Inventory 1.1

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Inventory ManagementInventory ManagementInventory ManagementInventory Management

Using Analytics for Better DecisionsUsing Analytics for Better Decisions

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Why We Want to Hold InventoriesWhy We Want to Hold InventoriesWhy We Want to Hold InventoriesWhy We Want to Hold Inventories

Improve customer serviceImprove customer service Reduce certain costs such asReduce certain costs such as

ordering costsordering costs stockout costsstockout costs acquisition costsacquisition costs start-up quality costsstart-up quality costs

Contribute to the efficient and effective operation of Contribute to the efficient and effective operation of the production systemthe production system

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Why We Want to Hold InventoriesWhy We Want to Hold InventoriesWhy We Want to Hold InventoriesWhy We Want to Hold Inventories

Finished GoodsFinished Goods Essential in produce-to-stock positioning strategiesEssential in produce-to-stock positioning strategies Necessary in level aggregate capacity plansNecessary in level aggregate capacity plans Products can be displayed to customersProducts can be displayed to customers

Work-in-ProcessWork-in-Process Necessary in process-focused productionNecessary in process-focused production May reduce material-handling & production costsMay reduce material-handling & production costs

Raw MaterialRaw Material Suppliers may produce/ship materials in batchesSuppliers may produce/ship materials in batches Quantity discounts and freight/handling lead to Quantity discounts and freight/handling lead to

savingssavings

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Why We Do Not Want to Hold InventoriesWhy We Do Not Want to Hold InventoriesWhy We Do Not Want to Hold InventoriesWhy We Do Not Want to Hold Inventories

Certain costs increase such asCertain costs increase such as carrying costscarrying costs cost of diluted return on investmentcost of diluted return on investment reduced-capacity costsreduced-capacity costs large-lot quality costlarge-lot quality cost cost of production problemscost of production problems

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Why We Do Not Want to Hold InventoriesWhy We Do Not Want to Hold InventoriesWhy We Do Not Want to Hold InventoriesWhy We Do Not Want to Hold Inventories

Difficult to ControlDifficult to Control Hides Production problemsHides Production problems

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Trade-offsTrade-offsTrade-offsTrade-offs

Lot-size – Inventory (Bullwhip effect)Lot-size – Inventory (Bullwhip effect) Inventory – Transportation (Ordering/Setup) costInventory – Transportation (Ordering/Setup) cost Lead Time – Transportation costLead Time – Transportation cost Product variety – InventoryProduct variety – Inventory Cost – Customer serviceCost – Customer service

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How do we know we have a good inventory How do we know we have a good inventory management system?management system?

How do we know we have a good inventory How do we know we have a good inventory management system?management system?

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Effective Inventory ManagementEffective Inventory ManagementEffective Inventory ManagementEffective Inventory Management

A system to keep track of inventoryA system to keep track of inventory A reliable forecast of demandA reliable forecast of demand Knowledge of lead timeKnowledge of lead time Reasonable estimate ofReasonable estimate of

- Holding cost- Holding cost- Ordering cost- Ordering cost- Shortage cost- Shortage cost

A classification systemA classification system

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Nature of InventoryNature of InventoryNature of InventoryNature of Inventory

Two Fundamental Inventory DecisionsTwo Fundamental Inventory Decisions Terminology of InventoriesTerminology of Inventories Independent Demand Inventory SystemsIndependent Demand Inventory Systems Dependent Demand Inventory SystemsDependent Demand Inventory Systems Inventory CostsInventory Costs

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Two Fundamental Inventory DecisionsTwo Fundamental Inventory DecisionsTwo Fundamental Inventory DecisionsTwo Fundamental Inventory Decisions

How muchHow much to order of each material when orders are to order of each material when orders are placed with either outside suppliers or production placed with either outside suppliers or production departments within organizationsdepartments within organizations

WhenWhen to place the orders to place the orders

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Independent Demand Inventory SystemsIndependent Demand Inventory SystemsIndependent Demand Inventory SystemsIndependent Demand Inventory Systems

Demand for an item carried in inventory is Demand for an item carried in inventory is independent of the demand for any other item in independent of the demand for any other item in inventoryinventory

Finished goods inventory is an exampleFinished goods inventory is an example Demands are estimated from forecasts and/or Demands are estimated from forecasts and/or

customer orderscustomer orders

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Dependent Demand Inventory SystemsDependent Demand Inventory SystemsDependent Demand Inventory SystemsDependent Demand Inventory Systems

Items whose demand depends on the demands for Items whose demand depends on the demands for other itemsother items

For example, the demand for raw materials and For example, the demand for raw materials and components can be calculated from the demand for components can be calculated from the demand for finished goodsfinished goods

The systems used to manage these inventories are The systems used to manage these inventories are different from those used to manage independent different from those used to manage independent demand itemsdemand items

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Inventory CostsInventory CostsInventory CostsInventory Costs

Costs associated with ordering too much (represented Costs associated with ordering too much (represented by carrying costs)by carrying costs)

Costs associated with ordering too little (represented Costs associated with ordering too little (represented by ordering costs)by ordering costs)

These costs are opposing costs, i.e., as one increases These costs are opposing costs, i.e., as one increases the other decreasesthe other decreases

. . . more. . . more

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Carrying costCarrying costCarrying costCarrying cost

ObsolescenceObsolescence InsuranceInsurance Extra staffingExtra staffing InterestInterest PilferagePilferage DamageDamage Warehousing Warehousing Etc.Etc.

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Carrying CostCarrying Cost(Approximate ranges)(Approximate ranges)

Carrying CostCarrying Cost(Approximate ranges)(Approximate ranges)

CategoryCategory Cost as % ofCost as % of

Inventory Inventory ValueValue

Investments costs 11%

(6 – 24%)

Labour cost from extra handling 3%

(3 – 5%)

Housing cost 6%

(3 – 10%)

Other costs also range between 1 – 5%.

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Ordering costOrdering costOrdering costOrdering cost

Processing SuppliesProcessing Supplies FormsForms Order processingOrder processing Clerical supportClerical support Etc.Etc.

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Set-up costsSet-up costsSet-up costsSet-up costs

Clean-up costClean-up cost Re-tooling costRe-tooling cost Adjustment costAdjustment cost Etc.Etc.

