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**Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.**

12 Inventory Management For Operations Management, 9e by Krajewski/Ritzman/Malhotra © 2010 Pearson Education Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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**© 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458**

Learning Objectives Know the types of inventory and their associated costs Be able to do an ABC analysis for a list of inventory items Be able to determine the Economic Order Quantity (EOQ) for an inventory item and adapt this concept to volume pricing options Be able to determine the Economic Production Lot Size (ELS) for an inventory item based on its demand and production rates. (discussed in Supplement D in your text). Be able to determine reorder points and appropriate safety stock levels for a desired level of service Know the differences between ELS and EOQ lot-sizing rules Know the differences between continuous review and periodic review of inventory strategies and where each is best used. Be able to list the various approaches for reducing required inventory levels © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J Transparency Masters to accompany Heizer/Render – Principles of Operations Management, 5e, and Operations Management, 7e

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**Functions of Inventory**

Meet customer demands, provide selection De-couple production from distribution Decrease costs (all types) De-couple production processes Manage input demand (services, backlogs) Uncertainty protection

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**Disadvantages of Inventory (All measured by increased cost)**

Increased costs for storage, handling, tracking, and protection Hides production problems Risk of deterioration or obsolescence Risk of lost business (demand inventory)

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**Types of Inventory – Where Did It Come From?**

Cycle inventory –> result of lot size = (Q/2)+SS Raw –> fixed order sizes WIP and FG –> fixed production batch sizes Safety stock –> just in case something goes wrong – all types Demand/Backorders -> insufficient capacity Anticipation inventory –> all types Pipeline –> demand rate times transit time = d×L Vendor Managed Inventory (VMI) Maintenance/Repair/Operating Supplies

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**Inventory Notation and Definitions**

IP = inventory position = OH + SR – BO T = target inventory position OH = on-hand inventory SR = scheduled receipts (open orders) BO = back orders (delayed delivery) ROP = reorder point (inventory level when order is placed) Q = order quantity TBO = time between orders for continuous review P = fixed time between orders for periodic review L = lead time from order until delivery Protection interval = P + L ADDLT = average demand during lead time SO = stock out (no material to satisfy demand) SS = safety stock (cushion to prevent stock out)

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**Physical Inventory Classifications**

Process stage Number and Value Demand Type Other Raw Material WIP Finished Goods A Items B Items C Items Independent Dependent Maintenance Operating

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**Demand Inventory Classifications**

Waiting Lines Booked Appointments Reservations Backlogs

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**Pressures on inventory**

Pressure for lower inventory Inventory investment Inventory holding cost Pressure for higher inventory Customer service Other costs related to inventory

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**ABC Analysis Enables effective focus, prioritization**

Divides on-hand inventory into 3 classes A class, B class, C class Basis is usually annual dollar volume (investment) $ volume = Annual demand x Unit cost Policies based on ABC analysis Develop class A suppliers more Establish tighter physical control of A items Forecast A items more carefully It might be helpful here to discuss some of the differences in the ways we would manage items in the three different levels. What actions would we actually take in managing A versus in managing C?

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**ABC Analysis Class % $Vol % Items A 80 20 B 15 30 C 5 50**

10 20 30 40 50 60 70 80 90 100 Percentage of items Percentage of dollar value 100 — 90 — 80 — 70 — 60 — 50 — 40 — 30 — 20 — 10 — 0 — Class C Class B Class A Class % $Vol % Items A 80 20 B 15 30 C 5 50

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Inventory Costs Holding costs - associated with holding or “carrying” inventory over time Ordering costs - associated with costs of placing order and receiving goods Setup costs - cost to prepare a machine or process for manufacturing an order Transportation costs – associated with distribution (pipeline inventory).

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**Holding (Carrying) Costs**

Obsolescence Insurance Extra staffing for handling, management Interest Pilferage Damage Warehousing Taxes Tracking IBM Tracking RFID Etc. You might ask students if they can identify an industry for which the cost of obsolescence is particularly important. Is the number of such industries likely to grow or decline? The same question could be asked regarding pilferage. The question could be asked in a more general manner: Are there industries for which one or another of the areas listed is of particular or unusual importance?

