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**Service Levels and Lead Times in Supply Chains: The Order-up-to Inventory Model Chapter 13**

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**Medtronic’s InSync pacemaker supply chain and objectives**

One distribution center (DC) in Mounds View, MN. About 500 sales territories throughout the country. Consider Susan Magnotto’s territory in Madison, Wisconsin. Objective: Because the gross margins are high, develop a system to minimize inventory investment while maintaining a very high service target, e.g., a 99.9% in-stock probability or a 99.9% fill rate.

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**Timing in the order up-to model**

Time is divided into periods of equal length, e.g., one hour, one month. During a period the following sequence of events occurs: A replenishment order can be submitted. Inventory is received. Random demand occurs. Lead times: An order is received after a fixed number of periods, called the lead time. Let l represent the length of the lead time. Fig 13.4: An example with l = 1

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**Order up-to model vs. newsvendor model**

Both models have uncertain future demand, but there are differences… Newsvendor applies to short life cycle products with uncertain demand and the order up-to applies to long life cycle products with uncertain, but stable, demand. Newsvendor Order up-to Inventory obsolescence After one period Never Number of replenishments One (maybe two or three with some reactive capacity) Unlimited Demand occurs during replenishment No Yes

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**InSync demand and inventory at the DC**

Average monthly demand = 349 units Standard deviation of monthly demand = Average weekly demand = 349/4.33 = 80.6 Standard deviation of weekly demand = (The evaluations for weekly demand assume 4.33 weeks per month and demand is independent across weeks.) Fig 13.2 Monthly implants (columns) and end of month inventory (line)

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**InSync demand and inventory in Susan’s territory**

Total annual demand = 75 units Average daily demand = 0.29 units (75/260), assuming 5 days per week. Poisson demand distribution works better for slow moving items Fig 13.3 Monthly implants (columns) and end of month inventory (line)

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**Order up-to model definitions**

On-order inventory / pipeline inventory = the number of units that have been ordered but have not been received. On-hand inventory = the number of units physically in inventory ready to serve demand. Backorder = the total amount of demand that has has not been satisfied: All backordered demand is eventually filled, i.e., there are no lost sales. Inventory level = On-hand inventory - Backorder. Inventory position = On-order inventory + Inventory level. Order up-to level, S the maximum inventory position we allow. sometimes called the base stock level.

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**Order up-to model implementation**

Each period’s order quantity = S – Inventory position Suppose S = 4. If a period begins with an inventory position = 1, then three units are ordered. (4 – 1 = 3 ) If a period begins with an inventory position = -3, then seven units are ordered (4 – (-3) = 7) A period’s order quantity = the previous period’s demand: If demand were 10 in period 1, then the inventory position at the start of period 2 is 4 – 10 = -6, which means 10 units are ordered in period 2. The order up-to model is a pull system because inventory is ordered in response to demand. The order up-to model is sometimes referred to as a 1-for-1 ordering policy.

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**What determines the inventory level?**

Short answer: Inventory level at the end of a period = S minus demand over l +1 periods. Explanation via an example with S = 6, l = 3, and 2 units on-hand at the start of period 1 Keep in mind: At the start of a period the Inventory level + On-order equals S. All inventory on-order at the start of period 1 arrives before the end of period 4 Nothing ordered in periods 2-4 arrives by the end of period 4 All demand is satisfied so there are no lost sales. Period 1 Period 2 Period 3 Period 4 Time D1 D2 D3 D4 ? Inventory level at the end of period four = 6 - D1 – D2 – D3 – D4

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**Expected on-hand inventory and backorder**

… Period 1 S Period 4 S – D > 0, so there is on-hand inventory D = demand over l +1 periods … D Time S – D < 0, so there are backorders This is like a Newsvendor model in which the order quantity is S and the demand distribution is demand over l +1 periods. So … Expected on-hand inventory at the end of a period can be evaluated like Expected left over inventory in the Newsvendor model with Q = S. Expected backorder at the end of a period can be evaluated like Expected lost sales in the Newsvendor model with Q = S.

