1 IE&M Shanghai Jiao Tong University Supply Chain Management One's work may be finished some day,...

1IE&MShanghai Jiao

Tong University

Supply Chain Management

One's work may be finished some day, but one's education never.

– Alexandre Dumas

2IE&MShanghai Jiao

Tong University

Inventory is the Lifeblood of ManufacturingInventory is the Lifeblood of Manufacturing Plays a role in almost all operations decisions

shop floor control scheduling aggregate planning capacity planning, …

Links to most other major strategic decisions quality assurance product design facility design marketing organizational management, …

Managing inventory is close to managing the entire system…

3IE&MShanghai Jiao

Tong University

Plan of AttackPlan of Attack

Classification:raw materialswork-in-process (WIP)finished goods inventory (FGI)spare parts

Justification:Why is inventory being held?benchmarking

4IE&MShanghai Jiao

Tong University

Plan of Attack (cont.)Plan of Attack (cont.)

Structural Changes:major reorganization (e.g., eliminate stockpoints,

change purchasing contracts, alter product mix, focused factories, etc.)

reconsider objectives (e.g., make-to-stock vs. make-to-order, capacity strategy, time-based-competition, etc.)

Modeling:What to model – identify key tradeoffs.How to model – EOQ, (Q,r), optimization,

simulation, etc.

5IE&MShanghai Jiao

Tong University

Raw MaterialsRaw Materials

Reasons for Inventory: batching (quantity discounts, purchasing capacity, … ) safety stock (buffer against randomness in

supply/production) obsolescence (changes in demand or design, also called

obsolete inventory) Improvement Policies:

ABC classification (stratify parts management) JIT deliveries (expensive and/or bulky items) vendor monitoring/management

6IE&MShanghai Jiao

Tong University

Raw Materials (cont.)Raw Materials (cont.)

Models: EOQ power-of-two service constrained optimization model

7IE&MShanghai Jiao

Tong University

ABCABC Analysis (1)Analysis (1)

A Parts —— The first 5% to 10% of the parts, accounting for 75% to 80% of the total annual cost

B Parts —— The next 10% to 15% of the parts, accounting for 10% to 15% of the total annual cost

C Parts —— The bottom 80% or so of the

parts,

accounting for only 10% of the total annual cost

8IE&MShanghai Jiao

Tong University


Basic Idea ： classify the parts according to their values per year

Conclusion: we should differentiate the parts in inventory and use different inventory control strategies for different parts.

Class Value per year (%)

Percentage to the total inventory

A Parts

B Parts

C parts

75%-80%

10%-15%

10%

5%-10%

10%~15%

80%

9IE&MShanghai Jiao

Tong University


Example ： A semiconductor manufacturer

Part Number Percentage Annual Value($) Ratio Classes

12

345

678910

20%

30%

50%

90,00077,000

26,35015,00112,500

8,2501200850504150

72%

23%

5%

A

B

C

10IE&MShanghai Jiao

Tong University


0

10

20

30

40

50

60

70

80

11IE&MShanghai Jiao

Tong University

Multiproduct EOQ ModelsMultiproduct EOQ Models

Notation:N = total number of distinct part numbers in the

system

Di = demand rate (units per year) for part i

ci = unit production cost of part i

Ai = fixed cost to place an order for part i

hi = cost to hold one unit of part i for one year

Qi = the size of the order or lot size for part i (decision variable)

12IE&MShanghai Jiao

Tong University

Multiproduct EOQ Models (cont.)Multiproduct EOQ Models (cont.)

Cost-Based EOQ Model: For part i,

Frequency Constrained EOQ Model:

min Inventory holding cost

subject to:

Average order frequency F

i

ii h

ADQ

2*

13IE&MShanghai Jiao

Tong University

Multiproduct EOQ Solution ApproachMultiproduct EOQ Solution Approach

Constraint Formulation:

Cost Formulation:

FQ

D

N

Qh

N

i i

i

N

i

ii

1

1

1:subject to

2min

N

i i

iN

i

ii

Q

ADQhQY

11 2)(

14IE&MShanghai Jiao

Tong University

Multiproduct EOQ Solution Approach Multiproduct EOQ Solution Approach (cont.)(cont.)

