Inventory Control Part 1 Subject to Known Demand By Ming Dong

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Inventory Control Part 1 Subject to Known Demand

By Ming Dong

Department of Industrial Engineering & Management

Shanghai Jiao Tong University

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ContentsContents

Types of Inventories Motivation for Holding Inventories; Characteristics of Inventory Systems; Relevant Costs; The EOQ Model; EOQ Model with Finite Production Rate

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IntroductionIntroduction Definition: Inventory is the stock of any item or resource

used in an organization. An inventory system is the set of policies and controls that

monitors levels of inventory or determines what levels should be maintained.

Generally, inventory is being acquired or produced to meet the need of customers;

Dependant demand system - the demand of components and subassemblies (lower levels depend on higher level) -MRP;

The fundamental problem of inventory management : When to place order for replenishing the stock ? How much to order?

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IntroductionIntroduction

Inventory: plays a key role in the logistical behavior of virtually all manufacturing systems.

The classical inventory results: are central to more modern techniques of manufacturing management, such as MRP, JIT, and TBC.

The complexity of the resulting model depends on the assumptions about the various parameters of the system -the major distinction is between models for known demand and random demand.

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IntroductionIntroduction The current investment in inventories in USA is enormous; It amounted up to $1.37 trillion in the last quarter of 1999; It accounts for 20-25% of the total annual GNP (general net product); There exists enormous potential for improving the efficiency of

economy by scientifically controlling inventories;

Breakdown of total investment in inventories

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Types of InventoriesTypes of Inventories A natural classification is based on the value added

from manufacturing operations Raw materials: Resources required in the production

or processing activity of the firm. Components: Includes parts and subassemblies. Work-in-process (WIP): the inventory either waiting

in the system for processing or being processed. The level of WIP is taken as a measure of the

efficiency of a production scheduling system. JIT aims at reducing WIP to zero.

Finished good: also known as end items or the final products.

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Why Hold Inventories (1)Why Hold Inventories (1) For economies of scale

It may be economical to produce a relatively large number of items in each production run and store them for future use.

Coping with uncertainties Uncertainty in demand Uncertainty in lead time Uncertainty in supply

For speculation Purchase large quantities at current low prices and

store them for future use. Cope with labor strike

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Why Hold Inventories (2)Why Hold Inventories (2) Transportation

Pipeline inventories is the inventory moving from point to point, e.g., materials moving from suppliers to a plant, from one operation to the next in a plant.

Smoothing Producing and storing inventory in anticipation of

peak demand helps to alleviate the disruptions caused by changing production rates and workforce level.

Logistics To cope with constraints in purchasing, production,

or distribution of items, this may cause a system maintain inventory.

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Characteristics of Inventory SystemsCharacteristics of Inventory Systems Demand (patterns and characteristics)

Constant versus variable Known versus random

Lead Time Ordered from the outside Produced internally

Review Continuous: e.g., supermarket Periodic: e.g., warehouse

Excess demand demand that cannot be filled immediately from stock backordered or lost.

Changing inventory Become obsolete: obsolescence

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Relevant Costs - Relevant Costs - Holding CostHolding Cost

Holding cost (carrying or inventory cost) The sum of costs that are proportional to the amount

of inventory physically on-hand at any point in time Some items of holding costs

Cost of providing the physical space to store the items

Taxes and insurance Breakage, spoilage, deterioration, and obsolescence Opportunity cost of alternative investment

Inventory cost fluctuates with time inventory as a function of time

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Relevant Costs - Relevant Costs - Holding CostHolding Cost

Inventory as a Function of Time

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Relevant Costs - Relevant Costs - Order CostOrder Cost

It depend on the amount of inventory that is ordered or produced.

Two components The fixed cost K: independent of size of order The variable cost c: incurred on per-unit basis

0 0;( )

0

if xC x

K cx if x

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Relevant Costs - Relevant Costs - Order CostOrder Cost

Order Cost Function

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Relevant Costs - Relevant Costs - Penalty CostPenalty Cost Also know as shortage cost or stock-out cost

The cost of not having sufficient stock on-hand to satisfy a demand when it occurs.

Two interprets Backorder case: include delay costs may be involved Lost-sale case: include “loss-of-goodwill” cost, a measure of

customer satisfaction Two approaches

Penalty cost, p, is charged per-unit basis. Each time a demand occurs that cannot be satisfied immediately,

a cost p is incurred independent of how long it takes to eventually fill the demand.

Charge the penalty cost on a per-unit-time basis.

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EOQ HistoryEOQ History

Introduced in 1913 by Ford W. Harris, “How Many Parts to Make at Once” Product types: A, B and C A-B-C-A-B-C: 6 times of setup A-A-B-B-C-C: 3 times of setup

A factory producing various products and switching between products causes a costly setup (wages, material and overhead). Therefore, a trade-off between setup cost and production lot size should be determined.

Early application of mathematical modeling to Scientific Management

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EOQ Modeling AssumptionsEOQ Modeling Assumptions

1. Production is instantaneous – there is no capacity constraint and the entire lot is produced simultaneously.

2. Delivery is immediate – there is no time lag between production and availability to satisfy demand.

3. Demand is deterministic – there is no uncertainty about the quantity or timing of demand.

4. Demand is constant over time – in fact, it can be represented as a straight line, so that if annual demand is 365 units this translates into a daily demand of one unit.

5. A production run incurs a fixed setup cost – regardless of the size of the lot or the status of the factory, the setup cost is constant.

6. Products can be analyzed singly – either there is only a single product or conditions exist that ensure separability of products.

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NotationNotation

demand rate (units per year)

c proportional order cost at c per unit ordered (dollars per unit)

K fixed or setup cost to place an order (dollars)

h holding cost (dollars per year); if the holding cost consists entirely of interest on money tied up in inventory, then h = ic where i is an annual interest rate.

Q the unknown size of the order or lot size

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Inventory vs Time in EOQ ModelInventory vs Time in EOQ Model

Q/ 2Q/ 3Q/ 4Q/

Q

Inve

nto

ry

Time

slope = -

Order cycle: T=Q/

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CostsCosts

Holding Cost:

Setup Costs: K per lot, so

The average annual cost:

2cost holdingunit

2cost holding annual

2inventory average

hQ

hQ

Q

,cost setupunit Q

K

Q

KhQQG

2)(

Q

Kcost setup annual

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MedEquip ExampleMedEquip Example

Small manufacturer of medical diagnostic equipment. Purchases standard steel “racks” into which components

are mounted. Metal working shop can produce (and sell) racks more

cheaply if they are produced in batches due to wasted time setting up shop.