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**Inventory Holding Costs (Approximate Ranges)**

Category Housing costs (building rent, depreciation, operating cost, taxes, insurance) Material handling costs (equipment, lease or depreciation, power, operating cost) Labor cost from extra handling Investment costs (borrowing costs, taxes, and insurance on inventory) Pilferage, scrap, and obsolescence Overall carrying cost Cost as a % of Inventory Value 6% (3 - 10%) 3% ( %) (3 - 5%) 11% (6 - 24%) (2 - 5%) 26% Note that this slide suggest holding costs are, on average, about 26% of the inventory value

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**Ordering Costs Supplies Forms**

Order placement (submitting, tracking, receiving) Clerical support Packaging waste disposal Etc.

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**Setup Costs Clean-up costs Re-tooling costs Adjustment costs**

Initial yield losses Documentation and tracking WIP management Routine maintenance Etc.

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Inventory Management Effective inventory management requires information about: Expected demand (accurate forecasting) Amounts on hand Amounts on order Appropriate timing and size of the reorder quantities Acquisition and holding costs

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**Inventory Models Continuous review models (Q)**

Economic order quantity (EOQ) Economic production order size (ELS) Quantity discount Periodic review models (P) Probabilistic models Help answer the inventory planning questions! This slide simply introduces some of the available models. Additional details are provided in subsequent slides. © T/Maker Co.

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**Economic Order Quantity**

The lot size, Q, that minimizes total annual inventory holding and ordering costs Five assumptions Demand rate is constant and known with certainty No constraints are placed on the size of each lot The only two relevant costs are the inventory holding cost and the fixed cost per lot for ordering or setup Decisions for one item can be made independently of decisions for other items The lead time is constant and known with certainty Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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**Economic Order Quantity**

Don’t use the EOQ Make-to-order strategy Order size is constrained Modify the EOQ Quantity discounts Replenishment not instantaneous Use the EOQ Make-to-stock Carrying and setup costs are known and relatively stable Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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**Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.**

Calculating EOQ Receive order Inventory depletion (demand rate) On-hand inventory (units) Time Q Average cycle inventory Q 2 1 cycle Figure 12.2 – Cycle-Inventory Levels Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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**Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.**

Calculating EOQ Annual holding cost Annual holding cost = (Average cycle inventory) (Unit holding cost) Annual ordering cost Annual ordering cost = (Number of orders/Year) (Ordering or setup costs) Total annual cycle-inventory cost Total costs = Annual holding cost + Annual ordering or setup cost Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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**Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.**

Calculating EOQ Annual cost (dollars) Lot Size (Q) Total cost Holding cost Ordering cost Figure 12.3 – Graphs of Annual Holding, Ordering, and Total Costs Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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**Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.**

Calculating EOQ Total annual cycle-inventory cost C = (H) (S) Q 2 D where C = total annual cycle-inventory cost Q = lot size H = holding cost per unit per year D = annual demand S = ordering or setup costs per lot Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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**The Cost of a Lot-Sizing Policy**

EXAMPLE 12.1 A museum of natural history opened a gift shop which operates 52 weeks per year. Managing inventories has become a problem. Top-selling SKU is a bird feeder. Sales are 18 units per week, the supplier charges $60 per unit. Ordering cost is $45. Annual holding cost is 25 percent of a feeder’s value. Management chose a 390-unit lot size. What is the annual cycle-inventory cost of the current policy of using a 390-unit lot size? Would a lot size of 468 be better? Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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**The Cost of a Lot-Sizing Policy**

SOLUTION We begin by computing the annual demand and holding cost as D = H = (18 units/week)(52 weeks/year) = 936 units 0.25($60/unit) = $15 The total annual cycle-inventory cost for the current policy is C = (H) (S) Q 2 D = ($15) ($45) = $2,925 + $108 = $3,033 390 2 936 The total annual cycle-inventory cost for the alternative lot size is ($15) ($45) = $3,510 + $90 = $3,600 468 2 936 C = Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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**The Cost of a Lot-Sizing Policy**

Figure 12.4 – Total Annual Cycle-Inventory Cost Function for the Bird Feeder Current cost 3000 – 2000 – 1000 – 0 – | | | | | | | | Lot Size (Q) Annual cost (dollars) Total cost = (H) (S) Q 2 D Holding cost = (H) Q 2 Lowest cost Ordering cost = (S) D Q Best Q (EOQ) Current Q Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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**Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.**

Calculating EOQ The EOQ formula: EOQ = 2DS H Time between orders TBOEOQ = (12 months/year) EOQ D Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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**Finding the EOQ, Total Cost, TBO**

EXAMPLE 12.2 For the bird feeders in Example 12.1, calculate the EOQ and its total annual cycle-inventory cost. How frequently will orders be placed if the EOQ is used? SOLUTION Using the formulas for EOQ and annual cost, we get EOQ = = 2DS H = or 75 units 2(936)(45) 15 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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**Finding the EOQ, Total Cost, TBO**