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**Stockout and in-stock probabilities, on-order inventory and fill rate**

The stockout probability is the probability at least one unit is backordered in a period: The in-stock probability is the probability all demand is filled in a period: Expected on-order inventory = Expected demand over one period x lead time This comes from Little’s Law. Note that it equals the expected demand over l periods, not l +1 periods. The fill rate is the fraction of demand within a period that is NOT backordered:

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**Demand over l+1 periods DC: Susan’s territory:**

The period length is one week, the replenishment lead time is three weeks, l = 3 Assume demand is normally distributed: Mean weekly demand is 80.6 (from demand data) Standard deviation of weekly demand is (from demand data) Expected demand over l +1 weeks is (3 + 1) x 80.6 = 322.4 Standard deviation of demand over l +1 weeks is Susan’s territory: The period length is one day, the replenishment lead time is one day, l =1 Assume demand is Poisson distributed: Mean daily demand is 0.29 (from demand data) Expected demand over l+1 days is 2 x 0.29 = 0.58 Recall, the Poisson is completely defined by its mean (and the standard deviation is always the square root of the mean)

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**DC’s Expected backorder assuming S = 625**

Expected backorder is analogous to the Expected lost sales in the Newsvendor model: Suppose S = 625 at the DC Normalize the order up-to level: Lookup L(z) in the Standard Normal Loss Function Table: Convert expected lost sales, L(z), for the standard normal into the expected backorder with the actual normal distribution that represents demand over l+1 periods: Therefore, if S = 625, then on average there are 0.19 backorders at the end of any period at the DC.

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**Other DC performance measures with S = 625**

Fill rate On-hand inventory Units on order

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**Performance measures in Susan’s territory**

Look up in the Poisson Loss Function Table expected backorders for a Poisson distribution with a mean equal to expected demand over l+1 periods: Suppose S = 3: Expected backorder In-stock Fill rate Expected on-hand Expected on-order inventory

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**Choosing S Target fill rate w/normal demand**

Target in-stock w/normal demand Target fill rate w/Poisson demand Target in-stock w/Poisson demand

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**Justifying a service level via cost minimization**

Let h equal the holding cost per unit per period e.g. if p is the retail price, the gross margin is 75%, the annual holding cost is 35% and there are 260 days per year, then h = p x ( ) x 0.35 / 260 = x p Let b equal the penalty per unit backordered e.g., let the penalty equal the 75% gross margin, then b = 0.75 x p “Too much-too little” challenge: If S is too high, then there are holding costs, Co = h If S is too low, then there are backorders, Cu = b Cost minimizing order up-to level satisfies

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**The optimal in-stock probability is usually quite high**

Suppose the annual holding cost is 35%, the backorder penalty cost equals the gross margin and inventory is reviewed daily. (Table 13.3 Graphed)

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**Impact of the period length**

Increasing the period length leads to larger and less frequent orders: The average order quantity = expected demand in a single period. The frequency of orders approximately equals 1/length of period. Suppose there is a cost to hold inventory and a cost to submit each order (independent of the quantity ordered)… … then there is a tradeoff between carrying little inventory (short period lengths) and reducing ordering costs (long period lengths)

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**Example with mean demand per week = 100 and standard deviation of weekly demand = 75.**

Period lengths of 1, 2, 4 and 8 weeks result in average inventory of 597, 677, 832 and 1130 respectively (Fig 13.7)

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**Tradeoff between inventory holding costs and ordering costs**

Ordering costs = $275 per order Holding costs = 25% per year Unit cost = $50 Holding cost per unit per year = 25% x $50 = 12.5 Period length of 4 weeks minimizes costs: This implies the average order quantity is 4 x 100 = 400 units EOQ model: Total costs Inventory holding costs Ordering costs Fig 13.8

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**Better service requires more inventory at an increasing rate**

More inventory is needed as demand uncertainty increases for any fixed fill rate. The required inventory is more sensitive to the fill rate level as demand uncertainty increases Fig 13.9: The tradeoff between inventory and fill rate with Normally distributed demand and a mean of 100 over (l+1) periods. The curves differ in the standard deviation of demand over (l+1) periods: 60, 50, 40, 30, 20, 10 from top to bottom.

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**Shorten lead times and you will reduce inventory**

Reducing the lead time reduces expected inventory, especially as the target fill rate increases Fig 13.10: The impact of lead time on expected inventory for four fill rate targets, 99.9%, 99.5%, 99.0% and 98%, top curve to bottom curve respectively. Demand in one period is Normally distributed with mean 100 and standard deviation 60.