Cost Solution: Differentiate Y(Q) with respect to Qi, set equal to zero, and solve:

Constraint Solution: For a given A we can find Qi(A) using the above formula. The resulting average order frequency is:

If F(A) < F then penalty on order frequency is too high and should be decreased. If F(A) > F then penalty is too low and needs to be increased.

i

ii

i

ii

i

h

ADAQ

Q

ADh

Q

Y

2)(

2 2

N

i i

i

AQ

D

NAF

1 )(

1)(

No surprise - regularEOQ formula

15IE&MShanghai Jiao

Tong University

Multiproduct EOQ Procedure – Constrained CaseMultiproduct EOQ Procedure – Constrained Case

Step (0) Establish a tolerance for satisfying the constraint (i.e., a sufficiently small number that represents “close enough” for the order frequency) and guess a value for A.

Step (1) Use A in previous formula to compute Qi(A) for i = 1, … , N. Step (2) Compute the resulting order frequency:

If |F(A) - F| < , STOP; Qi* = Qi(A), i = 1, … , N. ELSE,

If F(A) < F, decrease A

If F(A) > F, increase A

Go to Step (1). Note: The increases and decreases in A can be made by trial and error, or some

more sophisticated search technique, such as interval bisection .

N

i i

i

AQ

D

NAF

1 )(

1)(

16IE&MShanghai Jiao

Tong University

Multiproduct EOQ ExampleMultiproduct EOQ Example

Input Data:

Part Di ci

1 1000 1002 1000 103 100 1004 100 10

Target F value: F = 12

17IE&MShanghai Jiao

Tong University

Multiproduct EOQ Example (cont.)Multiproduct EOQ Example (cont.)

Calculations:

Iteration Q1(A) Q2(A) Q3(A) Q4(A) F(A)Inventory

Investment1 1.000 4.47 14.14 1.41 4.47 96.85 387.392 100.000 44.72 141.42 14.14 44.72 9.68 3,873.893 50.000 31.62 100.00 10.00 31.2 13.70 2,739.254 75.000 38.73 122.47 12.25 38.73 11.18 3,354.895 62.500 35.36 111.80 11.18 35.36 12.25 3,062.586 68.750 37.08 117.26 11.73 37.08 11.68 3,212.067 65.625 36.23 114.56 11.46 36.23 11.96 3,138.218 64.065 35.80 113.19 11.32 25.80 12.10 3,100.689 64.845 36.01 113.88 11.39 36.01 12.03 3,119.5010 65.235 36.12 114.22 11.42 36.12 11.99 3,128.8711 65.040 36.07 114.05 11.41 36.07 12.01 3,124.1912 65.138 36.09 114.14 11.41 36.09 12.00 3,126.53

18IE&MShanghai Jiao

Tong University

Powers-of-Two AdjustmentPowers-of-Two Adjustment

Rounding Order Intervals:T1

* = Q1*/D1 = 36.09/1000 = 0.03609 yrs = 13.17 16 days

T2* = Q2

*/D2 = 114.14/1000 = 0.11414 yrs = 41.66 32 days

T3* = Q3

*/D3 = 11.41/100 = 0.11414 yrs = 41.66 32 days

T4* = Q4

*/D4 = 36.09/100 = 0.3609 yrs = 131.73 128 days

Rounded Order Quantities:Q1' = D1 T1'/365 = 1000 16/365 = 43.84

Q2' = D2 T2' /365 = 1000 32/365 = 87.67

Q3' = D3 T3' /365 = 100 32/365 = 8.77

Q4' = D4 T4' /365 = 100 128/365 = 35.07

19IE&MShanghai Jiao

Tong University

Powers-of-Two Adjustment (cont.)Powers-of-Two Adjustment (cont.)