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Inventory Costs (continued)Inventory Costs (continued)Inventory Costs (continued)Inventory Costs (continued)

The sum of the two costs is the total stocking cost The sum of the two costs is the total stocking cost (TSC)(TSC)

When plotted against order quantity, the TSC When plotted against order quantity, the TSC decreases to a minimum cost and then increasesdecreases to a minimum cost and then increases

This cost behavior is the basis for answering the first This cost behavior is the basis for answering the first fundamental question: how much to orderfundamental question: how much to order

It is known as the economic order quantity (EOQ)It is known as the economic order quantity (EOQ)

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Balancing Carrying against Ordering CostsBalancing Carrying against Ordering CostsBalancing Carrying against Ordering CostsBalancing Carrying against Ordering Costs

Annual Cost ($)Annual Cost ($)

Order QuantityOrder Quantity

MinimumMinimumTotal AnnualTotal Annual

Stocking CostsStocking Costs

AnnualAnnualCarrying CostsCarrying Costs

AnnualAnnualOrdering CostsOrdering Costs

Total AnnualTotal AnnualStocking CostsStocking Costs

SmallerSmaller LargerLarger

Low

erL

ower

Hig

her

Hig

her

EOQEOQ

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Fixed Order Quantity SystemsFixed Order Quantity SystemsFixed Order Quantity SystemsFixed Order Quantity Systems

Behavior of Economic Order Quantity (EOQ) Behavior of Economic Order Quantity (EOQ) SystemsSystems

Determining Order QuantitiesDetermining Order Quantities Determining Order PointsDetermining Order Points

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Behavior of EOQ SystemsBehavior of EOQ SystemsBehavior of EOQ SystemsBehavior of EOQ Systems

As demand for the inventoried item occurs, the As demand for the inventoried item occurs, the inventory level dropsinventory level drops

When the inventory level drops to a critical point, the When the inventory level drops to a critical point, the order point, the ordering process is triggered order point, the ordering process is triggered

The amount ordered each time an order is placed is The amount ordered each time an order is placed is fixed or constantfixed or constant

When the ordered quantity is received, the inventory When the ordered quantity is received, the inventory level increaseslevel increases


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Behavior of EOQ SystemsBehavior of EOQ SystemsBehavior of EOQ SystemsBehavior of EOQ Systems

An application of this type system is the two-bin An application of this type system is the two-bin systemsystem

Two bin systemTwo bin system: Two containers of inventory; order : Two containers of inventory; order when one is emptywhen one is empty

A perpetual inventory accounting system is usually A perpetual inventory accounting system is usually associated with this type of systemassociated with this type of system

Perpetual inventory systemPerpetual inventory system: System that keeps track : System that keeps track of removals from inventory continuously; thus of removals from inventory continuously; thus monitoring current levels of each item. monitoring current levels of each item.

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Determining Order QuantitiesDetermining Order QuantitiesDetermining Order QuantitiesDetermining Order Quantities

Basic EOQBasic EOQ EOQ for Production LotsEOQ for Production Lots EOQ with Quantity DiscountsEOQ with Quantity Discounts

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Model I: Basic EOQ Model I: Basic EOQ Model I: Basic EOQ Model I: Basic EOQ

Typical assumptions madeTypical assumptions made annual demand (D), carrying cost (C) and ordering annual demand (D), carrying cost (C) and ordering

cost (S) can be estimatedcost (S) can be estimated average inventory level is the fixed order quantity average inventory level is the fixed order quantity

(Q) divided by 2 which implies(Q) divided by 2 which implies no safety stockno safety stock orders are received all at onceorders are received all at once demand occurs at a uniform ratedemand occurs at a uniform rate no inventory when an order arrivesno inventory when an order arrives


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Model I: Basic EOQModel I: Basic EOQModel I: Basic EOQModel I: Basic EOQ

Assumptions (continued)Assumptions (continued) Stockout, customer responsiveness, and other costs Stockout, customer responsiveness, and other costs

are inconsequentialare inconsequential acquisition cost is fixed, i.e., no quantity discountsacquisition cost is fixed, i.e., no quantity discounts

Annual carrying cost = (average inventory level) x Annual carrying cost = (average inventory level) x (carrying cost) = (Q/2)C(carrying cost) = (Q/2)C

Annual ordering cost = (average number of orders per Annual ordering cost = (average number of orders per year) x (ordering cost) = (D/Q)Syear) x (ordering cost) = (D/Q)S


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CDS /2 = EOQ CDS /2 = EOQ

Model I: Basic EOQModel I: Basic EOQModel I: Basic EOQModel I: Basic EOQ

Total annual stocking cost (TSC) = annual carrying Total annual stocking cost (TSC) = annual carrying cost + annual ordering cost = (Q/2)C + (D/Q)Scost + annual ordering cost = (Q/2)C + (D/Q)S

The order quantity where the TSC is at a minimum The order quantity where the TSC is at a minimum (EOQ) can be found using calculus (take the first (EOQ) can be found using calculus (take the first derivative, set it equal to zero and solve for Q) derivative, set it equal to zero and solve for Q)

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Example: Basic EOQExample: Basic EOQExample: Basic EOQExample: Basic EOQ

Zartex Co. stocks fertilizer to sell to retailers. One item – calcium nitrate – Zartex Co. stocks fertilizer to sell to retailers. One item – calcium nitrate – is purchased from a nearby manufacturer at Rs. 22.50 per ton. Zartex is purchased from a nearby manufacturer at Rs. 22.50 per ton. Zartex estimates it will need 5,750,000 tons of calcium nitrate next year.estimates it will need 5,750,000 tons of calcium nitrate next year.

The annual carrying cost for this material is 40% of the The annual carrying cost for this material is 40% of the acquisition cost, and the ordering cost is Rs. 595. acquisition cost, and the ordering cost is Rs. 595. a) What is the most economical order quantity?a) What is the most economical order quantity?b) How many orders will be placed per year?b) How many orders will be placed per year?c) How much time will elapse between orders?c) How much time will elapse between orders?

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Economical Order Quantity (EOQ)Economical Order Quantity (EOQ)

D = 5,750,000 tons/yearD = 5,750,000 tons/year

C = .40(22.50) = Rs. 9.00/ton/yearC = .40(22.50) = Rs. 9.00/ton/year

S = Rs. 595/orderS = Rs. 595/order

= 27,573.135 tons per order= 27,573.135 tons per order

EOQ = 2DS/CEOQ = 2DS/CEOQ = 2(5,750,000)(595)/9.00EOQ = 2(5,750,000)(595)/9.00

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Total Annual Stocking Cost (TSC)Total Annual Stocking Cost (TSC)

TSC = (Q/2)C + (D/Q)STSC = (Q/2)C + (D/Q)S

= (27,573.135/2)(9.00) = (27,573.135/2)(9.00)

+ (5,750,000/27,573.135)(595)+ (5,750,000/27,573.135)(595)

= 124,079.11 + 124,079.11= 124,079.11 + 124,079.11

= Rs.248,158.22= Rs.248,158.22

Note: Total Carrying CostNote: Total Carrying Costequals Total Ordering Costequals Total Ordering Cost

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Number of Orders Per YearNumber of Orders Per Year= D/Q = D/Q = 5,750,000/27,573.135 = 5,750,000/27,573.135 = 208.5 orders/year= 208.5 orders/year

Time Between OrdersTime Between Orders= Q/D= Q/D= 1/208.5= 1/208.5= .004796 years/order= .004796 years/order= .004796(365 days/year) = 1.75 days/order= .004796(365 days/year) = 1.75 days/order

Note: This is the inverseNote: This is the inverse of the formula above.of the formula above.