MedEquip doesn't want to hold too much capital in inventory.

Question: how many racks should MedEquip order at once?

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MedEquip Example CostsMedEquip Example Costs

= 1000 racks per year

c = $250

K = $500 (estimated from supplier’s pricing)

h = i*c + floor space cost = (0.1)($250) + $10 =

$35 per unit per year

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Costs in EOQ ModelCosts in EOQ Model

0.00

2.00

4.00

6.00

8.00

10.00

12.00

14.00

16.00

18.00

20.00

0 100 200 300 400 500

Order Quantity (Q)

Co

st (

$/u

nit

)

hQ/2D

A/Qc

Y(Q)

Q* =169

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Economic Order QuantityEconomic Order Quantity

Q

KhQQG

2)(

16935

)1000)(500(2* Q MedEquip Solution

EOQ Square Root Formula

Solution (by taking derivative and setting equal to zero):

h

KQ

QifQ

KQG

Q

Kh

dQ

QdGQG

2

002

)(''

02

)()('

*

3

2

Since Q”>0 ， G(Q) is convex function of Q

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Another ExampleAnother Example Example 2

Pencils are sold at a fairly steady rate of 60 per week; Pencils cost 2 cents each and sell for 15 cents each; Cost $12 to initiate an order, and holding costs are based on

annual interest rate of 25%. Determine the optimal number of pencils for the book store to

purchase each time and the time between placement of orders Solutions

Annual demand rate =6052=3,120; The holding cost is the product of the variable cost of the

pencil and the annual interest-h=0.02 0.25=0.05

* 2 2 12 3,1203,870

0.05

KQ

h

3,8701.24

3,120

QT yr

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The EOQ Model-The EOQ Model-Considering Lead TimeConsidering Lead Time

Since there exists lead time (4 moths for Example 2), order should be placed some time ahead of the end of a cycle;

Reorder point R-determines when to place order in term of inventory on hand, rather than time.

Reorder Point Calculation for Example 2

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The EOQ Model-The EOQ Model-Considering Lead TimeConsidering Lead Time Determine the reorder point when the lead time

exceeds a cycle.

Example:Q=25;=500/yr;=6 wks;T=25/500=2.6 wks;/T=2.31---2.31 cycles are included in LT.Action: place every order 2.31 cycles in advance.

Reorder Point Calculation for Lead Times Exceeding One Cycle

Computing R for placing order 2.31 cycles ahead is the same as that 0.31 cycle ahead.

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EOQ Modeling AssumptionsEOQ Modeling Assumptions

1. Production is instantaneous – there is no capacity constraint and the entire lot is produced simultaneously.

2. Delivery is immediate – there is no time lag between production and availability to satisfy demand.

3. Demand is deterministic – there is no uncertainty about the quantity or timing of demand.

4. Demand is constant over time – in fact, it can be represented as a straight line, so that if annual demand is 365 units this translates into a daily demand of one unit.

5. A production run incurs a fixed setup cost – regardless of the size of the lot or the status of the factory, the setup cost is constant.

6. Products can be analyzed singly – either there is only a single product or conditions exist that ensure separability of products.

relax via EOQ Model for Finite Production Rate

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The EOQ Model for Finite Production RateThe EOQ Model for Finite Production Rate

The EOQ model with finite production rate is a variation of the basic EOQ model

Inventory is replenished gradually as the order is produced (which requires the production rate to be greater than the demand rate)

Notice that the peak inventory is lower than Q since we are using items as we produce them

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Notation – Notation – EOQ Model for Finite Production RateEOQ Model for Finite Production Rate

demand rate (units per year)

P production rate (units per year), where P>

c unit production cost, not counting setup or inventory costs (dollars per unit)

K fixed or setup cost (dollars)

h holding cost (dollars per year); if the holding cost is consists entirely of interest on money tied up in inventory, then h = ic where i is an annual interest rate.

Q the unknown size of the production lot size decision variable

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Inventory vs TimeInventory vs Time

Inve

nto

ry

Time

- P-

1. Production run of Q takes Q/P time units

(P-)(Q/P)

(P-)(Q/P)/2

2. When the inventory reaches 0, production begins until Q products are produced (it takes Q/P time units). During the Q/P time units, the inventory level will increases to (P-)(Q/P)

Inventory increase rate

Time

slope = -

slope = P-

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Solution to Solution to EOQ Model with Finite Production RateEOQ Model with Finite Production Rate

Annual Cost Function:

Solution (by taking first derivative and setting equal to zero):

2

)/1()(

QPh

Q

KQG

setup holding

)/1(

2*

Ph

KQ

• tends to EOQ as P

• otherwise larger than EOQ because replenishment takes longer

h

KQ

2* EOQ model

00

2

2

dQ

QGd

dQ

QdG

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Example: Non-Slip Tile Co.Example: Non-Slip Tile Co.

Non-Slip Tile Company (NST) has been using production runs of 100,000 tiles, 10 times per year to meet the demand of 1,000,000 tiles annually. The set-up cost is $5000 per run and holding cost is estimated at 10% of the manufacturing cost of $1 per tile. The production capacity of the machine is 500,000 tiles per month. The factory is open 365 days per year.

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Example: Non-Slip Tile Co. (Cont.)Example: Non-Slip Tile Co. (Cont.)

This is a “EOQ Model with Finite Production Rate” problem with = 1,000,000

P = 500,000*12 = 6,000,000

h = 0.1

K = 5,000

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Find the Optimal Production Lot Size

How many runs should they expect per year?

How much will they save annually using EOQ Model with Finite Production Rate?

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Optimal Production Lot Size

346410)6000000/10000001(1.0

)1000000)(5000(2

)/1(

2*

Ph

KQ

Number of Production Runs Per Year The number of runs per year = /Q* = 2.89 times per year

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Annual Savings:

**1

Q

KhQ

PTC

Setupcost

Holdingcost

TC = 0.04167Q + 5,000,000,000/Current TC = 0.04167(1,000,000) + 5,000,000,000/(1,000,000)

= $54,167

Optimal TC = 0.04167(346,410) + 5,000,000,000/(346,410)

= $28,868

Difference = $54,167 - $28,868 = $25,299

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Sensitivity of EOQ Model to QuantitySensitivity of EOQ Model to Quantity

Optimal Unit Cost:

Optimal Annual Cost: Multiply G* by and simplify,

hK

K

hK

KhKh

Q

KhQQGG

2

2

22

2

2)(

*

***

hK2Cost Annual

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Sensitivity of EOQ Model to Quantity (cont.)Sensitivity of EOQ Model to Quantity (cont.)