Figure 12.5 shows that the total annual cost is much less than the $3,033 cost of the current policy of placing 390-unit orders. Figure 12.5 – Total Annual Cycle-Inventory Costs Based on EOQ Using Tutor 12.2 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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**Finding the EOQ, Total Cost, TBO**

When the EOQ is used, the TBO can be expressed in various ways for the same time period. TBOEOQ = EOQ D = = year 75 936 TBOEOQ = (12 months/year) EOQ D = (12) = 0.96 month 75 936 TBOEOQ = (52 weeks/year) EOQ D = (52) = 4.17 weeks 75 936 TBOEOQ = (365 days/year) EOQ D = (365) = days 75 936 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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**Noninstantaneous Replenishment**

Maximum cycle inventory Item used or sold as it is completed Usually production rate, p, exceeds the demand rate, d, so there is a buildup of (p – d) units per time period Both p and d expressed in same time interval Buildup continues for Q/p days Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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**Noninstantaneous Replenishment**

On-hand inventory Time Production quantity Q Demand during production interval Maximum inventory Imax p – d Production and demand Demand only TBO Figure D.1 – Lot Sizing with Noninstantaneous Replenishment Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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**Noninstantaneous Replenishment**

Maximum cycle inventory is: where p = production rate d = demand rate Q = lot size Cycle inventory is no longer Q/2, it is Imax /2 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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**Noninstantaneous Replenishment**

Total annual cost = Annual holding cost + Annual ordering or setup cost D is annual demand and Q is lot size d is daily demand; p is daily production rate Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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**Noninstantaneous Replenishment**

Economic Production Lot Size (ELS): optimal lot size Derived by calculus Because the second term is greater than 1, the ELS results in a larger lot size than the EOQ Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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**Finding the Economic Production Lot Size**

EXAMPLE D.1 A plant manager of a chemical plant must determine the lot size for a particular chemical that has a steady demand of 30 barrels per day. The production rate is 190 barrels per day, annual demand is 10,500 barrels, setup cost is $200, annual holding cost is $0.21 per barrel, and the plant operates 350 days per year. a. Determine the economic production lot size (ELS) b. Determine the total annual setup and inventory holding cost for this item c. Determine the time between orders (TBO), or cycle length, for the ELS d. Determine the production time per lot What are the advantages of reducing the setup time by 10 percent? Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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**Finding the Economic Production Lot Size**

SOLUTION a. Solving first for the ELS, we get b. The total annual cost with the ELS is Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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**Finding the Economic Production Lot Size**

c. Applying the TBO formula to the ELS, we get d. The production time during each cycle is the lot size divided by the production rate: Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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**Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.**

Managerial Insights TABLE | SENSITIVITY ANALYSIS OF THE EOQ Parameter EOQ Parameter Change EOQ Change Comments Demand ↑ Increase in lot size is in proportion to the square root of D. Order/Setup Costs ↓ Weeks of supply decreases and inventory turnover increases because the lot size decreases. Holding Costs Larger lots are justified when holding costs decrease. 2DS H Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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**Inventory Control Systems**

Continuous review (Q) system Reorder point system (ROP) and fixed order quantity (FOQ) system — FOQ often = EOQ For independent demand items Tracks inventory position (IP) Includes scheduled receipts (SR), on-hand inventory (OH), and back orders (BO) Inventory position = On-hand inventory + Scheduled receipts – Backorders IP = OH + SR – BO Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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**Selecting the Reorder Point**

IP Time On-hand inventory Order received Q OH R Order placed L L TBO TBO TBO Figure 12.6 – Q System When Demand and Lead Time Are Constant and Certain Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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**Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.**

Placing a New Order EXAMPLE 12.3 Demand for chicken soup at a supermarket is always 25 cases a day and the lead time is always 4 days. The shelves were just restocked with chicken soup, leaving an on-hand inventory of only 10 cases. No backorders currently exist, but there is one open order in the pipeline for 200 cases. What is the inventory position? Should a new order be placed? SOLUTION R = Total demand during lead time = (25)(4) = 100 cases IP = OH + SR – BO = – 0 = 210 cases Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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**Continuous Review Systems**

Selecting the reorder point with variable demand and constant lead time Reorder point = Average demand during lead time + Safety stock = dL + safety stock where d = average demand per week (or day or months) L = constant lead time in weeks (or days or months) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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**Continuous Review Systems**