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Model II: EOQ for Production LotsModel II: EOQ for Production LotsModel II: EOQ for Production LotsModel II: EOQ for Production Lots

Used to determine the order size, production lot, if an Used to determine the order size, production lot, if an item is produced at one stage of production, stored in item is produced at one stage of production, stored in inventory, and then sent to the next stage or the inventory, and then sent to the next stage or the customercustomer

Differs from Model I because orders are assumed to Differs from Model I because orders are assumed to be supplied or produced at a uniform rate (p) rate be supplied or produced at a uniform rate (p) rate rather than the order being received all at oncerather than the order being received all at once


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Model II: EOQ for Production LotsModel II: EOQ for Production LotsModel II: EOQ for Production LotsModel II: EOQ for Production Lots

It is also assumed that the supply rate, p, is greater It is also assumed that the supply rate, p, is greater than the demand rate, dthan the demand rate, d

The change in maximum inventory level requires The change in maximum inventory level requires modification of the TSC equationmodification of the TSC equation

TSC = (Q/2)[(p-d)/p]C + (D/Q)STSC = (Q/2)[(p-d)/p]C + (D/Q)S The optimization results inThe optimization results in

dp

p

C

DS2 = EOQ

dp

p

C

DS2 = EOQ

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Example: EOQ for Production LotsExample: EOQ for Production LotsExample: EOQ for Production LotsExample: EOQ for Production Lots

Highland Electric Co. buys coal from Cedar Highland Electric Co. buys coal from Cedar Creek Coal Co. to generate electricity. CCCC can Creek Coal Co. to generate electricity. CCCC can supply coal at the rate of 3,500 tons per day for supply coal at the rate of 3,500 tons per day for $10.50 per ton. HEC uses the coal at a rate of 800 $10.50 per ton. HEC uses the coal at a rate of 800 tons per day and operates 365 days per year.tons per day and operates 365 days per year.

HEC’s annual carrying cost for coal is 20% of HEC’s annual carrying cost for coal is 20% of the acquisition cost, and the ordering cost is $5,000.the acquisition cost, and the ordering cost is $5,000.

a) What is the economical production lot size?a) What is the economical production lot size?

b) What is HEC’s maximum inventory level for coal?b) What is HEC’s maximum inventory level for coal?

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Economical Production Lot SizeEconomical Production Lot Size

d = 800 tons/day; D = 365(800) = 292,000 tons/yeard = 800 tons/day; D = 365(800) = 292,000 tons/year

p = 3,500 tons/dayp = 3,500 tons/day

S = $5,000/order C = .20(10.50) = $2.10/ton/yearS = $5,000/order C = .20(10.50) = $2.10/ton/year

= 42,455.5 tons per order= 42,455.5 tons per order

EOQ = (2DS/C)[p/(p-d)]EOQ = (2DS/C)[p/(p-d)]

EOQ = 2(292,000)(5,000)/2.10[3,500/(3,500-800)]EOQ = 2(292,000)(5,000)/2.10[3,500/(3,500-800)]

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Total Annual Stocking Cost (TSC)Total Annual Stocking Cost (TSC)

TSC = (Q/2)((p-d)/p)C + (D/Q)STSC = (Q/2)((p-d)/p)C + (D/Q)S

= (42,455.5/2)((3,500-800)/3,500)(2.10) = (42,455.5/2)((3,500-800)/3,500)(2.10)

+ (292,000/42,455.5)(5,000)+ (292,000/42,455.5)(5,000)

= 34,388.95 + 34,388.95= 34,388.95 + 34,388.95

= $68,777.90= $68,777.90Note: Total Carrying CostNote: Total Carrying Costequals Total Ordering Costequals Total Ordering Cost

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Maximum Inventory LevelMaximum Inventory Level

= Q(p-d)/p = Q(p-d)/p

= 42,455.5(3,500 – 800)/3,500= 42,455.5(3,500 – 800)/3,500

= 42,455.5(.771429)= 42,455.5(.771429)

= 32,751.4 tons= 32,751.4 tons Note: HEC will use 23%Note: HEC will use 23%of the production lot by theof the production lot by thetime it receives the full lot.time it receives the full lot.

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Key Points from EOQ ModelKey Points from EOQ Model

In deciding the optimal lot size, the tradeoff is between setup In deciding the optimal lot size, the tradeoff is between setup (order) cost and holding cost.(order) cost and holding cost.

If demand increases by a factor of 4, it is optimal to increase If demand increases by a factor of 4, it is optimal to increase batch size by a factor of 2 and produce (order) twice as often. batch size by a factor of 2 and produce (order) twice as often. Cycle inventory (in days of demand) should decrease as Cycle inventory (in days of demand) should decrease as demand increasesdemand increases..

If lot size is to be reduced, one has to reduce fixed order cost. If lot size is to be reduced, one has to reduce fixed order cost. To reduce lot size by a factor of 2, order cost has to be To reduce lot size by a factor of 2, order cost has to be reduced by a factor of 4.reduced by a factor of 4.

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Model III: EOQ with Quantity DiscountsModel III: EOQ with Quantity DiscountsModel III: EOQ with Quantity DiscountsModel III: EOQ with Quantity Discounts

Under quantity discounts, a supplier offers a lower Under quantity discounts, a supplier offers a lower unit price if larger quantities are ordered at one timeunit price if larger quantities are ordered at one time

This is presented as a price or discount schedule, i.e., This is presented as a price or discount schedule, i.e., a certain unit price over a certain order quantity rangea certain unit price over a certain order quantity range

This means this model differs from Model I because This means this model differs from Model I because the acquisition cost (ac) may vary with the quantity the acquisition cost (ac) may vary with the quantity ordered, i.e., it is not necessarily constantordered, i.e., it is not necessarily constant


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All-Unit Quantity DiscountsAll-Unit Quantity DiscountsAll-Unit Quantity DiscountsAll-Unit Quantity Discounts

Pricing schedule has specified quantity break points qPricing schedule has specified quantity break points q00, ,

qq11, …, q, …, qrr, where q, where q00 = 0 = 0

If an order is placed that is at least as large as qIf an order is placed that is at least as large as q ii but but

smaller than qsmaller than qi+1i+1, then each unit has an average unit cost , then each unit has an average unit cost

of Cof Cii

The unit cost generally decreases as the quantity The unit cost generally decreases as the quantity increases, i.e., Cincreases, i.e., C00>C>C11>…>C>…>Crr

The objective for the company (a retailer in our example) The objective for the company (a retailer in our example) is to decide on a lot size that will minimize the sum of is to decide on a lot size that will minimize the sum of material, order, and holding costsmaterial, order, and holding costs

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Under this condition, acquisition cost becomes an Under this condition, acquisition cost becomes an incremental cost and must be considered in the incremental cost and must be considered in the determination of the EOQdetermination of the EOQ

The total annual material costs (TMC) = Total annual The total annual material costs (TMC) = Total annual stocking costs (TSC) + annual acquisition coststocking costs (TSC) + annual acquisition cost

TSC = (Q/2)C + (D/Q)S + (D)acTSC = (Q/2)C + (D/Q)S + (D)ac


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All-Unit Quantity Discounts: ExampleAll-Unit Quantity Discounts: Example

Cost/Unit

Rs. 3Rs. 2.96

Rs.2.92

Order Quantity

5,000 10,000

Order Quantity

5,000 10,000

Total Material Cost

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To find the EOQ, the following procedure is used:To find the EOQ, the following procedure is used:

1.1. Compute the EOQ using the lowest acquisition cost. Compute the EOQ using the lowest acquisition cost. If the resulting EOQ is feasible (the quantity can If the resulting EOQ is feasible (the quantity can

be purchased at the acquisition cost used), this be purchased at the acquisition cost used), this quantity is optimal and you are finished.quantity is optimal and you are finished.