Annual Cost from Using Q':

Ratio:

Q

KQhQG

2)(

Q

Q

Q

Q

hK

QKQh

QY

QY

Q

Q

*

*

**

2

1

......2

2

)(

)(

)(Cost

)(Cost

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Sensitivity of EOQ Model to Quantity (cont.)Sensitivity of EOQ Model to Quantity (cont.)

Q

Q

Q

Q

Q

Q *

** 2

1

)(Cost

)(Cost

Example: If Q' = 2Q*, then the ratio of the actual cost to optimal cost is

(1/2)[2 + (1/2)] = 1.25

If Q' = Q*/2, then the ratio of the actual cost to optimal cost is

(1/2)[(1/2)+2] = 1.25

A 100% error in lot size results in a 25% error in cost.

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EOQ TakeawaysEOQ Takeaways

Batching causes inventory (i.e., larger lot sizes translate into more stock).

Under specific modeling assumptions the lot size that optimally balances holding and setup costs is given by the square root formula:

Total cost is relatively insensitive to lot size (so rounding for other reasons, like coordinating shipping, may be attractive).

h

ADQ

2*

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Inventory Control Inventory Control Part 2 Inventory Control Subject to Part 2 Inventory Control Subject to

Unknown DemandUnknown Demand

By Ming DongBy Ming Dong

Department of Industrial Engineering & ManagementDepartment of Industrial Engineering & Management

Shanghai Jiao Tong UniversityShanghai Jiao Tong University

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The Wagner-Whitin Model

Change is not made without inconvenience, even from worse to better.

– Robert Hooker

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EOQ AssumptionsEOQ Assumptions

1. Instantaneous production.

2. Immediate delivery.

3. Deterministic demand.

4. Constant demand.

5. Known fixed setup costs.

6. Single product or separable products.

WW model relaxes this one

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Dynamic Lot Sizing NotationDynamic Lot Sizing Notationt a period (e.g., day, week, month); we will consider t = 1, … ,T,

where T represents the planning horizon.

Dt demand in period t (in units)

ct unit production cost (in dollars per unit), not counting setup or inventory costs in period t

At fixed or setup cost (in dollars) to place an order in period t

ht holding cost (in dollars) to carry a unit of inventory from period t to period t +1

Qt the unknown size of the order or lot size in period t

decision variables

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Wagner-Whitin ExampleWagner-Whitin Example

Data

Lot-for-Lot Solution

t 1 2 3 4 5 6 7 8 9 10Dt 20 50 10 50 50 10 20 40 20 30ct 10 10 10 10 10 10 10 10 10 10At 100 100 100 100 100 100 100 100 100 100ht 1 1 1 1 1 1 1 1 1 1

t 1 2 3 4 5 6 7 8 9 10 TotalDt 20 50 10 50 50 10 20 40 20 30 300Qt 20 50 10 50 50 10 20 40 20 30 300It 0 0 0 0 0 0 0 0 0 0 0Setup cost 100 100 100 100 100 100 100 100 100 100 1000Holding cost 0 0 0 0 0 0 0 0 0 0 0Total cost 100 100 100 100 100 100 100 100 100 100 1000

Since production cost c is constant, it can be ignored.

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Wagner-Whitin Example (cont.)Wagner-Whitin Example (cont.)

Fixed Order Quantity Solution

t 1 2 3 4 5 6 7 8 9 10 TotalDt 20 50 10 50 50 10 20 40 20 30 300Qt 100 0 0 100 0 0 100 0 0 0 300It 80 30 20 70 20 10 90 50 30 0 0Setup cost 100 0 0 100 0 0 100 0 0 0 300Holding cost 80 30 20 70 20 10 90 50 30 0 400Total cost 180 30 20 170 20 10 190 50 30 0 700

t 1 2 3 4 5 6 7 8 9 10Dt 20 50 10 50 50 10 20 40 20 30ct 10 10 10 10 10 10 10 10 10 10At 100 100 100 100 100 100 100 100 100 100ht 1 1 1 1 1 1 1 1 1 1

Data

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Wagner-Whitin PropertyWagner-Whitin Property

Under an optimal lot-sizing policy

(1) either the inventory carried to period t+1 from a previous period will be zero (there is a production in t+1)

(2) or the production quantity in period t+1 will be zero (there is no production in t+1)

A key observationIf we produce items in t (incur a setup cost) for use to satisfy demand in t+1, then it cannot possibly be economical to produce in t+1 (incur another setup cost) .

Either it is cheaper to produce all of period t+1’s demand in period t, or all of it in t+1; it is never cheaper to produce some in each.

Does fixed order quantity solution violate this property? Why?

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Basic Idea of Wagner-Whitin AlgorithmBasic Idea of Wagner-Whitin Algorithm

By WW Property, either Qt=0 or Qt=D1+…+Dk for some k.

If jk* = last period of production in a k period problem,

then we will produce exactly Dk+…DT in period jk*. Why?

We can then consider periods 1, … , jk*-1 as if they are an independent jk*-1 period problem.

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Wagner-Whitin ExampleWagner-Whitin Example

Step 1: Obviously, just satisfy D1 (note we are neglecting production cost, since it is fixed).

Step 2: Two choices, either j2* = 1 or j2* = 2.

1

100*1

1*1

j

AZ

1

150

200100100

150)50(1100min

2in produce ,Z

1in produce ,min

*2

2*1

211*2

j

A

DhAZ

t 1 2 3 4 5 6 7 8 9 10Dt 20 50 10 50 50 10 20 40 20 30ct 10 10 10 10 10 10 10 10 10 10At 100 100 100 100 100 100 100 100 100 100ht 1 1 1 1 1 1 1 1 1 1

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Step3: Three choices, j3* = 1, 2, 3.

1

170

250 100150210 10)1(10010017010)11()50(1100

min

3in produce ,Z2in produce ,Z1in produce ,)(

min

*3

3*2

322*1

321211*3

j

ADhA

DhhDhAZ

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Step 4: Four choices, j4* = 1, 2, 3, 4.