Resulting Inventory and Order Frequency: Optimal inventory investment is $3,126.53 and order frequency is 12. After rounding to nearest powers-of-two, we get:

12.124

1FrequencyOrder Average

84.243,3$2

InvestmentInventory

4

1

4

1

i i

i

i

ii

Q

D

Qc

20IE&MShanghai Jiao

Tong University

Questions – Raw MaterialsQuestions – Raw Materials Do you track vendor performance (i.e., as to variability)?

Do you have a vendor certification program?

Do your vendor contracts have provisions (防备 ) for varying quantities?

Are purchasing procedures different for different part categories?

Do you make use of JIT deliveries?

Do you have excessive “wait to match” inventory? (May need more safety stock of inexpensive parts.)

Do you have too many vendors?

Is current order frequency rationalized?

21IE&MShanghai Jiao

Tong University

Work-in-ProcessWork-in-Process Despite the JIT goal of zero inventories, we can never

operate a manufacturing system with zero WIP, since zero WIP implies zero throughput.

WIP Inventory will be in one of 5 states: queueing (if it is waiting for a resource) processing (if it is being worked on by a resource) waiting to move (if it has to wait for other jobs to arrive

in order to form a batch) moving (if it is actually being transported between

resources) waiting to match (synchronization)

22IE&MShanghai Jiao

Tong University

Work-in-Process (Improvement Policies)Work-in-Process (Improvement Policies) pull systems (will achieve the same level of throughput

with a lower average WIP – reduce ququeing and wait-for-match)

synchronization schemes (– reduce wait-for-match) lot splitting (process lots and move lots do not have to be

the same – reduce wait-for-batch) flow-oriented layout (more frequent moves can be

facilitated by the plant layout – reduce wait-for-batch) floating work (cross-trained workers can increase the

effective capacity of the production line – reduce ququeing)

23IE&MShanghai Jiao

Tong University

Work-in-Process (Improvement Policies)Work-in-Process (Improvement Policies)

setup reduction (reducing setups increases effective capacity and more frequent setups will decrease effective variability at machines – reduce ququeing)

reliability/maintainability upgrades (reduce effective variability at machines – reduce ququeing)

focused factories (Not all WIP need be treated equally: low-volume parts could be produced in a job shop environment; high-volume parts could be assigned to lines with few setups and steady flow by using high efficient pull system – reduce ququeing)

improved yield/rework (enhanced quality– reduce ququeing)

better finite-capacity scheduling (– reduce ququeing)

24IE&MShanghai Jiao

Tong University

Work-in-Process (cont.)Work-in-Process (cont.)

Benchmarks: WIP below 10 times critical WIP (smallest WIP level

required by a line to achieve full throughput under the best conditions)

relative benchmarks depend on position in supply chain

Models: queueing models simulation

25IE&MShanghai Jiao

Tong University

Science Behind WIP ReductionScience Behind WIP Reduction Cycle Time (can be approximated as):

WIP (by Little’s Law):

Conclusion: CT and WIP can be reduced by reducing utilization, variability, or both.

eeea tt

u

ucc

12CT

22

uuu

ucc ea

12THCTWIP

22

te: mean processing time; ce: coefficient of variation of processing time; ca: coefficient of variation of arrivals; u: utilization rate

26IE&MShanghai Jiao

Tong University

Questions – WIPQuestions – WIP Are you using production leveling and due date negotiation to smooth releases?

Do you have long, infrequent outages (储运损耗 ) on machines?

Do you have long setup times on highly utilized machines?

Do you move product infrequently in large batches?

Do some machines have utilizations in excess of 95%?

Do you have significant yield/rework problems?

Do you have significant waiting inventory at assembly stations (i.e., synchronization problems)?