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**Don’t forget about pipeline inventory**

Reducing the lead time reduces expected inventory and pipeline inventory The impact on pipeline inventory can be even more dramatic that the impact on expected inventory Fig 13.11: Expected inventory (diamonds) and total inventory (squares), which is expected inventory plus pipeline inventory, with a 99.9% fill rate requirement and demand in one period is Normally distributed with mean 100 and standard deviation 60

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**Risk-Pooling Strategies to Reduce and Hedge Uncertainty Chapter 14**

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**Risk pooling strategies**

Redesign of the: supply chain production process product to reduce uncertainty or to hedge uncertainty Four versions of risking pooling: location lead time delayed differentiation consolidated distribution capacity

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**Location pooling at Medtronic**

Current operations: Each sales representative has her own inventory to serve demand in her own territory. Lead time is 1 day from Mounds View DC e.g., 3 territories, 3 stockpiles of inventory The location pooling strategy: A single location stores inventory used by several sales reps. Sales reps no longer hold their own inventory, they must pull inventory from the pooled location. Inventory is automatically replenished at the pooled location as depleted by demand. Lead time to pooled location is still 1 day from Mounds View DC. e.g., 3 pooled territories, 1 stockpile of inventory Territory 1 DC Territory 2 Territory 3 Territory 1 Territory 2 Territory 3 DC

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**The impact of location pooling on inventory**

Suppose each territory’s expected daily demand is 0.29, the required in-stock probability is 99.9% and the lead time is 1 day with individual territories or pooled territories. Pooling 8 territories reduces expected inventory from 11.7 days-of-demand down to 3.6. But pooling has no impact on pipeline inventory.

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**Location pooling & the inventory-service curve**

Location pooling shifts the inventory-service tradeoff curve For a single product, location pooling decreases inventory with service constant increases service with holding cost constant a combination of inventory reduction and service increase. Location pooling can be used to broaden the product line. Fig 14.2: Inventory-service tradeoff curve for different levels of location pooling. The curves represent, from highest to lowest, individual territories, two pooled territories, four pooled territories, and eight pooled territories. Daily demand in each territory is Poisson with mean 0.29 and the lead time is one day.

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**Why does location pooling work?**

Location pooling reduces demand uncertainty (measured by CV) Reduced demand uncertainty reduces the inventory (with constant service level) But there are declining marginal returns to risk pooling! Fig 14.1: The relationship between expected inventory (diamonds) and the coefficient of variation (squares) as territories are pooled. Daily demand in each territory is Poisson with mean 0.29 units, the target in-stock probability is 99% and the lead time is one day.

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**Location pooling pros, cons and alternatives**

Reduces demand uncertainty Cons: Location pooling moves inventory away from customers: Alternatives: Virtual pooling: Drop shipping:

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**Product pooling – universal design**

O’Neill sells two Hammer 3/2 wetsuits that are identical except for the logo silk screened on the chest. Instead of having two Hammer 3/2 suits, O’Neill could consolidate its product line into a single Hammer 3/2 suit, i.e., a universal design, which we will call the “Universal Hammer”. Surf Hammer 3/2 logo Dive Hammer 3/2 logo

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**Product pooling analysis assumptions**

Demand for the Surf Hammer is Normally distributed with mean 3192 and standard deviation 1181. Demand for the Dive Hammer has the same distribution as the Surf Hammer. Surf and Dive demands are independent then the Universal Hammer’s demand has mean 2 x 3192 = 6384 and std deviation = sqrt(2) x 1181 = 1670. Price, cost and salvage value for the Universal Hammer are the same as for the other two: Hence, Co is 110 – 90 = 20, Cu = = 70 Same critical ratio = 70 /( ) = Same optimal z statistic, 0.77

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**Product pooling analysis results**

Performance of the two suits (Surf and Dive) Total order quantity Total profit Universal Hammer Order quantity: Profit: Reduces inventory investment by Increase profit by The profit increase of

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Demand correlation Fig Random demand for two products (x-axis is product 1, y-axis is product 2). Correlations are -0.9, 0.0, and +0.9 In all scenarios demand is ~N(10,3).