Time On-hand inventory TBO1 TBO2 TBO3 L1 L2 L3 R Order received Q placed IP Figure 12.7 – Q System When Demand Is Uncertain Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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**Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.**

Reorder Point Choose an appropriate service-level policy Select service level or cycle service level Protection interval Determine the demand during lead time probability distribution Determine the safety stock and reorder point levels Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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**Demand During Lead Time**

Specify mean and standard deviation Standard deviation of demand during lead time σdLT = σd2L = σd L Safety stock and reorder point Safety stock = zσdLT where z = number of standard deviations needed to achieve the cycle-service level σdLT = standard deviation of demand during lead time Reorder point = R = dL + safety stock Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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**Demand During Lead Time**

+ 75 Demand for week 1 + 75 Demand for week 2 σt = 15 = 75 Demand for week 3 σt = 15 σt = 25.98 225 Demand for 3-week lead time Figure 12.8 – Development of Demand Distribution for the Lead Time Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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**Demand During Lead Time**

Cycle-service level = 85% Probability of stockout (1.0 – 0.85 = 0.15) Average demand during lead time R zσdLT Figure 12.9 – Finding Safety Stock with a Normal Probability Distribution for an 85 Percent Cycle-Service Level Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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**Reorder Point for Variable Demand**

EXAMPLE 12.4 Let us return to the bird feeder in Example The EOQ is 75 units. Suppose that the average demand is 18 units per week with a standard deviation of 5 units. The lead time is constant at two weeks. Determine the safety stock and reorder point if management wants a 90 percent cycle-service level. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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**Reorder Point for Variable Demand**

SOLUTION In this case, σd = 5, d = 18 units, and L = 2 weeks, so σdLT = σd L = = Consult the body of the table in the Normal Distribution appendix for , which corresponds to a 90 percent cycle-service level. The closest number is , which corresponds to 1.2 in the row heading and 0.08 in the column heading. Adding these values gives a z value of With this information, we calculate the safety stock and reorder point as follows: Safety stock = zσdLT = 1.28(7.07) = 9.05 or 9 units Reorder point = dL + Safety stock = 2(18) + 9 = 45 units Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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**Reorder Point for Variable Demand and Lead Time**

Often the case that both are variable The equations are more complicated Safety stock = zσdLT R = (Average weekly demand Average lead time) + Safety stock = dL + Safety stock where d = Average weekly (or daily or monthly) demand L = Average lead time σd = Standard deviation of weekly (or daily or monthly) demand σLT = Standard deviation of the lead time σdLT = Lσd2 + d2σLT2 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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**Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.**

Reorder Point EXAMPLE 12.5 The Office Supply Shop estimates that the average demand for a popular ball-point pen is 12,000 pens per week with a standard deviation of 3,000 pens. The current inventory policy calls for replenishment orders of 156,000 pens. The average lead time from the distributor is 5 weeks, with a standard deviation of 2 weeks. If management wants a 95 percent cycle-service level, what should the reorder point be? Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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**Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.**

Reorder Point SOLUTION We have d = 12,000 pens, σd = 3,000 pens, L = 5 weeks, and σLT = 2 weeks σdLT = Lσd2 + d2σLT2 = (5)(3,000)2 + (12,000)2(2)2 = 24, pens From the Normal Distribution appendix for , the appropriate z value = We calculate the safety stock and reorder point as follows: Safety stock = zσdLT = (1.65)(24,919.87) = 41, or 41,118 pens Reorder point = dL + Safety stock = (12,000)(5) = 101,118 pens Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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**Continuous Review (Q) Systems**

Two-Bin system Visual system An empty first bin signals the need to place an order Calculating total systems costs Total cost = Annual cycle inventory holding cost + Annual ordering cost + Annual safety stock holding cost C = (H) (S) + (H) (Safety stock) Q 2 D Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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**Periodic Review System (P)**

Fixed interval reorder system or periodic reorder system Four of the original EOQ assumptions maintained No constraints are placed on lot size Holding and ordering costs Independent demand Lead times are certain Order is placed to bring the inventory position up to the target inventory level, T, when the predetermined time, P, has elapsed Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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**Periodic Review System (P)**

Time On-hand inventory T Order placed received IP OH Q1 Q2 Q3 IP3 IP1 IP2 L P Protection interval Figure – P System When Demand Is Uncertain Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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**How Much to Order in a P System**