27IE&MShanghai Jiao

Tong University

Finished Goods InventoryFinished Goods Inventory Reasons for Inventory:

respond to variable customer demand absorb variability in cycle times build for seasonality forecast errors

Improvement Policies: dynamic lead time quoting (instead of fixed lead time) cycle time reduction cycle time variability reduction (more variability in cycle times, the

more safety stock we must build) late customization (Semi-finished inventory is more flexible, if can

be used to produce more than one finished product, which makes it possible to carry less total inventory)

balancing labor, capacity and inventory (product is produced during low-demand periods and held as FGI to meet demand during peak periods)

improved forecasting

28IE&MShanghai Jiao

Tong University

Finished Goods Inventory (cont.)Finished Goods Inventory (cont.) Benchmarks:

seasonal products make-to-order products make-to-stock products

Models: reorder point models queueing models simulation

29IE&MShanghai Jiao

Tong University

Questions – FGIQuestions – FGI

All the WIP questions apply here as well.

Are lead times negotiated dynamically?

Have you exploited opportunities for late customization (e.g., product standardization, etc.)?

Have you adequately considered variable labor (seasonal hiring, cross-trained workers, overtime)?

Have you evaluated your forecasting procedures against past performance?

30IE&MShanghai Jiao

Tong University

Spare Parts InventorySpare Parts Inventory Reasons for Inventory:

customer service purchasing/production lead times batch replenishment

Improvement Policies: separate scheduled/unscheduled repairing demand increase order frequency eliminate unnecessary safety stock differentiate parts with respect to fill rate/order

frequency forecast life cycle effects on demand balance hierarchical inventories

31IE&MShanghai Jiao

Tong University

Spare Parts Inventory (cont.)Spare Parts Inventory (cont.)

2 distinct types of spare parts Scheduled preventive maintenance Unscheduled emergency repairs

Scheduled maintenance represents a predictable demand source

This demand is much more stable than customer demand MRP logic is applicable to these parts

Unscheduled emergency repairs are unpredictable (Q, r) model can be used

32IE&MShanghai Jiao

Tong University

Spare Parts Inventory (cont.)Spare Parts Inventory (cont.)

Benchmarks: scheduled demand parts unscheduled demand parts

Models: (Q,r) distribution requirements planning (DRP) multi-echelon models

33IE&MShanghai Jiao

Tong University

Multi-Product Multi-Product (Q,r)(Q,r) Systems Systems Many inventory systems (including most spare parts

systems) involve multiple products (parts)

Products are not always separable because: average service is a function of all products cost of holding inventory is different for different products

Different formulations are possible, including: constraint formulation (usually more intuitive) cost formulation (easier to model, can be equivalent to

constraint approach)

34IE&MShanghai Jiao

Tong University

Model Inputs and OutputsModel Inputs and Outputs

MODEL

Inputs (by part)Cost (c)Mean LT demand ()Std Dev of LT demand ()

CostsOrder (A)Backorder (b) or Stockout (k)Holding (h)

Stocking Parameters (by part)Order Quantity (Q)Reorder Point (r)

Performance Measures (by part and for system)Order Frequency (F)Fill Rate (S)Backorder Level (B)Inventory Level (I)

35IE&MShanghai Jiao

Tong University

Multi-Prod (Multi-Prod (QQ,,rr) Systems – Constraint Formulations) Systems – Constraint Formulations

Backorder model

min Inventory investment

subject to:


Average backorder level B

Fill rate model (or Stockout model)


subject to:


Average fill rate S

Two different waysto represent customerservice.

36IE&MShanghai Jiao

Tong University

Multi ProductMulti Product (Q,r) (Q,r) Notation Notation

parts) (allstockout per cost

parts) (allcost backorder unit annual

orderper cost fixed

part for cost holdingunit annual

part for ofcost unit

part for timelead during demand of cdf)(

part for timelead during demand of pmf

part for timeleadent replenishm during demand ofdeviation standard

part for timeleadent replenishm during demand expected

part for constant) (assumed timeleadent replenishm

demand total

part for year per demand expected

system in the partsdistinct ofnumber

1

k

b

A

ih

ic

ixG

i(x)p

i

i

i

DD

iD

N

i

i

i

i

i

i

i

N

iitot

i

37IE&MShanghai Jiao

Tong University

Multi-ProductMulti-Product (Q,r) (Q,r) Notation (cont.) Notation (cont.)

