that results from purchasing or producing in larger lots than are needed for immediate consumption or sale. Average cycle inventory = (Maximum inventory + Minimum inventory)/2 = Q/2 [12.2]

Exhibit 12.8 Cycle Inventory Pattern for the EOQ Model

The EOQ Model Inventory Holding Cost The cost of storing one unit in inventory for the year, Ch, is: Ch = (I)(C) [12.3] Where: I = Annual inventory-holding charge expressed as a percent of unit cost. C = Unit cost of the inventory item or SKU.

( ) The EOQ Model Annual inventory-holding cost is computed as: [12.4] average inventory annual holding cost per unit = ( ) 1 2 QCh

( ) The EOQ Model Ordering Cost If D = Annual demand and we order Q units each time, then we place D/Q orders/year. Annual ordering cost is computed as: ( ) annual ordering cost = number of orders per year cost per order D Co [12.5] Q Where C0 is the cost of placing one order.

The EOQ Model Total Annual Cost Total annual cost is the sum of the inventory holding cost plus the order or setup cost: TC 2 Q = QCh Co 1 + D [12.6]

√ The EOQ Model Economic Order Quantity The EOQ is the order quantity that minimizes the total annual cost: Ch Q* = √ 2DCo [12.7]

The EOQ Model Calculating the Reorder Point The reorder point, r, depends on the lead time and demand rate. Multiply the fixed demand rate d by the length of the lead time L (making sure they are expressed in the same units, e.g., days or months): r = Lead time demand = (demand rate)(lead time) [12.8] = (d)(L)

√ Solved Problem D = 24,000 cases per year. Co = $38.00 per order. I = 18 percent. C = $12.00 per case. Ch = IC = $2.16. TC = Q ($2.16) + ($38.00) 1 2 24,000 Q EOQ = = 919 cases rounded to a whole number. 2(24,000)(38) 2.16 √

Exhibit 12.9 Chart of Holding, Ordering, and Total Costs

Safety Stock and Uncertain Demand in a Fixed Order Quantity System When demand is uncertain, using EOQ based on the average demand will result in a high probability of a stockout. Safety stock is additional planned on-hand inventory that acts as a buffer to reduce the risk of a stockout. A service level is the desired probability of not having a stockout during a lead-time period.

Safety Stock and Uncertain Demand in a Fixed Order Quantity System When a normal probability distribution provides a good approximation of lead time demand, the general expression for reorder point is: r = mL + zsL [12.9] Where: mL = Average demand during the lead time. sL = Standard deviation of demand during the lead time. z = The number of standard deviations necessary to achieve the acceptable service level. “zsL” represents the amount of safety stock.

Safety Stock and Uncertain Demand in a Fixed Order Quantity System We may not know the mean and standard deviation of demand during the lead time, but only for some other length of time, t, such as a month or year. Suppose that mt and st are the mean and standard deviation of demand for some time interval t. If the distributions of demand for all time intervals are identical to and independent of each other, then: mL = mtL [12.10] sL = st √L [12.11]

Solved Problem Southern Office Supplies, Inc. distributes laser printer paper. Ordering costs are $45.00 per order. One ream of paper costs $3.80. Annual inventory-holding cost rate is 20%. The average annual demand is 15,000 reams, or about 15,000/52 = 288.5 per week. The standard deviation of weekly demand is about 71. The lead time from the manufacturer is two weeks. Inventory-holding cost is Ch = IC = 0.20($3.80) = $0.76 per ream per year.

Solved Problem The average demand during the lead time is (288.5)(2) = 577 reams. The standard deviation of demand during the lead time is approximately 71√2 = 100 reams. The EOQ model results in an order quantity of 1333, reorder point of 577, and total annual cost of $1,012.92.

Solved Problem Desired service level of 95%, which results in a stockout of roughly once every 2 years. For a normal distribution, this corresponds to a standard normal z-value of 1.645. This policy increases the reorder point by 742 – 577 = 165 reams, which represents the safety stock. The cost of the additional safety stock is Ch times the amount of safety stock, or ($0.76/ream)(165 reams) = $125.40. r = mL + zsL = 577 = 1.645(100) = 742 reams

EOQ and Sustainability Practices Commonly hidden in inventory management decisions are the costs to dispose of obsolete, hazardous materials. Environmental considerations, material losses, and waste disposal can be included in the EOQ model to improve inventory decisions. If a company examines its hazardous waste disposals and observed that some percentage of its material is eventually disposed of instead of used, then the company should incorporate this into their inventory decisions. Example: Annual demand for paint is 4000 lbs., item cost is $3/lb., order cost is $50, inventory-holding cost rate is 10%, 5% of paint is not sold and eventually disposed of, and the disposal cost is $1/lb.