If the resulting EOQ is If the resulting EOQ is notnot feasible, go to Step 2 feasible, go to Step 22.2. Identify the Identify the nextnext higher acquisition cost. higher acquisition cost.

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3.3. Compute the EOQ using the acquisition cost from Compute the EOQ using the acquisition cost from Step 2.Step 2. If the resulting EOQ is feasible, go to Step 4.If the resulting EOQ is feasible, go to Step 4. Otherwise, go to Step 2. Otherwise, go to Step 2.

4.4. Compute the TMC for the feasible EOQ (just found Compute the TMC for the feasible EOQ (just found in Step 3) and its corresponding acquisition cost.in Step 3) and its corresponding acquisition cost.

5.5. Compute the TMC for each of the lower acquisition Compute the TMC for each of the lower acquisition costs using the minimum allowed order quantity for costs using the minimum allowed order quantity for each cost. each cost.

6.6. The quantity with the lowest TMC is optimal. The quantity with the lowest TMC is optimal.

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All-Unit Quantity Discount: ExampleAll-Unit Quantity Discount: ExampleAll-Unit Quantity Discount: ExampleAll-Unit Quantity Discount: Example

Order quantityOrder quantity Unit PriceUnit Price0-50000-5000 Rs. 3.00Rs. 3.005001-100005001-10000 Rs. 2.96Rs. 2.96Over 10000Over 10000 Rs. 2.92Rs. 2.92

q0 = 0, q1 = 5000, q2 = 10000q0 = 0, q1 = 5000, q2 = 10000C0 = Rs. 3.00, C1 = Rs. 2.96, C2 = Rs. 2.92C0 = Rs. 3.00, C1 = Rs. 2.96, C2 = Rs. 2.92D = 120000 units/year, S = Rs. 100/lot, D = 120000 units/year, S = Rs. 100/lot, h = 0.2h = 0.2

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All-Unit Quantity Discount: ExampleAll-Unit Quantity Discount: ExampleAll-Unit Quantity Discount: ExampleAll-Unit Quantity Discount: Example

Step 1: Calculate Q2* = Sqrt[(2DS)/hC2] Step 1: Calculate Q2* = Sqrt[(2DS)/hC2] = Sqrt[(2)(120000)(100)/(0.2)(2.92)] = 6410= Sqrt[(2)(120000)(100)/(0.2)(2.92)] = 6410Not feasible (6410 < 10001)Not feasible (6410 < 10001)Calculate TC2 using C2 = Rs. 2.92 and q2 = 10001Calculate TC2 using C2 = Rs. 2.92 and q2 = 10001TC2 = (120000/10001)(100)+(10001/2)(0.2)(2.92)+(120000)(2.92)TC2 = (120000/10001)(100)+(10001/2)(0.2)(2.92)+(120000)(2.92)= Rs. 354,520= Rs. 354,520Step 2: Calculate Q1* = Sqrt[(2DS)/hC1]Step 2: Calculate Q1* = Sqrt[(2DS)/hC1]=Sqrt[(2)(120000)(100)/(0.2)(2.96)] = 6367=Sqrt[(2)(120000)(100)/(0.2)(2.96)] = 6367Feasible (5000<6367Feasible (5000<6367<<10000) 10000) StopStopTC1 = (120000/6367)(100)+(6367/2)(0.2)(2.96)+(120000)(2.96)TC1 = (120000/6367)(100)+(6367/2)(0.2)(2.96)+(120000)(2.96)= Rs. 358,969= Rs. 358,969TC2 < TC1 TC2 < TC1 The optimal order quantity Q* is q2 = 10001The optimal order quantity Q* is q2 = 10001

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Example: EOQ with Quantity DiscountsExample: EOQ with Quantity DiscountsExample: EOQ with Quantity DiscountsExample: EOQ with Quantity Discounts

A-1 Auto Parts has a regional tire warehouse in A-1 Auto Parts has a regional tire warehouse in Atlanta. One popular tire, the XRX75, has estimated Atlanta. One popular tire, the XRX75, has estimated demand of 25,000 next year. It costs A-1 $100 to demand of 25,000 next year. It costs A-1 $100 to place an order for the tires, and the annual carrying place an order for the tires, and the annual carrying cost is 30% of the acquisition cost. The supplier cost is 30% of the acquisition cost. The supplier quotes these prices for the tire:quotes these prices for the tire:

QQ acac

1 – 4991 – 499 $21.60$21.60500 – 999500 – 999 20.9520.951,000 +1,000 + 20.9020.90

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Economical Order QuantityEconomical Order Quantity

This quantity is not feasible, so try ac = $20.95This quantity is not feasible, so try ac = $20.95

This quantity is feasible, so there is no reason to try This quantity is feasible, so there is no reason to try ac = $21.60ac = $21.60

i iEOQ = 2DS/Ci iEOQ = 2DS/C

3EOQ = 2(25,000)100/(.3(20.90) = 893.003EOQ = 2(25,000)100/(.3(20.90) = 893.00

2EOQ = 2(25,000)100/(.3(20.95) = 891.932EOQ = 2(25,000)100/(.3(20.95) = 891.93

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Compare Total Annual Material Costs (TMCs)Compare Total Annual Material Costs (TMCs)

TMC = (Q/2)C + (D/Q)S + (D)acTMC = (Q/2)C + (D/Q)S + (D)ac

Compute TMC for Q = 891.93 and ac = $20.95Compute TMC for Q = 891.93 and ac = $20.95

TMCTMC22 = (891.93/2)(.3)(20.95) + (25,000/891.93)100 = (891.93/2)(.3)(20.95) + (25,000/891.93)100

+ (25,000)20.95+ (25,000)20.95

= 2,802.89 + 2,802.91 + 523,750= 2,802.89 + 2,802.91 + 523,750

= $529,355.80= $529,355.80

… … moremore

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Compute TMC for Q = 1,000 and ac = $20.90Compute TMC for Q = 1,000 and ac = $20.90

TMCTMC33 = (1,000/2)(.3)(20.90) + (25,000/1,000)100 = (1,000/2)(.3)(20.90) + (25,000/1,000)100

+ (25,000)20.90+ (25,000)20.90

= 3,135.00 + 2,500.00 + 522,500= 3,135.00 + 2,500.00 + 522,500

= $528,135.00 (lower than TMC= $528,135.00 (lower than TMC22))

The EOQ is 1,000 tiresThe EOQ is 1,000 tires

at an acquisition cost of $20.90.at an acquisition cost of $20.90.

