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**Chapter 5: Inventory Management**

Department of Business Administration FALL Chapter 5: Inventory Management

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**Outline: What You Will Learn . . .**

Define the term inventory and list the major reasons for holding inventories; and list the main requirements for effective inventory management. Discuss the nature and importance of service inventories Discuss periodic and perpetual review systems. Discuss the objectives of inventory management. Describe the A-B-C approach and explain how it is useful.

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**Outline: What You Will Learn . . .**

Describe the basic EOQ model and its assumptions and solve typical problems. Describe the economic production quantity model and solve typical problems. Describe the quantity discount model and solve typical problems. Describe reorder point models and solve typical problems. Describe situations in which the single-period model would be appropriate, and solve typical problems.

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Inventory Management Inventory management is a core operations management activity. Good inventory management is important for the successful operation of most businesses and their supply chains. Operations, marketing, and finance have interests in good inventory management. Bad inventory management hampers operations, diminishes customer satisfaction, and increases operating costs.

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**Objective of Inventory Management**

The basic objective of Inventory Management has traditionally been to keep inventory at desired level that will meet product demand and be cost effective. The major objective of Inventory Management (control system) is to discover and maintain the best possible level of inventory in terms of both unit of product and least possible cost. In reaching such objectives firms seek out to avoid two common pitfalls. Management tries to avoid the problem inadequate levels of inventory since too little inventory disrupts production and may results in lost sales. The existence of too many inventories increases the risk of obsolescence and create unnecessary cost levels.

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Inventory Inventory is a stock or store of goods . Inventory can also be defined as a stock of materials created to satisfy to satisfy eventual demand. Inventories are idle resources of any kind that possess economic value and held for future use. Items in inventory ranges from small things such as pencils, paper clips, screws nuts and bolts to large items such as machines, trucks, construction equipment and air planes.

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Inventory Inventories are present whenever the inputs and outputs of a company are not used as soon as they become available . Inventory can be thought as a final product waiting to be sold to a retail customer e.g., a new car, canned foods or drinks, and baked goods. Inventories contain not only finished goods but also raw materials, supplies, and spare parts. Some very large firms have tremendous amounts of inventory. For example general motor was reported to have as much as $ 40 mn worth of materials, parts, cars and trucks in its supply chain. The ratio of inventories to sales in manufacturing, wholesale and retail sectors is one measure that is used to determine health of an economy. Inventories may represent a significant proportion of total asset. It is worth to mention that a reduction of inventories can results in a significant increase in return on investment (ROI is profit after taxes diveded by total asset).

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Inventory Inventory decisions in service organization can be especially critical. For example , hospital- being out of stock on some important supplies such as drugs imperil the well-being of a patient. Many inventory items have a limited shelf life so carring large quantities would mean having to dispose of unused, costly supplies. On-site repair services for computers, printers and fax machines also have to carefully consider which parts to bring to the site to avaid having to make extra trip to obtain parts. The major revenue for wholesale and retail business is sale of Inventory. In terms of dollar, the inventory of goods held for sale is the one of largest assets of merchandising business.

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Inventory Independent demand – finished goods, items that are ready to be sold E.g. a computer Dependent demand – components of finished products E.g. parts such as chip and system unit that make up the computer

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**Independent demand is uncertain. Dependent demand is certain.**

Inventory Independent Demand A B(4) C(2) D(2) E(1) D(3) F(2) Dependent Demand Independent demand is uncertain. Dependent demand is certain.

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**Types of Inventories Raw materials & purchased parts**

Partially completed goods called work in progress (WIP) Finished-goods inventories (manufacturing firms) or merchandise (retail stores) Replacement parts, tools, and supplies Goods-in-transit to warehouse or customers (pipeline inventory)

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**Functions of Inventory**

To meet anticipated demand A customer can be a person who walks in off the street to buy a new stereo system. These inventories are referred to as anticipation stocks because they are held to satisfy expected demand. To smooth production requirements Firms that experience seasonal patterns in demand often build up inventories during preseason period to meet overly high requirements during seasonal period. These inventories are aptly named seasonal inventories. i.e., fresh fruits and vegetables or x-mas cards or greeting cards.