irQI

irQB

irQS

irQF

rrs

ir

iQ

iii

iii

iii

iii

iiii

i

i

part for levelinventory average),(

part for levelbackorder average ),(

part for rate) (fill level service average ),(

part for frequency order average ),(

by impliedstock safety

part for point reorder

partfor quantity order

Decision Variables:

Performance Measures:

38IE&MShanghai Jiao

Tong University

Backorder Constraint FormulationBackorder Constraint Formulation Verbal Formulation:


subject to: Average order frequency F

Total backorder level B

Mathematical Formulation:

N

iiii

N

iiii

iiii

BrQB

FrQF

rQIc

1

1

N

1i

),(

),(N

1 :subject to

),( min

“Coupling” of Q and r makes this hard to solve.

39IE&MShanghai Jiao

Tong University

Backorder Cost FormulationBackorder Cost Formulation Verbal Formulation:

min Ordering Cost + Backorder Cost + Holding Cost


)},(),(),({ min N

1iiiiiiiiiii rQIhrQbBrQAF


40IE&MShanghai Jiao

Tong University

Fill Rate Constraint FormulationFill Rate Constraint Formulation

Verbal Formulation:min Inventory investment

subject to: Average order frequency F

Average fill rate S


N

iiiii

N

iiii

iiii

SrQSD

FrQF

rQIc

1tot

1

N

1i

),(D

1

),(N

1 :subject to

),( min


41IE&MShanghai Jiao

Tong University

Fill Rate Cost FormulationFill Rate Cost Formulation

Verbal Formulation:

min Ordering Cost + Stockout Cost + Holding Cost


)},()),(1(),({ min N

1iiiiiiiiiiii rQIhrQSkDrQAF

Note: a stockout cost penalizeseach order not filled from stockby k regardless of the duration ofthe stockout


42IE&MShanghai Jiao

Tong University

Relationship Between Cost and Constraint Formulations Method:

1) Use cost model to find Qi and ri, but keep track of average order frequency (F) and fill rate (S) using formulas from constraint model.

2) Vary order cost A until order frequency constraint is satisfied, then vary backorder cost b (stockout cost k) until backorder (fill rate) constraint is satisfied.

Problems: Even with cost model, these are often a large-scale integer

nonlinear optimization problems, which are hard. Because Bi(Qi,ri), Si(Qi,ri), Ii(Qi,ri) depend on both Qi and ri, solution

will be “coupled”, so step (2) above won’t work without iteration between A and b (or k).

43IE&MShanghai Jiao

Tong University

Type I (Base Stock) Approximation for Backorder Model

Approximation: replace Bi(Qi,ri) with base stock formula for average backorder

level, B(ri)

Note that this “decouples” Qi from ri because Fi(Qi,ri) = Di/Qi depends only on Qi and not ri

Resulting Model:

))}(2

1()({ min

N

1iiii

iii

i

i rBrQ

hrbBQ

DA

44IE&MShanghai Jiao

Tong University

Solution of Approximate Backorder ModelSolution of Approximate Backorder Model

Taking derivative with respect to Qi and solving yields:

Taking derivative with respect to ri and solving yields:

i

ii h

ADQ

2*

iiiii

i zrbh

brG

**)(

if Gi is normal(i,i),where (zi)=b/(hi+b)

EOQ formula again

base stock formula again

45IE&MShanghai Jiao

Tong University

Using Approximate Cost Solution to Get a Using Approximate Cost Solution to Get a Solution to the Constraint FormulationSolution to the Constraint Formulation

1) Pick initial A, b values. 2) Solve for Qi, ri using:

3) Compute average order frequency and backorder level:

4) Adjust A until

Adjust b until

iiii

i

ii

zr

h

ADQ

*

2*

N

iiiitot

N

i i

i

rQBB

Q

D

NF

1

1

),(

1

FF BBtot

Note: use exact formulafor B(Qi,ri) not approx.

Note: search canbe automated withSolver in Excel.