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Dynamics of Inventory PlanningDynamics of Inventory PlanningDynamics of Inventory PlanningDynamics of Inventory Planning

Continually review ordering practices and decisionsContinually review ordering practices and decisions Modify to fit the firm’s demand and supply patternsModify to fit the firm’s demand and supply patterns Constraints, such as storage capacity and available Constraints, such as storage capacity and available

funds, can impact inventory planningfunds, can impact inventory planning Computers and information technology are used Computers and information technology are used

extensively in inventory planningextensively in inventory planning

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Inventory cycle is the central focus of independent Inventory cycle is the central focus of independent demand inventory systemsdemand inventory systems

Production planning and control systems are Production planning and control systems are changing to support lean inventory strategieschanging to support lean inventory strategies

Information systems electronically link supply chainInformation systems electronically link supply chain

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Planning Supply Chain ActivitiesPlanning Supply Chain Activities

Anticipatory - allocate supply to each Anticipatory - allocate supply to each warehouse based on the forecastwarehouse based on the forecast

Response-based - replenish inventory with Response-based - replenish inventory with order sizes based on specific needs of each order sizes based on specific needs of each warehousewarehouse

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Integrated planning at ShellIntegrated planning at ShellIntegrated planning at ShellIntegrated planning at Shell

““The most successful e-supply initiatives so far have been in The most successful e-supply initiatives so far have been in industries where the components converge to create a product industries where the components converge to create a product and where prices are not volatile. Energy is different. The supply and where prices are not volatile. Energy is different. The supply chain is divergent; there are more products than raw materials chain is divergent; there are more products than raw materials and prices are highly volatile. Shell understands this and is and prices are highly volatile. Shell understands this and is aiming to create a reliable, real time, multi-point-optimized, an aiming to create a reliable, real time, multi-point-optimized, an overview of entire supply chain…..overview of entire supply chain…..Across the globe this initiative has the potential to generate new Across the globe this initiative has the potential to generate new value and drive savings to the tune of multiple of million US value and drive savings to the tune of multiple of million US dollars a day.”dollars a day.”

- A vice president of Shell Oil Products - A vice president of Shell Oil Products

54

Integrated planning at ShellIntegrated planning at ShellIntegrated planning at ShellIntegrated planning at Shell

Requirements for IT toolsets:Requirements for IT toolsets:Complete horizontal supply chain integrationComplete horizontal supply chain integrationConvergence of strategy, planning and schedulingConvergence of strategy, planning and schedulingModularity to enable phased implementation and customizationModularity to enable phased implementation and customizationScalabilityScalabilityInteractiveInteractiveConvenient User-interfacingConvenient User-interfacingReal time resultsReal time resultsDirect links to online refinery / plant optimizationDirect links to online refinery / plant optimization

55

56

Determining Order PointsDetermining Order PointsDetermining Order PointsDetermining Order Points

Basis for Setting the Order PointBasis for Setting the Order Point DDLT DistributionsDDLT Distributions Setting Order PointsSetting Order Points

57

Basis for Setting the Order PointBasis for Setting the Order PointBasis for Setting the Order PointBasis for Setting the Order Point

In the fixed order quantity system, the ordering In the fixed order quantity system, the ordering process is triggered when the inventory level drops to process is triggered when the inventory level drops to a critical point, the order pointa critical point, the order point

This starts the lead time for the item.This starts the lead time for the item. Lead time is the time to complete all activities Lead time is the time to complete all activities

associated with placing, filling and receiving the associated with placing, filling and receiving the order. order.


58


During the lead time, customers continue to draw During the lead time, customers continue to draw down the inventorydown the inventory

It is during this period that the inventory is vulnerable It is during this period that the inventory is vulnerable to stockout (run out of inventory)to stockout (run out of inventory)

Customer service level is the probability that a Customer service level is the probability that a stockout will not occur during the lead timestockout will not occur during the lead time


59


The order point is set based onThe order point is set based on the demand during lead time (DDLT) andthe demand during lead time (DDLT) and the desired customer service levelthe desired customer service level

Order point (OP) = Expected demand during lead Order point (OP) = Expected demand during lead time (EDDLT) + Safety stock (SS)time (EDDLT) + Safety stock (SS)

The amount of safety stock needed is based on the The amount of safety stock needed is based on the degree of uncertainty in the DDLT and the customer degree of uncertainty in the DDLT and the customer service level desiredservice level desired

60

DDLT DistributionsDDLT DistributionsDDLT DistributionsDDLT Distributions

If there is variability in the DDLT, the DDLT is If there is variability in the DDLT, the DDLT is expressed as a distributionexpressed as a distribution discretediscrete continuouscontinuous

In a discrete DDLT distribution, values (demands) In a discrete DDLT distribution, values (demands) can only be integerscan only be integers

A continuous DDLT distribution is appropriate when A continuous DDLT distribution is appropriate when the demand is very highthe demand is very high

61

Setting Order PointSetting Order Pointfor a Discrete DDLT Distributionfor a Discrete DDLT Distribution

Setting Order PointSetting Order Pointfor a Discrete DDLT Distributionfor a Discrete DDLT Distribution

Assume a probability distribution of actual DDLTs is Assume a probability distribution of actual DDLTs is given or can be developed from a frequency given or can be developed from a frequency distributiondistribution

Starting with the lowest DDLT, accumulate the Starting with the lowest DDLT, accumulate the probabilities. These are the service levels for DDLTsprobabilities. These are the service levels for DDLTs

Select the DDLT that will provide the desired Select the DDLT that will provide the desired customer level as the order pointcustomer level as the order point

62

Example: OP for Discrete DDLT DistributionExample: OP for Discrete DDLT DistributionExample: OP for Discrete DDLT DistributionExample: OP for Discrete DDLT Distribution

One of Sharp Retailer’s inventory items is now being analyzed One of Sharp Retailer’s inventory items is now being analyzed to determine an appropriate level of safety stock. The to determine an appropriate level of safety stock. The manager wants an 80% service level during lead time. The manager wants an 80% service level during lead time. The item’s historical DDLT is:item’s historical DDLT is:

DDLT (cases)DDLT (cases) OccurrencesOccurrences33 8844 6655 4466 22

63

OP for Discrete DDLT DistributionOP for Discrete DDLT DistributionOP for Discrete DDLT DistributionOP for Discrete DDLT Distribution

Construct a Cumulative DDLT DistributionConstruct a Cumulative DDLT Distribution

ProbabilityProbability Probability ofProbability of

DDLT (cases)DDLT (cases) of DDLTof DDLT DDLT or LessDDLT or Less22 00 0033 .4.4 .4.444 .3.3 .7.755 .2.2 .9.966 .1.1 1.01.0