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**Functions of Inventory**

To decouple operations Firms use inventories as buffers between successive operations to maintain continuity of production and breakdown of equipment and accidents that cause the operation to shut down temporarily.The buffers permit other operation to continue temporarily while the problem is solved. To protect against stock-outs Delayed deliveries and unexpected increase in demand rise the risk of shortages. Delays can occurs because of whether conditions, suppliers out-stocks, deliveries of wrong materials, quality problems and so on. This risk of shortage can be reduced by holding safety stocks.

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**Functions of Inventory**

To take advantage of order cycles To minimize purchasing and inventory costs, firms often buy in quantities that exceed immediate requirement. This necessites storing some or all of the purchased amount later use. Similarly, it is usually economical to produce in large rather than small quantities. This inventory storage can used later with demand requirements in short-run. To help hedge against price increases Firms will suspect that a substantial price increase is about to occur and purchase larger than normal amounts to beat the rise. The ability to store extra amount of goods also allows firms to take advantage of price discounts for larger orders

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**Functions of Inventory**

To permit operations The fact that production operations take a certain amount of time that there will be some work in process including raw materials, semifinished items, unfinished items and finished goods at production site as well as goods stored in warehouse. This leads to pipeline inventories throughout a production-distrubution system. Little’s Law can be useful in quantifying pipeline inventory. Little’s Law: the average amount of inventory in a system is equal to the product of average rate at which inventory units leave the system and average time a unit is in the system. To take advantage of quantity discounts Suppliers may give price discounts for larger orders.

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**Objective of Inventory Control (IC)**

Inaduqate control of inventories can results in both under and over stocking of items. Understocking results in missed deliveries, lost sales, dissatisfied customers and production bottle necks Overstocking results in unnecessarily ties up funds that might be more productive elsewhere Objective: To achieve satisfactory levels of customer service while keeping inventory costs within reasonable bounds Level of customer service (right time, right quantity) Costs of ordering and carrying inventory (right place, size) Inventory turnover is the ratio of average cost of goods sold to average inventory investment (performance of IC).

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**Effective Inventory Management**

To be effective, Management has two basic fuctions concerning inventory; 1- establishing system, 2- making decision. The following should be taken into account: A system to keep track of inventory A reliable forecast of demand Knowledge of lead times Reasonable estimates of Holding costs Ordering costs Shortage costs A classification system for inventory items.

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**Inventory Counting Systems**

Inventory counting system can be either periodic or perpentual: Periodic System Physical count of items in inventory made at periodic intervals such as weekly, monthly etc..many small retailers use this approach. i.e., shelves, stock room etc.. Perpetual Inventory System(continual system) A system that keeps track of removals from inventory continuously, thus monitoring current levels of each item Two-Bin System – It is very elementary system.Two containers of inventory; reorder when the first is empty. Second bin contains enough stock to satisfy expected demand. Universal Bar Code (UPC) - Bar code printed on a label that has information about the item to which it is attached. i.e., supermarkets, discount stores etc..

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Key Inventory Terms Lead time: time interval between ordering and receiving the order Holding (carrying) costs: cost to carry an item in inventory for a length of time, usually a year Ordering costs: costs of ordering and receiving inventory Shortage costs: costs when demand exceeds supply

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**ABC Classification System**

Classifying inventory according to some measure of importance and allocating control efforts accordingly. A more reasonable approach would be to allocate control efforts according to the relatively importance of various items in inventory. A - very important B - mod. important C - least important Annual $ value of items A B C High Low Percentage of Items

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**Cycle Counting A physical count of items in inventory**

The purpose of Cycle counting is to find out the real amount between the amount indicated by inventory records and actual quantities of inventory on hand Cycle counting management How much accuracy is needed? When should cycle counting be performed? Who should do it?