4

270

270 100170

300 50)1(100150

310 50)11(10)1(100100

32050)111(10)11()50(1100

min

4in produce ,Z

3in produce ,Z

2in produce ,)(Z

1in produce ,)()(

min

*4

4*3

433*2

432322*1

4321321211

*4

j

A

DhA

DhhDhA

DhhhDhhDhA

Z

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Planning Horizon PropertyPlanning Horizon Property

In the Example: Given fact: we produce in period 4 for period 4 of a 4 period

problem. Question: will we produce in period 3 for period 5 in a 5

period problem? Answer: We would never produce in period 3 for period 5 in

a 5 period problem.

If jt*=t, then the last period in which production occurs in an optimal t+1 period policy must be in the set t, t+1,…t+1. (this means that it CANNOT be t-1, t-2……)

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Wagner-Whitin Example (cont.)Wagner-Whitin Example (cont.) Step 5: Only two choices, j5* = 4, 5.

Step 6: Three choices, j6* = 4, 5, 6.

And so on.

4

320

370 100270

320)50(1100170min

5in produce ,Z

4in produce ,min

*5

5*4

544*3*

5

j

A

DhAZZ

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Wagner-Whitin Example SolutionWagner-Whitin Example Solution

Planning Horizon (t)Last Periodwith Production 1 2 3 4 5 6 7 8 9 10

1 100 150 170 3202 200 210 3103 250 3004 270 320 340 400 5605 370 380 420 5406 420 440 5207 440 480 520 6108 500 520 5809 580 610

10 620Zt 100 150 170 270 320 340 400 480 520 580

jt 1 1 1 4 4 4 4 7 7 or 8 8Produce in period 8 for 8, 9, 10 (40 + 20 + 30 = 90 units

Produce in period 4 for 4, 5, 6, 7 (50 + 50 + 10 + 20 = 130 units)

Produce in period 1 for 1, 2, 3 (20 + 50 + 10 = 80 units)

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Wagner-Whitin Example Solution (cont.)Wagner-Whitin Example Solution (cont.) Optimal Policy:

Produce in period 8 for 8, 9, 10 (40 + 20 + 30 = 90 units) Produce in period 4 for 4, 5, 6, 7 (50 + 50 + 10 + 20 = 130 units) Produce in period 1 for 1, 2, 3 (20 + 50 + 10 = 80 units)

t 1 2 3 4 5 6 7 8 9 10 Total Dt 20 50 10 50 50 10 20 40 20 30 300 Qt 80 0 0 130 0 0 0 90 0 0 300 It 60 10 0 80 30 20 0 50 30 0 0 Setup cost 100 0 0 100 0 0 0 100 0 0 300 Holding cost 60 10 0 80 30 20 0 50 30 0 280 Total cost 160 10 0 180 30 20 0 150 30 0 580

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Problems with Wagner-WhitinProblems with Wagner-Whitin

1. Fixed setup costs.

2. Deterministic demand and production (no uncertainty)

3. Never produce when there is inventory (WW Property). safety stock (don't let inventory fall to zero) random yields (can't produce for exact no. periods)

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Statistical Reorder Point Models

When your pills get down to four, Order more.

– Anonymous, from Hadley &Whitin

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EOQ AssumptionsEOQ Assumptions 1. Instantaneous production.

2. Immediate delivery.

3. Deterministic demand.

4. Constant demand.

5. Known fixed setup costs.

6. Single product or separable products.

WW model relaxes this one

newsvendor and (Q,r) relax this one

can use constraint approach

Chapter 17 extends (Q,r) to multiple product cases

lags can be added to EOQ or other models

EPL model relaxes this one

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Modeling Philosophies for Handling UncertaintyModeling Philosophies for Handling Uncertainty

1. Use deterministic model – adjust solution

- EOQ to compute order quantity, then add safety stock

- deterministic scheduling algorithm, then add safety lead time

2. Use stochastic model- news vendor model

- base stock and (Q,r) models

- variance constrained investment models

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The Newsvendor ApproachThe Newsvendor Approach Assumptions:

1. single period 2. random demand with known distribution 3. linear overage/shortage costs 4. minimum expected cost criterion

Examples: newspapers or other items with rapid obsolescence Christmas trees or other seasonal items

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Newsvendor Model NotationNewsvendor Model Notation

ariable.decision v theis thisunits);(in quantity /order production

shortage. ofunit per dollars)(in cost

realized. is demandafter over left unit per dollars)(in cost

demand. offunction density )()(

.)continuous (assumed

demand offunction on distributi cumulative ),()(

variable.random a units),(in demand

Q

c

c

xGdx

dxg

xXPxG

X

s

o

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Newsvendor ModelNewsvendor Model

Cost Function:

Qs

Q

o

so

so

dxxgQxcdxxgxQc

dxxgQxcdxxgxQc

EcEc

QY

)()()()(

)(0,max)(0,max

short unitsover units

cost shortage expected overage expected)(

0

00

Units over = max {Q-X, 0}

Units short = max {X-Q, 0}

Objective: find the value of Q that minimizes this expected cost.

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Newsvendor Model – Newsvendor Model – Leibnitz’s ruleLeibnitz’s rule

For Y(Q): taking its derivative and setting it to 0.To do this, we need to take the derivative of integrals with limits that

are functions of Q. A tool called Leibnitz's rule can do this.

)(

)(

11

22

)(

)(

2

1

2

1

)()),((

)()),(()],([),(

Qa

Qa

Qa

Qa dQ

QdaQQaf

dQ

QdaQQafdxQxf

QdxQxf

dQ

d

Applying this for Y(Q):

0)](1[)()()1()(1)(

00

0

QGcQGcdxxgcdxxgcdQ

QdYs

Qs

Q

so

s

cc

cQG

)( *

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Newsvendor Model (cont.)Newsvendor Model (cont.)

so

s

cc

cQXPQG

** )(

G(Q*) represents the probability that demand is less than or equal to Q*.

G(x)1

so

s

cc

c

Q*

Note:

Critical Ratio is thatprobability stockcovers demand

*

*

Qc

Qc

s

o

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Newsvendor Example – T ShirtsNewsvendor Example – T Shirts Scenario:

Demand for T-shirts is exponential with mean 1000 (i.e., G(x) = P(X x) = 1- e-x/1000). (Note - this is an odd demand distribution; Poisson or Normal would probably be better modeling choices.)