EXAMPLE 12.6 A distribution center has a backorder for five 36-inch color TV sets. No inventory is currently on hand, and now is the time to review. How many should be reordered if T = 400 and no receipts are scheduled? SOLUTION IP = OH + SR – BO = – 5 = –5 sets T – IP = 400 – (–5) = 405 sets That is, 405 sets must be ordered to bring the inventory position up to T sets. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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**Periodic Review System**

Selecting the time between reviews, choosing P and T Selecting T when demand is variable and lead time is constant IP covers demand over a protection interval of P + L The average demand during the protection interval is d(P + L), or T = d(P + L) + safety stock for protection interval Safety stock = zσP + L , where σP + L = Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Calculating P and T EXAMPLE 12.7 Again, let us return to the bird feeder example. Recall that demand for the bird feeder is normally distributed with a mean of 18 units per week and a standard deviation in weekly demand of 5 units. The lead time is 2 weeks, and the business operates 52 weeks per year. The Q system developed in Example 12.4 called for an EOQ of 75 units and a safety stock of 9 units for a cycle-service level of 90 percent. What is the equivalent P system? Answers are to be rounded to the nearest integer. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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**Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.**

Calculating P and T SOLUTION We first define D and then P. Here, P is the time between reviews, expressed in weeks because the data are expressed as demand per week: D = (18 units/week)(52 weeks/year) = 936 units P = (52) = EOQ D (52) = 4.2 or 4 weeks 75 936 With d = 18 units per week, an alternative approach is to calculate P by dividing the EOQ by d to get 75/18 = 4.2 or 4 weeks. Either way, we would review the bird feeder inventory every 4 weeks. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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**Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.**

Calculating P and T We now find the standard deviation of demand over the protection interval (P + L) = 6: Before calculating T, we also need a z value. For a 90 percent cycle-service level z = The safety stock becomes Safety stock = zσP + L = 1.28(12.25) = or 16 units We now solve for T: T = Average demand during the protection interval + Safety stock = d(P + L) + safety stock = (18 units/week)(6 weeks) + 16 units = 124 units Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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**Periodic Review System**

Use simulation when both demand and lead time are variable Suitable to single-bin systems Total costs for the P system are the sum of the same three cost elements as in the Q system Order quantity and safety stock are calculated differently C = (H) (S) + HzσP + L dP 2 D Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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**Comparative Advantages**

Primary advantages of P systems Convenient Orders can be combined Only need to know IP when review is made Primary advantages of Q systems Review frequency may be individualized Fixed lot sizes can result in quantity discounts Lower safety stocks Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Hybrid systems Optional replenishment systems Optimal review, min-max, or (s,S) system, like the P system Reviews IP at fixed time intervals and places a variable-sized order to cover expected needs Ensures that a reasonable large order is placed Base-stock system Replenishment order is issued each time a withdrawal is made Order quantities vary to keep the inventory position at R Minimizes cycle inventory, but increases ordering costs Appropriate for expensive items Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Quantity Discounts Price incentives to purchase large quantities create pressure to maintain a large inventory Item’s price is no longer fixed If the order quantity is increased enough, then the price per unit is discounted A new approach is needed to find the best lot size that balances: Advantages of lower prices for purchased materials and fewer orders Disadvantages of the increased cost of holding more inventory Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Quantity Discounts Total annual cost = Annual holding cost + Annual ordering or setup cost + Annual cost of materials where P = price per unit Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Quantity Discounts Unit holding cost (H) is usually expressed as a percentage of unit price The lower the unit price (P) is, the lower the unit holding cost (H) is The total cost equation yields U-shape total cost curves There are cost curves for each price level The feasible total cost begins with the top curve, then drops down, curve by curve, at the price breaks EOQs do not necessarily produce the best lot size The EOQ at a particular price level may not be feasible The EOQ at a particular price level may be feasible but may not be the best lot size Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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**Two-Step Solution Procedure**

Step 1. Beginning with lowest price, calculate the EOQ for each price level until a feasible EOQ is found. It is feasible if it lies in the range corresponding to its price. Each subsequent EOQ is smaller than the previous one, because P, and thus H, gets larger and because the larger H is in the denominator of the EOQ formula. Step 2. If the first feasible EOQ found is for the lowest price level, this quantity is the best lot size. Otherwise, calculate the total cost for the first feasible EOQ and for the larger price break quantity at each lower price level. The quantity with the lowest total cost is optimal. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Quantity Discounts C for P = $4.00 C for P = $3.50 C for P = $3.00 EOQ 4.00 EOQ 3.50 EOQ 3.00 Total cost (dollars) Purchase quantity (Q) Total cost (dollars) Purchase quantity (Q) First price break Second price break PD for P = $4.00 PD for P = $3.50 PD for P = $3.00 First price break Second price break (a) Total cost curves with purchased materials added (b) EOQs and price break quantities Figure D.3 – Total Cost Curves with Quantity Discounts Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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**Find Q with Quantity Discounts**