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Example-ABC approach Using the following annual demand & unit cost and calculate annual dollar value in each raw. Having computed the annual dollar values, use the concept of ABC classification and array from highest to lowest Item Number Annual demand Unit Cost Annual Dollar Value Classification 8 1000 $ 4000 5 3900 700 3 1900 500 6 915 1 2500 330 4 1500 100 12 400 300

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Example-ABC approach The first two items have relatively high annual dollar value, so it seems reasonable to classify them as A items. The next three items appear to have moderate annual dollar values and should be classified as B items. The remainder are C items due to their relatively low annual dollar value. Item Number Annual demand Unit Cost Annual Dollar Value Classification 8 1000 $ 4000 A 5 3900 700 3 1900 500 950000 B 6 915 915000 B 1 2500 330 825000 4 1500 100 150000 C 12 400 300 120000 C

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**Economic Order Quantity Models (Ford Harris, 1915)**

The question of how much to order is frequently determined by using an economic order quantity model (EOQ). (EOQ) models identify the optimal order quantity by minimizing the sum of certain annual cost that vary with order size. Three order size models are described here: The Basic Economic Order Quantity (EOQ) model The Economic Production Quantity Model Quantity Discount Model

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**Basic Economic Order Quantity Models**

The basic economic order quantity model (EOQ) is the simplest of the three model. It is used to identify a fixed order size that minimize the sum of annual cost of holding inventory and ordering inventory. The unit purchase price of items in inventory is not generally included in the total cost because the unit cost is unaffected by the order size unless quantity discounts are a factor. If holding costs are specified as a percentage of unit cost, then unit cost is indirectly included in the total cost as a part of holding costs.

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**Assumptions of the basic EOQ Model**

Only one product is involved Annual demand requirements are known Demand is spread evenly throughout the year so that the demand rate is reasonably constant Lead time does not vary Each order is received in a single delivery There are no quantity discounts

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**Profile of Inventory Level Over Time**

The Inventory Cycle Profile of Inventory Level Over Time Quantity on hand Q Receive order Place Lead time Reorder point Usage rate Time Order size=350 units Useage rate= 50 units per day Lead time= 2 days Reorder point= 100 units

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**Total Cost Annual carrying cost Annual ordering cost Total cost = + Q**

2 H D S + TC = Two basic inventory costs; Ordering Cost: are the basically the costs of getting the items into firm inventory, therefore these costs are the cost of replenishing inventory. Carring or holding cost: are the basically the costs incurred due to maintainance of inventories or are the costs of holding items in storage. As order size varies, one of type of cost will increase whilst the other decreases. The greater level of inventory over time, the higher the carring costs exist

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**The relationship between the type of costs**

The Total-Cost Curve is U-Shaped Annual Cost Ordering Costs Order Quantity (Q) QO (optimal order quantity)

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**The relationship between the type of costs**

Carring costs are linearly related to order size Ordering costs are inversely and non linearly related to order size Total cost curve is U-shape The total cost curve reaches its minimum where the carrying and ordering costs are equal. Q 2 H D S =

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Deriving the EOQ Using calculus, we take the derivative of the total cost function and set the derivative (slope) equal to zero and solve for Q.

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Example- EOQ model A local distributor for a national tire company expects to sell approximately 9600 steel-belted radial tires of a certain size and tread design next year. Annual carring cost is $ 16 per tire, and ordering cost is $ 75. The distributor operates 288 days a year. (a) What is the EOQ? (b) How many times per year does the store reorder? (c) What is the length of an order cycle? (d) What is the total annual cost if the EOQ quantity is ordered?

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**Example- EOQ model-Answer**

(a) What is the EOQ? (b) How many times per year does the store reorder? (c) What is the length of an order cycle? (d) What is the total annual cost if the EOQ quantity is order?

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**Example- Optimal Quantity**

Piddling manufacturing assembles security monitors. It purchases 3600 black and white cathode ray tubes a year at $ 65 each. Ordering costs are $ 31, and annual carring costs are 20 percent of the purchase price. (a) Compute the optimal quantity (b) What is the carring cost? (c) What is the ordering cost? (d) Calculate the total annual cost ?