To provide 80% service level, OP = 5 casesTo provide 80% service level, OP = 5 cases

.8.8

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OP for Discrete DDLT DistributionOP for Discrete DDLT DistributionOP for Discrete DDLT DistributionOP for Discrete DDLT Distribution

Safety Stock (SS)Safety Stock (SS)

OP = EDDLT + SSOP = EDDLT + SS

SS = OP SS = OP EDDLT EDDLT

EDDLT = .4(3) + .3(4) + .2(5) + .1(6) = 4.0EDDLT = .4(3) + .3(4) + .2(5) + .1(6) = 4.0

SS = 5 – 4 = 1SS = 5 – 4 = 1

65

The Role of Safety Inventory The Role of Safety Inventory in a Supply Chainin a Supply Chain

The Role of Safety Inventory The Role of Safety Inventory in a Supply Chainin a Supply Chain

Forecasts are rarely completely accurateForecasts are rarely completely accurate If average demand is 1000 units per week, then half the time If average demand is 1000 units per week, then half the time

actual demand will be greater than 1000, and half the time actual demand will be greater than 1000, and half the time actual demand will be less than 1000; what happens when actual demand will be less than 1000; what happens when actual demand is greater than 1000?actual demand is greater than 1000?

If you kept only enough inventory in stock to satisfy average If you kept only enough inventory in stock to satisfy average demand, half the time you would run outdemand, half the time you would run out

Safety inventory:Safety inventory: Inventory carried for the purpose of Inventory carried for the purpose of satisfying demand that exceeds the amount forecasted in a satisfying demand that exceeds the amount forecasted in a given periodgiven period

66

Role of Safety InventoryRole of Safety InventoryRole of Safety InventoryRole of Safety Inventory

Average inventory is therefore cycle inventory plus safety Average inventory is therefore cycle inventory plus safety inventoryinventory

There is a fundamental tradeoff:There is a fundamental tradeoff: Raising the level of safety inventory provides higher levels Raising the level of safety inventory provides higher levels

of product availability and customer serviceof product availability and customer service Raising the level of safety inventory also raises the level of Raising the level of safety inventory also raises the level of

average inventory and therefore increases holding costsaverage inventory and therefore increases holding costs Very important in high-tech or other industries where Very important in high-tech or other industries where

obsolescence is a significant risk (where the value of obsolescence is a significant risk (where the value of inventory, such as PCs, can drop in value)inventory, such as PCs, can drop in value)

Compaq and Dell in PCsCompaq and Dell in PCs

67

Two Questions to Answer in Planning Safety Two Questions to Answer in Planning Safety InventoryInventory

Two Questions to Answer in Planning Safety Two Questions to Answer in Planning Safety InventoryInventory

What is the appropriate level of safety inventory What is the appropriate level of safety inventory to carry?to carry?

What actions can be taken to improve product What actions can be taken to improve product availability while reducing safety inventory?availability while reducing safety inventory?

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Determining the AppropriateDetermining the AppropriateLevel of Safety InventoryLevel of Safety Inventory

Determining the AppropriateDetermining the AppropriateLevel of Safety InventoryLevel of Safety Inventory

Measuring demand uncertaintyMeasuring demand uncertainty Measuring product availabilityMeasuring product availability Replenishment policiesReplenishment policies Evaluating cycle service level and fill rateEvaluating cycle service level and fill rate Evaluating safety level given desired cycle service Evaluating safety level given desired cycle service

level or fill ratelevel or fill rate Impact of required product availability and Impact of required product availability and

uncertainty on safety inventoryuncertainty on safety inventory

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Determining the AppropriateDetermining the AppropriateLevel of Demand UncertaintyLevel of Demand UncertaintyDetermining the AppropriateDetermining the AppropriateLevel of Demand UncertaintyLevel of Demand Uncertainty

Appropriate level of safety inventory determined by:Appropriate level of safety inventory determined by: supply or demand uncertaintysupply or demand uncertainty desired level of product availabilitydesired level of product availability

Higher levels of uncertainty require higher levels of Higher levels of uncertainty require higher levels of safety inventory given a particular desired level of safety inventory given a particular desired level of product availabilityproduct availability

Higher levels of desired product availability require Higher levels of desired product availability require higher levels of safety inventory given a particular higher levels of safety inventory given a particular level of uncertaintylevel of uncertainty

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Measuring Demand UncertaintyMeasuring Demand UncertaintyMeasuring Demand UncertaintyMeasuring Demand Uncertainty

Demand has a systematic component and a random componentDemand has a systematic component and a random component The estimate of the random component is the measure of demand uncertaintyThe estimate of the random component is the measure of demand uncertainty Random component is usually estimated by the standard deviation of demandRandom component is usually estimated by the standard deviation of demand Notation:Notation:

D = Average demand per periodD = Average demand per period

D D = standard deviation of demand per period= standard deviation of demand per period

L = lead time = time between when an order is placed and when it L = lead time = time between when an order is placed and when it is receivedis received

Uncertainty of demand during lead time is what is importantUncertainty of demand during lead time is what is important

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Measuring Demand UncertaintyMeasuring Demand UncertaintyMeasuring Demand UncertaintyMeasuring Demand Uncertainty

P = demand during k periods = kDP = demand during k periods = kD = std dev of demand during k periods = = std dev of demand during k periods = RRSqrt(k)Sqrt(k) Coefficient of variation = cv = Coefficient of variation = cv = = mean/(std dev) = mean/(std dev)

= size of uncertainty relative to demand= size of uncertainty relative to demand

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Measuring Product AvailabilityMeasuring Product AvailabilityMeasuring Product AvailabilityMeasuring Product Availability

Product availability:Product availability: a firm’s ability to fill a customer’s a firm’s ability to fill a customer’s order out of available inventoryorder out of available inventory

Stockout:Stockout: a customer order arrives when product is not a customer order arrives when product is not availableavailable

Product fill rate (fr):Product fill rate (fr): fraction of demand that is satisfied fraction of demand that is satisfied from product in inventoryfrom product in inventory

Order fill rate:Order fill rate: fraction of orders that are filled from fraction of orders that are filled from available inventoryavailable inventory

Cycle service level:Cycle service level: fraction of replenishment cycles fraction of replenishment cycles that end with all customer demand metthat end with all customer demand met

73

Replenishment PoliciesReplenishment PoliciesReplenishment PoliciesReplenishment Policies

Replenishment policy:Replenishment policy: decisions regarding when to decisions regarding when to reorder and how much to reorderreorder and how much to reorder

Continuous review:Continuous review: inventory is continuously inventory is continuously monitored and an order of size Q is placed when the monitored and an order of size Q is placed when the inventory level reaches the reorder point ROPinventory level reaches the reorder point ROP

Periodic review:Periodic review: inventory is checked at regular inventory is checked at regular (periodic) intervals and an order is placed to raise the (periodic) intervals and an order is placed to raise the inventory to a specified threshold (the “order-up-to” inventory to a specified threshold (the “order-up-to” level)level)