Cost of shirts is $10. Selling price is $15. Unsold shirts can be sold off at $8.

Model Parameters:cs = 15 – 10 = $5

co = 10 – 8 = $2

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Newsvendor Example – T Shirts (cont.)Newsvendor Example – T Shirts (cont.)

Solution:

Sensitivity: If co = $10 (i.e., shirts must be discarded) then

253,1

714.052

51)(

*

1000*

Q

cc

ceQG

so

sQ

405

333.0510

51)(

*

1000*

Q

cc

ceQG

so

sQ

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Newsvendor Model with Normal DemandNewsvendor Model with Normal Demand

Suppose demand is normally distributed with mean and standard deviation . Then the critical ratio formula reduces to:

Note: Q* increases in both and if z is positive (i.e.,if ratio is greater than 0.5).

zQ

cc

czz

Q

cc

cQQG

so

s

so

s

*

)( where*

*)( *

z00.00

3.00

1 7 13 19 25 31 37 43 49 55 61 67 73 79 85 91 97 103 109 115 121 127 133 139 145 151 157

(z)

is the cumulative distribution function (cdf) of the standard normal distribution.

z is the value in the standard normal table.

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Multiple Period ProblemsMultiple Period Problems

Difficulty: Technically, Newsvendor model is for a single period.

Extensions: But Newsvendor model can be applied to multiple period situations, provided: demand during each period is iid distributed according to

G(x) there is no setup cost associated with placing an order stockouts are either lost or backordered

Key: make sure co and cs appropriately represent overage and shortage cost.

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ExampleExample Scenario:

GAP orders a particular clothing item every Friday mean weekly demand is 100, std dev is 25 wholesale cost is $10, retail is $25 holding cost has been set at $0.5 per week (to reflect obsolescence,

damage, etc.)

Problem: how should they set order amounts?

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Example (cont.)Example (cont.)

Newsvendor Parameters: c0 = $0.5

cs = $15

Solution:

146)25(85.1100

85.125

100

9677.025

100

9677.0155.0

15)( *

Q

Q

Q

QG

Every Friday, they shouldorder-up-to 146, that is, ifthere are x on hand, then order 146-x.

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Newsvendor TakeawaysNewsvendor Takeaways

Inventory is a hedge against demand uncertainty.

Amount of protection depends on “overage” and “shortage” costs, as well as distribution of demand.

If shortage cost exceeds overage cost, optimal order quantity generally increases in both the mean and standard deviation of demand.

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The (The (QQ, , rr) Approach) Approach

Decision Variables:Reorder Point: r – affects likelihood of stockout

(safety stock).Order Quantity: Q – affects order frequency (cycle

inventory).

Assumptions:1. Continuous review of inventory.

2. Demands occur one at a time.

3. Unfilled demand is backordered.

4. Replenishment lead times are fixed and known.

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Inventory vs Time in (Inventory vs Time in (QQ,,rr) Model) Model

Q

Inve

nto

ry

Time

r

l

Delivery lead time

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The Base-Stock PolicyThe Base-Stock Policy

Start with an initial amount of inventory R. Each time a new demand arrives, place a replenishment order with the supplier.

An order placed with the supplier is delivered l units of time after it is placed.

Because demand is stochastic, we can have multiple orders that have been placed but not delivered yet.

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Base Stock ModelBase Stock Model

What the policy looks like Choose a base-stock level stock level R Each period, order to bring inventory position up to

the base-stock level R That order will arrive after a lead time l Costs are incurred based on net inventory Balance holding costs vs. backorder costs

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Base Stock Model AssumptionsBase Stock Model Assumptions

1. There is no fixed cost associated with placing an order.

2. There is no constraint on the number of orders that can be placed per year.

That is, we can replenishone at a time (Q=1).

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Base Stock NotationBase Stock Notation Q = 1, order quantity (fixed at one) r = reorder point R = r +1, base stock level l = delivery lead time = mean demand during l = std dev of demand during l p(x) = Prob{demand during lead time l equals x} G(x) = Prob{demand during lead time l is less than

x} h = unit holding cost b = unit backorder cost S(R) = average fill rate (service level) B(R) = average backorder level I(R) = average on-hand inventory level

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Inventory Balance EquationsInventory Balance Equations

Balance Equation:inventory position = on-hand inventory - backorders + orders

Under Base Stock Policy inventory position = R

On-hand inventory

On-order quantity from suppliers

Back-orders from customers

Inventory position R

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Inventory Profile for Base Stock System (Inventory Profile for Base Stock System (RR=5)=5)

0

1

2

3

4

5

6

7

0 5 10 15 20 25 30 35

Time

On Hand Inventory

Backorders

Orders

Inventory Position

l

R

r

# of orders = # of demands

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Service Level (Fill Rate)Service Level (Fill Rate)

Let:

X = (random) demand during lead time l

so E[X] = . Consider a specific replenishment order. Since inventory position is always R, the only way this item can stock out is if X R.

Expected Service Level:

discrete is if ),()1(

continuous is if ),()()(

GrGRG

GRGRXPRS

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Backorder LevelBackorder Level Note: At any point in time, number of orders equals number demands

that have occurred during the last l time units, (X), so from our previous balance equation:

R = on-hand inventory - backorders + orderson-hand inventory - backorders = R - X

Note: on-hand inventory and backorders are never positive at the same time, so if X=x, then

Expected Backorder Level:

RxRx

Rx

if ,

if ,0backorders

)](1)[()()()()( RGRRpxpRxRBRx

simpler version forspreadsheet computing

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Inventory LevelInventory Level

Observe:

on-hand inventory - backorders = R-X

E[X] =

E[backorders] = B(R) from previous slide

Result:I(R) = R - + B(R)

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Base Stock ExampleBase Stock Example

l = one month

= 10 units (per month)

Assume Poisson demand, so

x

k

x

k

k

k

ekpxG

0 0

10

!

10)()( Note: Poisson

demand is a good choice when no variability data is available.