46IE&MShanghai Jiao

Tong University

Type I and II Approximation for Fill Rate (or Stockout) Model

Approximation: Use EOQ to compute Qi as before

Replace Bi(Qi,ri) with B(ri) (Type I approx.) in inventory cost term.

Replace Si(Qi,ri) with 1-B(ri)/Qi (Type II approx.) in stockout term

Resulting Model:

))}(2

1(

)({ min

N

1iiii

ii

i

i

i

i rBrQ

hQ

rBkD

Q

DA

Note: we use this approximatecost function to compute ri only,not Qi

47IE&MShanghai Jiao

Tong University

Solution of Approximate Fill Rate ModelSolution of Approximate Fill Rate Model

EOQ formula for Qi yields:

Taking derivative with respect to ri and solving yields:

i

ii h

ADQ

2*

iiiiii

ii zr

hQkD

kDrG

**)(

if Gi is normal(i,i),where (zi)=kDi/(kDi+hQi)

Note: modified versionof basestock formula,which involves Qi

48IE&MShanghai Jiao

Tong University

Using Approximate Cost Solution to Get a Using Approximate Cost Solution to Get a Solution to the Constraint FormulationSolution to the Constraint Formulation

1) Pick initial A, k values. 2) Solve for Qi, ri using:

3) Compute average order frequency and fill rate using:

4) Adjust A until

Adjust b until

iiii

i

ii

zr

h

ADQ

*

2*

N

iiii

N

i i

i

rQSDS

Q

D

NF

1tot

1

),(D

1

1

FF SS

Note: use exact formulafor S(Qi,ri) not approx.

Note: search canbe automated withSolver in Excel.

49IE&MShanghai Jiao

Tong University

Multi-Product Multi-Product (Q,r) (Q,r) ExampleExample

j hj ($/unit) Dj (units/yr) lj (days) j (units) j (units)

1

2

3

4

100

10

100

10

1000

1000

100100

60

30

100

15

164.4

82.2

27.4

4.1

12.8

9.1

5.2

2.0

Cost and demand data

50IE&MShanghai Jiao

Tong University


j Qj (units)

kDj/(kDj+hjQj)

(unitless)

rj (units)

Fj (Order Freq.)

Sj (Fill Rate)

Bj (Backo

rder Level)

Ij (Invent

ory Level)

($)

1

2

3

4

36.1

114.1

11.4

36.1

0.666

0.863

0.387

0.666

169.9

92.1

25.9

5.0

27.7

8.8

8.8

2.8

0.922

0.995

0.749

0.988

0.544

0.022

0.918

0.014

2410.6

670.24

512.52

189.33

12.0 0.95 1.497 3782.7

Results of multipart Stockout model (Q,r) calculations

Step 1. Assume a target average order frequency of F=12.

Step 2. Assume a target average fill rate of S=0.95.

Result: k = 7.213

Too low!!

51IE&MShanghai Jiao

Tong University

Conclusions of Multipart Stockout (Q,r) Model

This algorithm produce a very high fill rate (99.5%) for inexpensive, low-demand part 3.

Intuitively, the algorithm is trying to achieve an average fill rate of 95% as cheaply as possible, so it makes service high where it can do so cheaply (i.e., where the unit holding cost is low) and where it has a big impact on the overall average (i.e., where annual demand is high).

52IE&MShanghai Jiao

Tong University


j Qj (units)

bj/(bj +h)

(unitless)

rj (units) Fj (Order Freq.)

Sj (Fill Rate)

Bj (Backor

der Level)

Ij (Invento

ry Level)

($)

1

2

3

4

36.1

114.1

11.4

36.1

0.538

0.921

0.538

0.921

165.6

95.0

27.9

7.0

27.7

8.8

8.8

2.8

0.875

0.997

0.840

0.998

0.974

0.010

0.511

0.002

2024.8

698.76

671.85

209.1

12.0 0.934 1.497 3604.5

Results of multipart Backorder model (Q,r) calculations

1. Using backorder model algorithm to adjust the backorder cost b until the total backorder level achieves a specified target.