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**Example- Optimal Quantity-Answer**

(a) Compute the optimal quantity (b) What is the carring cost? (c) What is the ordering cost? (d) Calculate the total annual cost ?

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**Economic Production Quantity (EPQ)**

Production done in batches or lots. Even in assembly operations, portions of the work are done in batches. Capacity to produce a part exceeds the part’s usage or demand rate As long as production continious, inventory will continue to grow. This makes sense to periodically produce such items in batches or lots instead of producing continually. Assumptions of EPQ are similar to EOQ except orders are received incrementally during production

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**Economic Production Quantity Assumptions**

Only one item is involved Annual demand is known Usage rate is constant Usage occurs continually Production rate is constant Lead time does not vary No quantity discounts

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**Setup Cost and Economic Run Size (quantity)**

In the case of EPQ, there are no ordering cost, however there are setup costs-the costs required to prepare the equipment for the job, such as cleaning, adjusting changing tools etc. Setup costs are analogous to ordering costs because they are independent of the lot or run size. The larger run size, the fewer the number of runs needed as well as the lower the annual setup cost. D/Q is the number of batches per year. DS/Q0 is setup cost.

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Example- Run size A toy manufacturer uses rubber wheels per year for its popular damp truck series. The firm makes its own wheels which it can produce at the rate of 800 per day. The toy trucks are assembled uniformly over the entire year. Carring cost is $ 1 per wheel a year. Setup cost for a production run of wheels is $ 45. The firm operates 240 days per year. Determine (a) Optimal run size (b) Minimum total annual cost for carring and setup (c) Cycle time for the optimal run size (d) Run time

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**Example- Run size-Answer**

(a) Optimal run size (b) Minimum total annual cost for carring and setup (c) Cycle time for the optimal run size (d) Run time

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**The Quantity Discount Model**

Quantity discount rate are price reductions for large orders offered to customers to induce them to buy in large quantities. As prices decrease order of quantities increase. The buyer’s goal with quantity discount rate is to select the order quantity that minimize total cost in the following equation where p is unit price Annual carrying cost Purchasing TC = + Q 2 H D S ordering PD

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**The Quantity Discount Model**

In the basic EOQ model, determination of order size does not involve the purchasing cost. The rationale for not including unit price is that under the assumption of no discount discounts, price per unit is the same for all order sizes. Inclusion of unit price in the total cost computation in that case would merely increase the total cost by the amount P times D A graph of total annual purchase cost versus quantity would be a horizontal line. Hence, including purchasing costs would only raise the total cost curve by the same amount (PD) at every point. In the following graph, that would not change the EOQ.

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**Total Costs with PD Cost Adding Purchasing cost doesn’t change EOQ**

TC with PD TC without PD PD Quantity Adding Purchasing cost doesn’t change EOQ

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**The Quantity Discount Model**

When quantity discounts are offered, there is a separate U-shaped total cost curve for each unit price. Again, including unit prices merely raises each curve by a constant amount. However, because the prices are all different, each curve is raised by a different amount. Smaller unit prices will raise a total cost curve less than large unit prices In the following graph, no one curve applies to the entire range of quantities and each curve applies to only a portion of the range. Hence the applicable or feasible total cost is initially on the curve with highest unit price and then drops down curve by curve at the price breaks which are the minimum quantities needed to obtain the discount.

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**Total Cost with Constant Carrying Costs**

Comparison of TC curves for constant carrying costs and carrying costs that are a percentage of unit costs. When carrying costs are constant, all curves have their minimum points at the same quantity. When carrying costs are stated as percentage of unit price, the minimum points do not line up. TCa TCb Decreasing Price Total Cost TCc CC a,b,c OC EOQ Quantity