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Base Stock Example CalculationsBase Stock Example Calculations

R p(R) G(R) B(R) R p(R) G(R) B(R)0 0.000 0.000 10.000 12 0.095 0.792 0.5311 0.000 0.000 9.000 13 0.073 0.864 0.3222 0.002 0.003 8.001 14 0.052 0.917 0.1873 0.008 0.010 7.003 15 0.035 0.951 0.1034 0.019 0.029 6.014 16 0.022 0.973 0.0555 0.038 0.067 5.043 17 0.013 0.986 0.0286 0.063 0.130 4.110 18 0.007 0.993 0.0137 0.090 0.220 3.240 19 0.004 0.997 0.0068 0.113 0.333 2.460 20 0.002 0.998 0.0039 0.125 0.458 1.793 21 0.001 0.999 0.00110 0.125 0.583 1.251 22 0.000 0.999 0.00011 0.114 0.697 0.834 23 0.000 1.000 0.000

Summarize the results in Table

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Base Stock Example ResultsBase Stock Example Results

Service Level: For fill rate of 90%, we must set R-1= r =14, so R=15 and safety stock s = r- = 4. Resulting service is 91.7%.

Backorder Level:

B(R) = B(15) = 0.103

Inventory Level:

I(R) = R - + B(R) = 15 - 10 + 0.103 = 5.103

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““Optimal” Base Stock LevelsOptimal” Base Stock Levels

Objective Function:

Y(R) = hI(R) + bB(R)

= h(R-+B(R)) + bB(R)

= h(R- ) + (h+b)B(R)

Solution: if we assume G is continuous, we can get

)( *

bh

bRG

Implication: set base stocklevel so fill rate is b/(h+b).

Note: R* increases in b anddecreases in h.

holding plus backorder cost

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Base Stock Normal ApproximationBase Stock Normal Approximation

If G is normal(,), then

where (z)=b/(h+b). So

R* = + z

zR

bh

bRG

*

*)(

Note: R* increases in and also increases in provided z>0.

z is the value from the standard normal table.

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““Optimal” Base Stock ExampleOptimal” Base Stock Example

Data: Approximate Poisson with mean 10 by normal with mean 10 units/month and standard deviation 10 = 3.16 units/month. Set h=$15, b=$25.

Calculations:

since (0.32) = 0.625, z=0.32 and hence

R* = + z = 10 + 0.32(3.16) = 11.01 11

Observation: from previous table fill rate is G(10) = 0.583, so maybe backorder cost is too low.

0.6252515

25

bh

b

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Inventory PoolingInventory Pooling

Situation: n different parts with lead time demand normal(, ) z=2 for all parts (i.e., fill rate is around 97.5%)

Specialized Inventory: base stock level for each item = + 2 total safety stock = 2n

Pooled Inventory: suppose parts are substitutes for one another lead time demand is normal (n ,n ) base stock level (for same service) = n +2 n ratio of safety stock to specialized safety stock = 1/ n

cycle stock safety stock

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Effect of Pooling on Safety StockEffect of Pooling on Safety Stock

Conclusion: cycle stock isnot affected by pooling, butsafety stock falls dramatically.So, for systems with high safetystock, pooling (through productdesign, late customization, etc.)can be an attractive strategy.

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Pooling ExamplePooling Example

PC’s consist of 6 components (CPU, HD, CD ROM, RAM, removable storage device, keyboard)

3 choices of each component:

Each component costs $150 ($900 material cost per PC)

Demand for all models is Poisson distributed with mean 100 per year

Replenishment lead time is 3 months (0.25 years)

Use base stock policy with fill rate of 99%

36 = 729 different PC’s

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Pooling Example - Stock PC’sPooling Example - Stock PC’s

Base Stock Level for Each PC: = 100 0.25 = 25, so using Poisson formulas,

G(R-1) 0.99 R = 38 units

On-Hand Inventory for Each PC:

I(R) = R - + B(R) = 38 - 25 + 0.0138 = 13.0138 units

Total On-Hand Inventory:

13.0138 729 $900 = $8,538,358

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Pooling Example - Stock ComponentsPooling Example - Stock Components Necessary Service for Each Component: S = (0.99)1/6 = 0.9983

Base Stock Level for Components: = (100 729/3)0.25 = 6075, so G(R-1) 0.9983 R = 6306

On-Hand Inventory Level for Each Component:

I(R) = R - + B(R) = 6306-6075+0.0363 = 231.0363 units

Total On-Hand Inventory:

231.0363 18 $150 = $623,798

93% reduction!

729 models of PC3 types of each comp.

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Base Stock InsightsBase Stock Insights

1. Reorder points control prob of stockouts by establishing safety stock.

2. To achieve a given fill rate, the required base stock level (and hence safety stock) is an increasing function of mean and (provided backorder cost exceeds shortage cost) std dev of demand during replenishment lead time.

3. The “optimal” fill rate is an increasing in the backorder cost and a decreasing in the holding cost. We can use either a service constraint or a backorder cost to determine the appropriate base stock level.

4. Base stock levels in multi-stage production systems are very similar to kanban systems and therefore the above insights apply.

5. Base stock model allows us to quantify benefits of inventory pooling.

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The Single Product (The Single Product (QQ,,rr) Model) Model

Motivation: Either

1. Fixed cost associated with replenishment orders and cost per backorder.

2. Constraint on number of replenishment orders per year and service constraint.

Objective: Under (1)

costbackorder cost holdingcost setup fixed min Q,r

As in EOQ, this makesbatch production attractive.

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Summary of (Q,r) Model AssumptionsSummary of (Q,r) Model Assumptions

1. One-at-a-time demands.

2. Demand is uncertain, but stationary over time and distribution is known.

3. Continuous review of inventory level.

4. Fixed replenishment lead time.

5. Constant replenishment batch sizes.

6. Stockouts are backordered.

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(Q,r)(Q,r) Notation Notation

cost backorder unit annual

stockoutper cost

cost holdingunit annual

iteman ofcost unit

orderper cost fixed

timelead during demand of cdf ) ()(

timelead during demand of pmf )(

timeleadent replenishm during demand ofdeviation standard

timeleadent replenishm during demand expected ][

timeleadent replenishm during demand (random)

constant) (assumed timeleadent replenishm

yearper demand expected

b

k

h

c

A

xXPxG

xXPp(x)

XE

X

D

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(Q,r)(Q,r) Notation (cont.) Notation (cont.)

levelinventory average),(

levelbackorder average ),(

rate) (fill level service average ),(

frequencyorder average )(

by impliedstock safety

pointreorder

quantityorder

rQI

rQB

rQS

QF

rrs

r

Q

Decision Variables:

Performance Measures:

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Inventory and Inventory Position for Inventory and Inventory Position for QQ=4, =4, rr=4=4

-2

-1

0

1

2

3

4

5

6

7

8

9

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32

Time

Qu

anti

ty

Inventory Position Net Inventory

Inventory Positionuniformly distributedbetween r+1=5 and r+Q=8

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Costs in Costs in (Q,r)(Q,r) Model Model

Fixed Setup Cost: AF(Q)

Stockout Cost: kD(1-S(Q,r)), where k is cost per stockout

Backorder Cost: bB(Q,r)

Inventory Carrying Costs: cI(Q,r)


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Fixed Setup Cost in (Fixed Setup Cost in (QQ,,rr) Model) Model

Observation: since the number of orders per year is D/Q,

Q

DF(Q)


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Stockout Cost in Stockout Cost in (Q,r)(Q,r) Model Model

Key Observation: inventory position is uniformly distributed between r+1 and r+Q. So, service in (Q,r) model is weighted sum of service in base stock model.