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**Key driver of product pooling**

Most effective if the CVUniversal < CVindividual CVindividual = 1181/2192 = 0.37 CVUniversal = 1670/6384 = 0.26 Negative correlation in demand for the individual products is best for reducing CV The correlation between surf and dive demand for the Hammer 3/2 and the expected profit of the universal Hammer wetsuit (decreasing curve) and the coefficient of variation of total demand (increasing curve)

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**Limitations of product pooling/universal design**

May not provide key functionality to consumers with special needs May be more expensive to produce because additional functionality may require additional components. May be less expensive to produce/procure because each component is needed in a larger volume. May eliminate brand/price segmentation opportunities

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**Lead time pooling – consolidated distribution**

Consider the following two systems from Fig 14.9: In each case weekly demand at each store is ~P(0.5) and the target in-stock probability at each store is 99.5% DC demand is ~N(50,15) If demands were independent across stores, then DC demand would have a standard deviation of sqrt(50) = 7.07

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**Consolidated distribution results**

Table 14.4 Consolidated distribution … reduces retail inventory by more than 50%! not as effectively as location pooling… … but consolidated distribution keeps inventory near demand, reduces inventory even though the total lead time increases from 8 to 9 weeks!

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**Consolidated distribution summary**

Reduces inventory in a supply chain via lead time risk pooling Decide the total quantity to ship from the supplier, not a total quantity and its allocation across locations. Most effective if demands are negatively correlated across locations. Most effective if the supplier lead time is long and the DC to store lead time is short. Increases distance traveled & lead time from supplier to stores. Other benefits of consolidated distribution: Easier to obtain quantity discounts in purchasing. Easier to obtain economies of scale in transportation:

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**Lead time risk pooling – delayed differentiation**

A Universal Hammer 3/2 increases O’Neill’s profit, but does not allow O’Neill to differentiate between the Surf and the Dive markets. Delayed differentiation is an alternative to the Universal Hammer: O’Neill stocks “generic” Hammers with no logo. When demand occurs O’Neill quickly silk screens on the appropriate logo, This generates the same profit as the Universal Hammer! Wheee! When does delayed differentiation make sense? Customers demand variety. There is less uncertainty with total demand than demand for individual versions. Variety is created late in the production process. Variety can be added quickly and cheaply. Component needed for variety is inexpensive relative to the generic component.

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**Other examples of delayed differentiation**

Retail paint HP printer Private label canned goods manufacturer Black and Decker Nokia

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**Capacity pooling with flexible manufacturing**

Consider the following stylized situation faced by GM … 10 production facilities 10 vehicle models Each plant is capable of producing 100 units/day Demand for each product is ~N(100,40) Each plant can be configured to produce up to 10 products But flexibility is expensive. GM must decide which plants can produce which products before demand is realized. After demand is realized, GM can allocate its capacity to satisfy demand. If demand exceeds capacity, sales are lost.

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**Four possible capacity configurations: no flexibility to total flexibility**

Figs & The more links, the more flexibility constructed In the 16 link configuration plant 4 is flexible enough to produce 4 products but plant 5 has no flexibility (it produces a single product).

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**How is flexibility used**

Flexibility allows production shifts to avoid lost sales. Two plant, two product example If demand turns out to be 75 for product A, 115 for product B then.. Plant 2 1 Vehicle B A No flexibility Total flexibility Product Demand Plant 1 Plant 2 Sales A 75 B 115 100 Total Sales 175 Plant Utilization 88% Production With no flexibility Product Demand Plant 1 Plant 2 Sales A 75 B 115 15 100 Total Sales 190 Plant Utilization 95% Production With total flexibility

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**The value of flexibility**

Adding flexibility increases capacity utilization and expected sales: Fig 14.14: Note that 20 links can provide nearly the same performance as total flexibility! These data are collected via simulation

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**Chaining: how to add flexibility**

A chain is a group of plants and products connected via links. Flexibility is most effective if it is added to create long chains. A configuration with 20 links can produce nearly the results of total flexibility as long as it constructs one large chain: Hence, a little bit of flexibility is very useful as long as it is designed correctly

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**When is flexibility valuable?**

Flexibility is most valuable when capacity approximately equals expected demand. Flexibility is least valuable when capacity is very high or very low. A 20 link (1 chain) configuration with 1000 units of capacity produces the same expected sales as 1250 units of capacity with no flexibility. If flexibility is cheap relative to capacity, add flexibility. But if flexibility is expensive relative to capacity, add capacity. Fig C = total capacity of all ten plants

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**One way to make money with capacity pooling: contract manufacturing**

A fast growing industry: But one with low margins: Total revenue of six leading contract manufacturers by fiscal year: Solectron Corp, Flextronics International Ltd, Sanmina-SCI, Jabil Circuit Inc, Celestica Inc and Plexus Corp.

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CHAPTER 7 Managing Inventories

CHAPTER 7 Managing Inventories

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