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**Example- Discount Model**

The maintenance department of a large hospital uses about 816 cases of luquid cleanser annually. Ordering costs are $ 12, Carring cost are $ 4 per case a year and the new price schedule indicates that orders of less than 50 cases will cost $ 20 per case, 50 to 79 cases will cost $ 18 per case, 80 to 99 cases will cost $ 17 per cases and larger orders will cost $ 16 per case. Determine (a) the common minimum point for EOQ (b) the total cost if the feasible minimum point is on the lowest price range, that is the optimal order quantity. (c) the total cost if the feasible minimum point is in any other price range. Range Price 1 to 49 $ 20 50 to 79 18 80 to 99 17 100 or more 16

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**Example- Discount Model -Answer**

Range Price 1 to 49 $ 20 50 to 79 18 80 to 99 17 100 or more 16 (a) the common EOQ (b) total annual cost (c) total annual cost TC = Carrying cost + Order cost + Purchase cost The cases can be bought at $ 18 per case because 70 falls in the range of 50 to 79 cases.

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**Example 2- Discount Model**

Surge Electric uses 4000 toggle switches a year. Switch are priced in the following table. It costs approximately $ 30 to prepare an order and receive it, and carring costs are 40 percent of purchase price per unit on an annual basis. Range Price 1 to 449 $ 0.90 500 to 999 0.85 1000 or more 0.80 Determine (a) the common EOQ (b) the total cost if the feasible minimum point is on the lowest price range, that is the optimal order quantity. (c) the total cost if the feasible minimum point is in any other price range.

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**Example- Discount Model -Answer**

Range Price 1 to 449 $ 0.90 500 to 999 0.85 1000 or more 0.80 (a) the common minimum point or EOQ (b) total annual cost (c) total annual cost H= 0.40 P H= 0.40 (0.80)=0.32 H= 0.40 (0.85)=0.34 TC = Carrying cost + Order cost + Purchase cost

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**When to Reorder with EOQ Ordering**

Reorder Point (ROP) – This point occurs When the quantity on hand of an item drops to a predetermined amount, the item is reordered. Safety Stock - Stock that is held in excess of expected demand due to variable demand rate and/or lead time. Service Level - Probability that demand will not exceed supply during lead time.

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**Determinants of the Reorder Point**

The rate of demand The lead time Demand and/or lead time variability Stock out risk (safety stock) If demand and lead time are both constant, the reorder point is simply defined as ROP=(d) x (LT) Where d = demand rate units per day or week LT= lead time in days or weeks

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Example - reorder Tingly takes two a day vitamins which are delivered to his home by routeman seven days after an order is called in. At what point should Tingly reorder?. Usage is 2 vitamins per day Lead time is 7 days ROP=(2) (7)= 14 vitamins Thus, Tingly should reorder when 14 vitamin tables are left.

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**Safety Stock Safety stock reduces risk of stockout during lead time**

LT Time Expected demand during lead time Maximum probable demand ROP Quantity Safety stock ROP= Expected demand in (LT) + safety stock

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Reorder Point The ROP based on a normal Distribution of lead time demand ROP= Expected demand in (LT) + ZαdLT Where z is the number of standard deviationαdLT is the standard deviation lead time demand ROP Risk of a stockout Service level Probability of no stockout Expected demand Safety stock z Quantity z-scale

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Example – ROP and Z A manager of a construction supply house determined from historical records that demand for sand during lead time averages 50 tons. In addition, the manager determined that demand during lead time could be described by a normal distribution that has a mean of 50 tons and a standard deviation of 5 tons as well as the manager is willingly to accept a stockout risk of no more than 3 percent. (a) What value of Z is appropriate? (b) How much safety stock should be held? (c) What reorder point should be used?

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**Answer– ROP and Z (a) What value of Z is appropriate?**

Expected lead time is 50 tons α dLT=5 tons risk =3 percent 1-0.03=0.97 from z table (lead time), z= 1.88 (page 569, table 12.3) (b) How much safety stock should be held? Safety stock = Z αdLT= (1.88) (5)= 9.40 tons (c) What reorder point should be used? ROP= Expected demand in (LT) + ZαdLT = = tons ROP= Expected demand in (LT) + ZαdLT = = tons

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**Example – shortage and service lead**

Suppose standard deviation of lead time demand is known to be 20 units. Lead time demand is approximately normal. (a) For lead time service level of 90 percent, determine the expected number of units short for any other cycle. (b) What lead time service level would an expected shortage of 2 units imply?