Result:

)]()([1

1),(

)]1()([1

)1(1

),(1

QrBrBQ

rQS

QrGrGQ

xGQ

rQSQr

rx

Note: this form is easier to use in spreadsheets because it does not involve a sum.

R

xRx

xGRBxpRxRB0

)](1[)()()()(


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Service Level ApproximationsService Level Approximations

Type I (base stock):

Type II:

)(),( rGrQS

Q

rBrQS

)(1),(

Note: computes numberof stockouts per cycle, underestimates S(Q,r)

Note: neglects B(r,Q)term, underestimates S(Q,r)


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Backorder Costs in (Backorder Costs in (QQ,,rr) Model) Model

Key Observation: B(Q,r) can also be computed by averaging base stock backorder level function over the range [r+1,r+Q]. (Similar to the fill rate)

Result:

Qr

rx

QrBrBQ

xBQ

rQB1

)]()1([1

)(1

),(

Notes: 1. B(Q,r) B(r) is a base stock approximation for backorder level.

2. If we can compute B(x) (base stock backorder level function), then we can compute stockout and backorder costs in (Q,r) model.


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Inventory Costs in (Inventory Costs in (QQ,,rr) Model) Model

Approximate Analysis: on average inventory declines from Q+s to s+1 so

Exact Analysis: this neglects backorders, which add to average inventory since on-hand inventory can never go below zero. The corrected version turns out to be

rQ

sQssQ

rQI2

1

2

1

2

)1()(),(

),(2

1),( rQBr

QrQI


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Inventory vs Time in (Inventory vs Time in (QQ,,rr) Model) Model

Inve

nto

ry

Time

r

s+1=r-+1

s+Q

Expected Inventory Actual Inventory

Approx I(Q,r)

Exact I(Q,r) =Approx I(Q,r) + B(Q,r)


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Expected Inventory Level for Expected Inventory Level for QQ=4, =4, rr=4, =4, =2=2

0

1

2

3

4

5

6

7

0 5 10 15 20 25 30 35

Time

Inve

nto

ry L

evel

s+Q

s


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(Q,r)(Q,r) Model with Backorder Cost Model with Backorder Cost

Objective Function:

Approximation: B(Q,r) makes optimization complicated because it depends on both Q and r. To simplify, approximate with base stock backorder formula, B(r):

),(),(),( rQhIrQbBAQ

DrQY

))(2

1()(),(

~),( rBr

QhrbBA

Q

DrQYrQY


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Results of Approximate OptimizationResults of Approximate Optimization

Assumptions: Q,r can be treated as continuous variables

G(x) is a continuous cdf

Results:

zrbh

brG

h

ADQ

**)(

2*

if G is normal(,),where (z)=b/(h+b)

Note: this is just the EOQ formula

Note: this is just the base stock formula


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((QQ,,rr) Example) Example

Stocking Repair Parts:

D = 14 units per year

c = $150 per unit

h = $25 per unit

l = 45 days

= (14/365) × 45 = 1.726 units during replenishment lead time

A = $10

b = $40

Demand during lead time is Poisson


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Values for Poisson(Values for Poisson() Distribution) Distribution

111

r p(r) G(r) B(r)

0 0.178 0.178 1.7261 0.307 0.485 0.9042 0.265 0.750 0.3893 0.153 0.903 0.1404 0.066 0.969 0.0425 0.023 0.991 0.0116 0.007 0.998 0.0037 0.002 1.000 0.0018 0.000 1.000 0.0009 0.000 1.000 0.000

10 0.000 1.000 0.000


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Calculations for ExampleCalculations for Example

2107.2)314.1(29.0726.1*

29.0 so ,615.0)29.0(

615.04025

40

43.415

)14)(10(22*

zr

z

bh

b

h

ADQ


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Performance Measures for ExamplePerformance Measures for Example

823.2049.0726.122

14*)*,(*

2

1**)*,(

049.0]003.0011.0042.0140.0[4

1

)]6()5()4()3([1

)(*

1*)*,(

904.0]003.0389.0[4

11

)]42()2([1

1*)]*(*)([*

11**

5.34

14

**)(

**

1*

rQBrQ

rQI

BBBBQ

xBQ

rQB

BBQ

QrBrBQ

),rS(Q

Q

DQF

Qr

rx


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Observations on ExampleObservations on Example

Orders placed at rate of 3.5 per year

Fill rate fairly high (90.4%)

Very few outstanding backorders (0.049 on

average)

Average on-hand inventory just below 3 (2.823)


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Varying the ExampleVarying the Example

Change: suppose we order twice as often so F=7 per year, then Q=2 and:

which may be too low, so increase r from 2 to 3:

This is better. For this policy (Q=2, r=3) we can compute B(2,3)=0.026, I(Q,r)=2.80.

Conclusion: this has higher service and lower inventory than the original policy (Q=4, r=2). But the cost of achieving this is an extra 3.5 replenishment orders per year.