EOQ and Sustainability Practices Disposal costs, on a per unit basis, can be comparable to the initial item costs. For example, knowing that 5% of the paint is disposed of, then the inventory-holding should be calculated as: Ch = [(10%)($3/lb. item cost)] + [(5% of paint disposed of)($1/lb. disposal cost)] = $0.35/lb./year

Managing Fixed Period Inventory Systems An alternative to a fixed order quantity system is a fixed period system (FPS)—sometimes called a periodic review system—in which the inventory position is checked only at fixed intervals of time, T, rather than on a continuous basis. Two principal decisions in a FPS: The time interval between reviews (T), and The replenishment level (M)

Managing Fixed Period Inventory Systems Economic time interval: T = Q*/D [12.12] Optimal replenishment level without safety stock: M = d (T + L) [12.13] Where: d = Average demand per time period. L = Lead time in the same time units. M = Demand during the lead time plus review period.

Exhibit 12.10 Summary of Fixed Period Inventory Systems

Exhibit 12.11 Operation of a Fixed Period Systems (FPS)

Managing Fixed Period Inventory Systems Uncertain Demand Compute safety stock over the period T + L. The replenishment level is computed as: M = mT+L + zσT+L mT+L = mt (T + L) σT+L = σt √T + L [12.14] [12.15] [12.16]

Single-Period Inventory Model Applies to inventory situations in which one order is placed for a good in anticipation of a future selling season where demand is uncertain. At the end of the period, the product has either sold out or there is a surplus of unsold items to sell for a salvage value. Sometimes called a newsvendor problem, because newspaper sales are a typical example.

Single-Period Inventory Model Solve using marginal economic analysis. The optimal order quantity Q* must satisfy: cs = The cost per item of overestimating demand (salvage cost); this cost represents the loss of ordering one additional item and finding that it cannot be sold. cu = The cost per item of underestimating demand (shortage cost); this cost represents the opportunity loss of not ordering one additional item and finding that it could have been sold. P (demand ≤ Q*) = cu cu + cs [12.17]

Solved Problem A buyer orders fashion swimwear about six months before the summer season. Each piece costs $40 and sells for $60. At the sale price of $30, it is expected that any remaining stock can be sold during the August sale. The cost per item of overestimating demand is equal to the purchase cost per item minus the August sale price per item: cs = $40 – $30 = $10. The per-item cost of underestimating demand is the difference between the regular selling price per item and the purchase cost per item; that is, cu = $60 – $40 = $20.

Solved Problem Assume that a uniform probability distribution ranging from 350 to 650 items describes the demand. Exhibit 12.12 Probability Distribution for Single Period Model

Solved Problem The optimal order size Q must satisfy: P (demand ≤ Q*) = cu /(cu + cs) = 20/(20+10) = 2/3 Because the demand distribution is uniform, the value of Q* is two-thirds of the way from 350 to 650. This results in Q* = 550.

There’s More to Inventory Modeling Quantitative models have been developed for other practical situations: Backorder Models: It may be desirable from an economic point of view to plan for and allow shortages, such as when the value per unit of the inventory is very high, and hence the inventory-holding cost is high (a new-car dealer’s inventory). Most customers do not find the specific car they want in stock, but are willing to backorder it. Quantity Discount Models: Suppliers often offer discounts for purchasing larger quantities of goods. This often occurs because of economies of scale associated with larger quantities or simply as an incentive to increase total revenue.

Hardy Hospital Case Study What are good estimates of order cost and inventory holding cost? (State all assumptions and show all computations.) What is the EOQ and reorder point for Strike Disinfectant given your answer to Question 1? Compute the total order and inventory holding costs for a Fixed Quantity System (FQS) and compare to their current order Q's. Can you save money by adopting a FQS? What are your final recommendations? Clearly explain your reasoning.