EXAMPLE D.2 A supplier for St. LeRoy Hospital has introduced quantity discounts to encourage larger order quantities of a special catheter. The price schedule is Order Quantity Price per Unit 0 to 299 $60.00 300 to 499 $58.80 500 or more $57.00 The hospital estimates that its annual demand for this item is 936 units, its ordering cost is $45.00 per order, and its annual holding cost is 25 percent of the catheter’s unit price. What quantity of this catheter should the hospital order to minimize total costs? Suppose the price for quantities between 300 and 499 is reduced to $ Should the order quantity change? Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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**Find Q with Quantity Discounts**

SOLUTION Step 1: Find the first feasible EOQ, starting with the lowest price level: A 77-unit order actually costs $60.00 per unit, instead of the $57.00 per unit used in the EOQ calculation, so this EOQ is infeasible. Now try the $58.80 level: This quantity also is infeasible because a 76-unit order is too small to qualify for the $58.80 price. Try the highest price level: Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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**Find Q with Quantity Discounts**

This quantity is feasible because it lies in the range corresponding to its price, P = $60.00 Step 2: The first feasible EOQ of 75 does not correspond to the lowest price level. Hence, we must compare its total cost with the price break quantities (300 and 500 units) at the lower price levels ($58.80 and $57.00): Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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**Find Q with Quantity Discounts**

The best purchase quantity is 500 units, which qualifies for the deepest discount Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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One-Period Decisions Seasonal goods are a dilemma facing many retailers. Newsboy problem Step 1: List different demand levels and probabilities. Step 2: Develop a payoff table that shows the profit for each purchase quantity, Q, at each assumed demand level, D. Each row represents a different order quantity and each column represents a different demand. The payoff depends on whether all units are sold at the regular profit margin which results in two possible cases. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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One-Period Decisions If demand is high enough (Q ≤ D), then all of the cases are sold at the full profit margin, p, during the regular season Payoff = (Profit per unit)(Purchase quantity) = pQ If the purchase quantity exceeds the eventual demand (Q > D), only D units are sold at the full profit margin, and the remaining units purchased must be disposed of at a loss, l, after the season Payoff = – (Demand) Loss per unit Profit per unit sold during season Amount disposed of after Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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One-Period Decisions Step 3: Calculate the expected payoff of each Q by using the expected value decision rule. For a specific Q, first multiply each payoff by its demand probability, and then add the products. Step 4: Choose the order quantity Q with the highest expected payoff. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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**Finding Q for One-Period Decisions**

EXAMPLE D.3 One of many items sold at a museum of natural history is a Christmas ornament carved from wood. The gift shop makes a $10 profit per unit sold during the season, but it takes a $5 loss per unit after the season is over. The following discrete probability distribution for the season’s demand has been identified: Demand 10 20 30 40 50 Demand Probability 0.2 0.3 0.1 How many ornaments should the museum’s buyer order? Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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**Finding Q for One-Period Decisions**

SOLUTION Each demand level is a candidate for best order quantity, so the payoff table should have five rows. For the first row, where Q = 10, demand is at least as great as the purchase quantity. Thus, all five payoffs in this row are Payoff = pQ = ($10)(10) = $100 This formula can be used in other rows but only for those quantity–demand combinations where all units are sold during the season. These combinations lie in the upper-right portion of the payoff table, where Q ≤ D. For example, the payoff when Q = 40 and D = 50 is Payoff = pQ = ($10)(40) = $400 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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**Finding Q for One-Period Decisions**

The payoffs in the lower-left portion of the table represent quantity–demand combinations where some units must be disposed of after the season (Q > D). For this case, the payoff must be calculated with the second formula. For example, when Q = 40 and D = 30, Payoff = pD – l(Q – D) = ($10)(30) – ($5)(40 – 30) = $250 Using OM Explorer, we obtain the payoff table in Figure D.5 Now we calculate the expected payoff for each Q by multiplying the payoff for each demand quantity by the probability of that demand and then adding the results. For example, for Q = 30, Payoff = 0.2($0) + 0.3($150) + 0.3($300) + 0.1($300) + 0.1($300) = $195 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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**Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.**

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