2. To make a comparison of stockout model and backorder model, we set B=1.497 units.

3. Use backorder algorithm to find the backorder penalty that causes total backorders to achieve this backorder level (B=1.497 units), it turns out when b=116.5, B=1.497 units.

53IE&MShanghai Jiao

Tong University


j Qj (units) bj/(bj +h)

(unitless)


Sj (Fill Rate)

Bj (Backorde

r Level)

Ij (Inventory Level)

($)

1

2

3

4

36.1

114.1

11.4

36.1

0.538

0.921

0.538

0.921

165.6

95.0

27.9

7.0

27.7

8.8

8.8

2.8

0.875

0.997

0.840

0.998

0.974

0.010

0.511

0.002

2024.8

698.76

671.85

209.1

12.0 0.934 1.497 3604.5


Conclusions:

• This algorithm results in low backorder levels for inexpensive parts 2 and 4, but higher backorder levels for expensive parts 1 and 3.

• Higher backorder levels for high-demand parts (i.e., parts 1 and 2) because higher demand produces more backorders when all other

things are equal.

54IE&MShanghai Jiao

Tong University


j Qj (units) bj/(bj +h)

(unitless)


Sj (Fill Rate)

Bj (Backorde

r Level)

Ij (Inventory Level)

($)

1

2

3

4

36.1

114.1

11.4

36.1

0.538

0.921

0.538

0.921

165.6

95.0

27.9

7.0

27.7

8.8

8.8

2.8

0.875

0.997

0.840

0.998

0.974

0.010

0.511

0.002

2024.8

698.76

671.85

209.1

12.0 0.934 1.497 3604.5


Conclusions:

3. Difference between 2 solutions: the fill rates and inventory levels are different. Given same backorder level, the backorder model generates smaller inventory investment ($3604.5 vs $3782.7). But it does so at the price of a lower fill rate (93.4% vs 95%).

4. If fill rate is the right measure of service, the stockout model is appropriate. If backorder level (or time delay) is a better representation of service, then the backorder model makes more sense.

55IE&MShanghai Jiao

Tong University

Multi-Product Multi-Product (Q,r)(Q,r) Insights Insights All other things being equal, an optimal solution will

hold less inventory (i.e., smaller Q and r) for an expensive part than for an inexpensive one.

Reduction in total inventory investment resulting from use of “optimized” solution instead of constant service (i.e., same fill rates for all parts) can be substantial.

Aggregate service may not always be valid:– could lead to undesirable impacts on some customers– additional constraints (minimum stock or service) may be

appropriate

56IE&MShanghai Jiao

Tong University

Questions – Spare Parts InventoryQuestions – Spare Parts Inventory Is scheduled demand handled separately from unscheduled

demand? Are stocking rules sensitive to demand, replenishment lead

time, and cost? Can you predict life-cycle demand better? Are you relying

on historical usage only? Are your replenishment lead times accurate? Is excess distributed inventory returned from regional

facilities to central warehouse? How are regional facility managers evaluated against

inventory? Frequency of inspection? Are lateral transhipments between regional facilities being

used effectively? Officially?

57IE&MShanghai Jiao

Tong University

Multi-Echelon Inventory SystemsMulti-Echelon Inventory Systems Many supply chains

Involve multiple levels as well as multiple parts Because of variability pooling, stocking inventory in a central

location, such as a warehouse or distribution center, allow holding less safety stock than holding separate inventories at individual demand sites.

The basic challenge in multiechelon supply chains is to balance the efficiency of central inventories with the responsiveness of distributed inventories so as to provide high system performance without excessive investment in inventory.

Questions: How much to stock? Where to stock it? How to coordinate levels?

58IE&MShanghai Jiao

Tong University

Types of Multi-Echelon Systems

Level 1 Level 2 Level 3

Stocking Site Inventory Flow

SerialSystem

GeneralArborescent

System

59IE&MShanghai Jiao

Tong University

System ConfigurationsSystem Configurations

Feature of a multiechelon supply chain Lower-level locations are supplied by higher-level locations.