74

Continuous Review Policy: Safety Inventory Continuous Review Policy: Safety Inventory and Cycle Service Leveland Cycle Service Level

Continuous Review Policy: Safety Inventory Continuous Review Policy: Safety Inventory and Cycle Service Leveland Cycle Service Level

LL:: Lead time for replenishmentLead time for replenishment

D:D: Average demand per unit timeAverage demand per unit time

D:D:Standard deviation of demand Standard deviation of demand

per periodper period

DDLL:: Mean demand during lead Mean demand during lead

time time

LL: Standard deviation of demand : Standard deviation of demand

during lead timeduring lead time

CSLCSL: Cycle service level: Cycle service level

ssss:: Safety inventorySafety inventory

ROPROP: Reorder point: Reorder point),,(

)(1

LL

L

LS

DL

L

DD

F

D

ROPFCSL

ssROP

CSLss

L

DL

Average Inventory = Q/2 + ss

75

Auto Zone sells auto parts and supplies including a Auto Zone sells auto parts and supplies including a popular multi-grade motor oil. When the stock of popular multi-grade motor oil. When the stock of this oil drops to 20 gallons, a replenishment order is this oil drops to 20 gallons, a replenishment order is placed. The store manager is concerned that sales placed. The store manager is concerned that sales are being lost due to stockouts while waiting for an are being lost due to stockouts while waiting for an order. It has been determined that lead time demand order. It has been determined that lead time demand is normally distributed with a mean of 15 gallons is normally distributed with a mean of 15 gallons and a standard deviation of 6 gallons. and a standard deviation of 6 gallons.

The manager would like to know the probability of a The manager would like to know the probability of a stockout during lead time. stockout during lead time.

Example: OP - Continuous DDLT DistributionExample: OP - Continuous DDLT DistributionExample: OP - Continuous DDLT DistributionExample: OP - Continuous DDLT Distribution

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EDDLT = 15 gallonsEDDLT = 15 gallons

DDLTDDLT = 6 gallons = 6 gallons

OP = EDDLT + Z(OP = EDDLT + Z(DDLTDDLT ) )

20 = 15 + Z(6)20 = 15 + Z(6)

5 = Z(6)5 = Z(6)

Z = 5/6Z = 5/6

Z = .833Z = .833

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Standard Normal DistributionStandard Normal Distribution

00 .833.833

Area = .2967Area = .2967

Area = .5Area = .5

Area = .2033Area = .2033

zz

78


The Standard Normal table shows an area ofThe Standard Normal table shows an area of

.2967 for the region between the .2967 for the region between the zz = 0 line and the = 0 line and the

z z = .833 line. The shaded tail area is .5 - .2967 =.2033.= .833 line. The shaded tail area is .5 - .2967 =.2033.

The probability of a stock-out during lead time is .2033The probability of a stock-out during lead time is .2033

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Setting Order PointSetting Order Pointfor a Continuous DDLT Distributionfor a Continuous DDLT Distribution


The resulting DDLT distribution is a normal The resulting DDLT distribution is a normal distribution with the following parameters:distribution with the following parameters:

EDDLT = LT(d) EDDLT = LT(d)

DDLTDDLT = = LT d( ) 2

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The customer service level is converted into a Z value using the normal The customer service level is converted into a Z value using the normal distribution tabledistribution table

The safety stock is computed by multiplying the Z value by The safety stock is computed by multiplying the Z value by DDLTDDLT.. The order point is set using OP = EDDLT + SS, or by substitutionThe order point is set using OP = EDDLT + SS, or by substitution

2dOP = LT(d) + z LT(σ )2dOP = LT(d) + z LT(σ )

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ExampleExampleExampleExample

Q = 5;Q = 5; σσdd = 1.5; = 1.5; SL = 95%SL = 95%

R = d + Z R = d + Z σσdd = 5 + 1.645*1.5 = 5 + 2.5 = 5 + 1.645*1.5 = 5 + 2.5

= 7.5= 7.5

Order 5 (Q) whenever the inventory level is below Order 5 (Q) whenever the inventory level is below 7.5 (8).7.5 (8).

So, what does this mean?So, what does this mean?

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Example A: Estimating Safety Inventory Example A: Estimating Safety Inventory (Continuous Review Policy)(Continuous Review Policy)

D =D = 2,500/week; 2,500/week; D D = 500 = 500

LL = 2 weeks; = 2 weeks; QQ = 10,000; = 10,000; ROPROP = 6,000 = 6,000

DDLL = D = DL = (2500)(2) = 5000L = (2500)(2) = 5000

ssss = = ROP - RROP - RLL = = 6000 - 5000 = 1000 6000 - 5000 = 1000

Cycle inventory = Q/2 = 10000/2 = 5000Cycle inventory = Q/2 = 10000/2 = 5000

Average Inventory = cycle inventory + ss = 5000 + 1000 = 6000Average Inventory = cycle inventory + ss = 5000 + 1000 = 6000

Average Flow Time = Avg inventory / throughput = 6000/2500 = Average Flow Time = Avg inventory / throughput = 6000/2500 = 2.4 weeks2.4 weeks

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Example B: Estimating Cycle Service Level Example B: Estimating Cycle Service Level (Continuous Review Policy)(Continuous Review Policy)

Example B: Estimating Cycle Service Level Example B: Estimating Cycle Service Level (Continuous Review Policy)(Continuous Review Policy)

D =D = 2,500/week; 2,500/week; D =D = 500 500

LL = 2 weeks; = 2 weeks; QQ = 10,000; = 10,000; ROPROP = 6,000 = 6,000

Cycle service level, Cycle service level, CSLCSL = = F(DF(DLL + ss, D + ss, DLL, , LL)) = =

= NORMDIST (D= NORMDIST (DLL + ss, D + ss, DLL, , LL) = NORMDIST(6000,5000,707,1)) = NORMDIST(6000,5000,707,1)

= 0.92 (This value can also be determined from a Normal = 0.92 (This value can also be determined from a Normal probability distribution table)probability distribution table)

7072)500( LRL

84

Impact of Supply UncertaintyImpact of Supply UncertaintyImpact of Supply UncertaintyImpact of Supply Uncertainty

D:D: Average demand per period Average demand per period D:D: Standard deviation of demand per period Standard deviation of demand per period LL: Average lead time: Average lead time ssLL: Standard deviation of lead time: Standard deviation of lead time

sD

D

LDL

L

L

DL

222

85


D =D = 2,500/day; 2,500/day; D =D = 500 500

LL = 7 days; = 7 days; QQ = 10,000; = 10,000; CSL = 0.90; sCSL = 0.90; sLL = 7 days = 7 days

DDLL = DL = (2500)(7) = 17500 = DL = (2500)(7) = 17500

ss = Fss = F-1-1ss(CSL)(CSL)LL = NORMSINV(0.90) x 17550 = NORMSINV(0.90) x 17550

= 22,491= 22,491

17500)7()2500(500)7( 222

222

sDL LDL

86


Safety inventory when Safety inventory when ssLL = 0 is 1,695 = 0 is 1,695







87

Information CentralizationInformation CentralizationInformation CentralizationInformation Centralization