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**Answer – shortage and service lead**

(a) For lead time service level of 90 percent, determine the expected number of units short for any other cycle. αdLT= 20 units lead time service level is 0.90 from z table (lead time), (E) z= (page 569, table 12.3) E(n) =(E)z αdL= (0.048) (20)= 0.96 or about 1 unit. or E(N) =E (n) (D/Q) (b) What lead time service level would an expected shortage of 2 units imply? E(n) = 2 E(n) =(E)z αdL or (E)z = E(n) / αdL =(2)/(20)= from the table, lead time service level is percent or 87%

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**Fixed-Order-Interval Model**

Orders are placed at fixed time intervals Order quantity is for next interval Suppliers might encourage fixed intervals May require only periodic checks of inventory levels Risk of stockout Fill rate – the percentage of demand filled by the stock on hand

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**Fixed-Interval Benefits**

Tight control of inventory items Items from same supplier may yield savings in: Ordering Packing Shipping costs May be practical when inventories cannot be closely monitored

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**Fixed-Interval Disadvantages**

Requires a larger safety stock Increases carrying cost Costs of periodic reviews

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**Example – Amount to order**

Given the following information: (a) determine the amount to order Amount to order = expected time during the production interval + safety stock - Amount on hand at reorder time

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Single Period Model Single period model: model for ordering of perishables and other items with limited useful lives Shortage cost: generally the unrealized profits per unit Excess cost: difference between purchase cost and salvage value of items left over at the end of a period

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**Single Period Model Continuous stocking levels**

Identifies optimal stocking levels Optimal stocking level balances unit shortage and excess cost Discrete stocking levels Service levels are discrete rather than continuous Desired service level is equaled or exceeded

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**Optimal Stocking Level**

Service level = Cs Cs + Ce Cs = Shortage cost per unit Ce = Excess cost per unit Service Level So Quantity Ce Cs Balance point

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**Example-optimal stocking level**

Sweet cider is delivered weekly to Cindy’s bar. Demand varies uniformly between 300 lt and 500 lt per week. Cindy pays 20 cents per liter and charges 80 percent per liter for it. Unsold cider has no salvage value and cannot be carried over into the week due to spoilage. Find the optimal stocking level and its stockout risk for that quantity where Ce = Cost per unit- Salvage value per unit =$ = $0.20 per unit Cs = Revenue per unit-Cost per unit =$0.80-$0.20=$0.60 per unit

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**Answer-optimal stocking level**

Ce = $ 0.20 per unit Cs = $ 0.60 per dozen Service level = Cs/(Cs+Ce) = 0.60/( ) Service level = .75 S= ( )=450 lt Service Level = 75% Quantity Ce Cs Stockout risk = 1.00 – 0.75 = 0.25

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**Example-Discrete-optimal stocking level**

Demand for long-stemmed red roses at a small flower shop can be approximated using a poisson distribution that has a mean of four dozen per. Profit on the roses is $ 3 per dozen. Leftover flowers are marked down and sold the next day at a loss of $2 per dozen. Assume that all marked flowers are sold. Find the optimal stocking level and its stockout risk for that quantity where Ce = Cost per unit- Salvage value per unit =$ 2 per dozen Cs = Revenue per unit-Cost per unit =$ 3 per dozen

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**Answer-Discrete-optimal stocking level**

Ce = $ 2 per dozen Cs = $ 3 per dozen Service level = Cs/(Cs+Ce) = 3/(3+2) Service level = .60 It is neccessary to stock 4 dozan. (D) Dozen per day Cum.Freq. 0.018 1 0.092 2 0.238 3 0.433 4 0.629 5 0.785 Service Level = 60% Quantity Ce Cs Appendix B, Table C Stockout risk = 1.00 – 0.60 = 0.40

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