826.0]042.0389.0[2

11)]()([

11),( QrBrB

QrQS

936.0]011.0140.0[2

11)]()([

11),( QrBrB

QrQS


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(Q,r)(Q,r) Model with Stockout Cost Model with Stockout Cost

Objective Function:

Approximation: Assume we can still use EOQ to compute Q* but replace S(Q,r) by Type II approximation and B(Q,r) by base stock approximation:

),()),(1(),( rQhIrQSkDAQ

DrQY

))(2

1(

)(),(

~),( rBr

Qh

Q

rBkDA

Q

DrQYrQY


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Results of Approximate OptimizationResults of Approximate Optimization

Assumptions: Q,r can be treated as continuous variablesG(x) is a continuous cdf

Results:

zrhQkD

kDrG

h

ADQ

**)(

2*

if G is normal(,),where (z)=kD/(kD+hQ)

Note: this is just the EOQ formula

Note: another version of base stock formula(only z is different)


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Backorder vs. Stockout ModelBackorder vs. Stockout Model

Backorder Model when real concern is about stockout time because B(Q,r) is proportional to time customers wait for

backorders useful in multi-level systems

Stockout Model when concern is about fill rate better approximation of lost sales situations (e.g., retail)

Note: We can use either model to generate solutions Keep track of all performance measures regardless of model B-model will work best for backorders, S-model for stockouts


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Lead Time VariabilityLead Time Variability

Problem: replenishment lead times may be variable, which increases variability of lead time demand.

Notation:

L = replenishment lead time (days), a random variable

l = E[L] = expected replenishment lead time (days)

L= std dev of replenishment lead time (days)

Dt = demand on day t, a random variable, assumed

independent and identically distributed

d = E[Dt] = expected daily demand

D= std dev of daily demand (units)


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Including Lead Time Variability in FormulasIncluding Lead Time Variability in Formulas

Standard Deviation of Lead Time Demand:

Modified Base Stock Formula (Poisson demand case):

22222LLD dd

22LdzzR

Inflation term due tolead time variability

Note: can be used in anybase stock or (Q,r) formulaas before. In general, it willinflate safety stock.

if demand is Poisson


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Single Product (Single Product (QQ,,rr) Insights) Insights

Basic Insights: Safety stock provides a buffer against stockouts. Cycle stock increases as replenishment frequency decreases.

Other Insights:1. Increasing D tends to increase optimal order quantity Q.

2. Increasing tends to increase the optimal reorder point. (Note: either increasing D or l increases .)

3. Increasing the variability of the demand process tends to increase the optimal reorder point (provided z > 0).

4. Increasing the holding cost tends to decrease the optimal order quantity and reorder point.


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Safety StockSafety Stock

Stock carried to provide a level of protection against costly stockouts due to uncertainty of demand during lead time

Stock outs occur when Demand over the lead time is larger than

expected service level (1 - )100%.

Service LevelService Level


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Computing Computing Inventory PositionInventory Position … …

Probability {Demand over lead-time < s} = 1-

Service LevelService Level

1-

Assumption: Demand over lead-time is normally distributedAssumption: Demand over lead-time is normally distributed

s

Probability distribution of demand over L

Probability distribution of demand over L


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Computing Computing Inventory PositionInventory Position: Normal Distribution: Normal Distribution

1-

s

Probability distribution of demand over L:

Mean = ; Std Dev =

Probability distribution of demand over L:

Mean = ; Std Dev =

sz s z

sz s z

1-

z0

1 - .90 .95 .98 .99 .999

z 1.28 1.65 2.05 2.33 3.09

From normal table or, in Excel, use: =normsinv (0.90)


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Computation of Variance for Demand over Lead Time:Computation of Variance for Demand over Lead Time:Variability Comes From Two SourcesVariability Comes From Two Sources

LD L AVG

2 2 2{ } { } { }LVar D Var L AVG AVG Var L AVG STDL

3: Adding the two terms, we get to our result

2 2 2{ }LVar D AVGL STD AVG STDL

1 2 ...L AVGLD d d d timesAVGL

1: Suppose only demand di in day i is variable; lead time is constant at AVGL

2: Now, suppose only lead time is variable; daily demand is constant at AVG

21 2 1 2{ } { ... } { } { } ... { }L AVGL AVGLVar D Var d d d Var d Var d Var d AVGL STD

di’s are independent di’s are identically distributed


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Safety stock SS

More specifically….More specifically….

2 2 2s AVG AVGL z STDL AVG STD AVGL

Note:•If lead time is constant,• If demand is constant,

0STDL 0STD

Standard deviation of demand over LT

Safety factor (std

normal table)

Mean demand over LT


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Example:Example: Consider inventory management for a certain SKU at

Home Depot. Supply lead time is variable (since it depends on order consolidation with other stores) and has a mean of 5 days and std deviation of 2 days. Daily demand for the item is variable with a mean of 30 units and std deviation of 6 units. Find the reorder point for 95% service level.

5; 2AVGL STDL

95% service level z = 1.64

2 2 2

2 2 2 30 5 1.64 2 30 6 5 150 100.8 251

s AVG AVGL z STDL AVG STD AVGL

AVG = 30, STD = 6


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The (s,S) Policy: Fixed Ordering CostsThe (s,S) Policy: Fixed Ordering Costs

s should be set to cover the lead time demand and together with a safety stock that insures the stock out probability is within the specific limit (When to reorder).

S depends on the fixed order cost – EOQ (How much)

Time

Inve

ntor

y

L

R

Orderplaced

Orderarrives

Average demandduring lead time

Safety Stock

sS


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The (s,S) Policy: Fixed Ordering CostsThe (s,S) Policy: Fixed Ordering Costs

Compute s exactly as in the base-stock model:

• Compute Q using the EOQ formula, using mean demand D = AVG (be careful about units…):

• Set S = s + Q

• Order when: inventory position (IP) drops below s• Order how much: bring IP to S

2 2 2s AVG AVGL z STDL AVG STD AVGL 2 2 2s AVG AVGL z STDL AVG STD AVGL

h

ADQ

2*


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Example: (s,S) ModelExample: (s,S) Model Consider previous Home Depot example, however, there

are fixed ordering costs, which are estimated at $50. Assume that holding costs are 15% of the product cost ($80) per year. Also, assume that the store is open 360 days a year.

2 2(30)50300

0.0333AVG K

Qh

251 300 551S s Q

251s (from previous calculations)

h = (0.15)*80/360=0.0333; A = K = 50, D = AVG = 30


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Summary of Inventory ModelsSummary of Inventory Models

Is demand rate and lead time constant?

Use EOQ•How much: EOQ formula•When: d*L

no

yesAre there fixed ordering costs?

no

yesUse (s, S) policy

•How much: Q = S – s (Q is from EOQ formula)•When: IP drops below s (base-stock policy formula)

Use base stock (s) policy•When: IP drops below s•How much: necessary to bring IP back to s

Date post:	03-Jan-2016
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Inventory Control Part 1 Subject to Known Demand By Ming Dong

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