Virtual aggregationVirtual aggregation Information system that allows access to current inventory Information system that allows access to current inventory

records in all warehouses from each warehouserecords in all warehouses from each warehouse Most orders are filled from closest warehouseMost orders are filled from closest warehouse In case of a stockout, another warehouse can fill the orderIn case of a stockout, another warehouse can fill the order Better responsiveness, lower transportation cost, higher Better responsiveness, lower transportation cost, higher

product availability, but reduced safety inventoryproduct availability, but reduced safety inventory Examples: McMaster-Carr, Gap, Wal-MartExamples: McMaster-Carr, Gap, Wal-Mart

88

PostponementPostponementPostponementPostponement

The ability of a supply chain to delay product The ability of a supply chain to delay product differentiation or customization until closer to the differentiation or customization until closer to the time the product is soldtime the product is sold

Goal is to have common components in the Goal is to have common components in the supply chain for most of the push phase and supply chain for most of the push phase and move product differentiation as close to the pull move product differentiation as close to the pull phase as possiblephase as possible

Examples: Dell, BenettonExamples: Dell, Benetton

89

Impact of ReplenishmentImpact of ReplenishmentPolicies on Safety InventoryPolicies on Safety Inventory

Impact of ReplenishmentImpact of ReplenishmentPolicies on Safety InventoryPolicies on Safety Inventory

Continuous review policiesContinuous review policies Periodic review policiesPeriodic review policies

90

Estimating and ManagingEstimating and ManagingSafety Inventory in PracticeSafety Inventory in PracticeEstimating and ManagingEstimating and Managing

Safety Inventory in PracticeSafety Inventory in Practice

Account for the fact that supply chain demand is Account for the fact that supply chain demand is lumpylumpy

Adjust inventory policies if demand is seasonalAdjust inventory policies if demand is seasonal Use simulation to test inventory policiesUse simulation to test inventory policies Start with a pilotStart with a pilot Monitor service levelsMonitor service levels Focus on reducing safety inventoriesFocus on reducing safety inventories

91



Assume that the lead time (LT) is constantAssume that the lead time (LT) is constant Assume that the demand per day is normally distributed with Assume that the demand per day is normally distributed with

the mean (d ) and the standard deviation (the mean (d ) and the standard deviation (dd ) ) The DDLT distribution is developed by “adding” together the The DDLT distribution is developed by “adding” together the

daily demand distributions across the lead timedaily demand distributions across the lead time . . . more. . . more

92

Customer Service CriterionCustomer Service CriterionCustomer Service CriterionCustomer Service Criterion

The number of units short in one year (time period) is equal to The number of units short in one year (time period) is equal to the percentage short times the annual demand.the percentage short times the annual demand.

(1 – SL) * D(1 – SL) * D

This is equal to the number of units short per order (This is equal to the number of units short per order (σσdd E(Z)) E(Z))

times the number of orders per year (time period).times the number of orders per year (time period).

σσdd E(Z) [D/Q] E(Z) [D/Q]

SL = SL = σσdd E(Z)/Q E(Z)/Q

E(Z) = Q(1- SL)/E(Z) = Q(1- SL)/σσdd

93

Example TextExample TextExample TextExample Text

Q = 5;Q = 5; σσdd = 1.5 = 1.5 SL = 95%SL = 95%

E(Z) = Q(1 - .05)/E(Z) = Q(1 - .05)/σσdd = 5*.05/1.5 = 5*.05/1.5

= .167= .167

From the tables Z = 0.6From the tables Z = 0.6

R = d + ZR = d + Zσσdd = 5 + 0.6*1.5 = 5 + 0.9 = 5.9 = 5 + 0.6*1.5 = 5 + 0.9 = 5.9

Order 5 when the inventory level reaches 5.9Order 5 when the inventory level reaches 5.9

94

ExampleExampleExampleExample

A service station is located right across campus. His gas sales have A service station is located right across campus. His gas sales have been going down. To improve his sales he is considering been going down. To improve his sales he is considering utilizing some available space to place some soda vending utilizing some available space to place some soda vending machines. When he orders, he usually orders 10 cases (240 machines. When he orders, he usually orders 10 cases (240 cans). He estimates that the daily demand can be approximated cans). He estimates that the daily demand can be approximated by a Normal distribution with a mean of 75 cans and a by a Normal distribution with a mean of 75 cans and a standard deviation of 10 cans. He also feels that an 85% (very standard deviation of 10 cans. He also feels that an 85% (very sophisticated gas station owner) service level would be sophisticated gas station owner) service level would be adequate. His soda supplier promises that his lead time will be adequate. His soda supplier promises that his lead time will be exactly 4 days.exactly 4 days.

a)a) What should his reorder point be?What should his reorder point be?b)b) What is the safety stock?What is the safety stock?

95

Solution in terms of probability of stock-outSolution in terms of probability of stock-outSolution in terms of probability of stock-outSolution in terms of probability of stock-out

a) N(75, 10) Time period correction factora) N(75, 10) Time period correction factor

N(75*4, 10√4)N(75*4, 10√4)

R R = d + Z = d + Z σσdd

= 4*75 + (1.04)*(10 √4)= 4*75 + (1.04)*(10 √4)

= 300 + 20.8 = 321= 300 + 20.8 = 321

b) Safety Stockb) Safety Stock

SS = 20.8SS = 20.8

96

Another ExampleAnother ExampleAnother ExampleAnother Example

D = annual demand 1000 units LT=15daysD = annual demand 1000 units LT=15days

Q = 200 unitsQ = 200 units Service Level = 95 % (.95)Service Level = 95 % (.95)

Working days/yr = 250 Working days/yr = 250 σσdd=50 units=50 units

Average demand/day=1000/250 = 4 units/dayAverage demand/day=1000/250 = 4 units/day

R = d + Z R = d + Z σσdd = 4*15 + Z(50) = 4*15 + Z(50)

E(Z)=(1-.95)200/50 = 0.2 from tables E(Z)=(1-.95)200/50 = 0.2 from tables

Z = 0.49Z = 0.49

R = 4(15) + 0.49*50 = 84.5R = 4(15) + 0.49*50 = 84.5

R = 4(15) + 1.645*50 = 142.25R = 4(15) + 1.645*50 = 142.25

97

Example (Cont.)Example (Cont.)Example (Cont.)Example (Cont.)

Policy:Policy:

When inventory level gets to 85 or less then order 200.When inventory level gets to 85 or less then order 200.

What is the expected number of units short per order?What is the expected number of units short per order?

E(Z) E(Z) σσdd = 0.2 * 50 = 10 = 0.2 * 50 = 10

How many orders per year?How many orders per year?

(1000/200)= 5(1000/200)= 5

Total number of units short?Total number of units short?

10*5 = 50 (Service level is 95%; 950/1000)10*5 = 50 (Service level is 95%; 950/1000)

Business

Inventory 1.1