Some variations Transshipment between locations at the same level For example, regional warehouses can supply one another

In short, multiechelon systems can be extremely complex.

System configuration itself is a decision variable. Determining the # of inventory levels, the locations of warehouses,

and the policies for interconnecting them can be among the most important logistics decisions for a distribution center design.

Research indicates that applying directly single-level approaches to multiechelon supply chains may work poorly.

60IE&MShanghai Jiao

Tong University

Performance MeasuresPerformance Measures Fill rate: the fraction of demands that are met out of stock.

This could apply at any level in the supply chain Backorder level: average # of orders waiting to be filled.

This measure applies to systems where backordering occurs. Backorder level is closely related to the average backorder delay, by using

Little’s Law: average backorder delay = average backorder level/ average demand rate

Lost sales: # of potential orders lost due to stockout. This measure applies to systems in which customers go elsewhere rather

than wait for a backordered item. Lost sales = (1-Fill rate) × average demand rate If fill rate is 95% and annual demand is 100 parts a year, then (1-0.95)(100)

=5 parts per year will be lost due to stockout. Probability of delay: likelihood that an activity (e.g., a machine

repair) will be delayed for lack of inventory. In general, the probability of delay in a multipart, multilevel system is a

function of the fill rates of various parts.

In summary, fill rate and average backorder level are key measures, since other measures can be computed from them.

61IE&MShanghai Jiao

Tong University

Two Echelon SystemTwo Echelon System

Warehouse evaluate with (Q,r) model compute stocking parameters and performance measures

Facilities evaluate with base stock model (ensures one-at-a-time

demands at warehouse consider delays due to stockouts at warehouse in

replenishment lead times

W

F

F

F

62IE&MShanghai Jiao

Tong University

Facility NotationFacility Notation

variablerandom a

,facility from part oforder an for delay)backorder (including timelead

ngbackorderi todue seat warehou waitspart for order an timeexpected

facility at part for timelead during demand of cdf)(

facility at part for timelead during demand of pmf

facility at part for demanddaily ofdeviation standard)(

facility at part for demanddaily mean

facility at

leadtimeent replenishm during part for demand ofdeviation standard

facility at timeleadent replenishm during part for demand expected

facility at part for year per demand expected

miL

iW

mixG

mi(x)p

miD

mid

m

i

mi

miD

im

i

im

im

im

im

im

im

im

63IE&MShanghai Jiao

Tong University

Warehouse NotationWarehouse Notation

cost setup fixed

part for cost backorder unit

part for cost holdingunit

part for ofcost unit

seat warehou part for timelead during demand of cdf)(

seat warehou part for timelead during demand of pmf

seat warehou timeleadent replenishm during part for demand of dev std

seat warehou timeleadent replenishm during part for demand expected

warehouse the topart for timeleadent replenishm

warehouseat the part for demand annual

seby warehou serviced facilities ofnumber

system in the partsdistinct ofnumber total

1

A

ib

ih

ic

ixG

i(x)p

i

i

i

iDD

M

N

i

i

i

i

i

i

i

i

M

mimi

64IE&MShanghai Jiao

Tong University

Variables and Measures in Two Echelon ModelVariables and Measures in Two Echelon Model

mi)(RI

mi)(RB

mi)(RS

irQI

irQB

irQS

irQF

miR

mir

ir

iQ

imim

imim

imim

iii

iii

iii

iii

im

im

i

i

facility at part for rate) (fill levelinventory average

facility at part for rate) (fill levelbackorder average

facility at part for rate) (fill level service average

seat warehou part for levelinventory average),(

seat warehou part for levelbackorder average ),(

seat warehou part for rate) (fill level service average ),(

seat warehou part for frequency order average ),(

facility at part for levelstock base

facility at part for point reorder

seat warehou part for point reorder

seat warehou partfor quantity order

Decision Variables:

Performance Measures:

65IE&MShanghai Jiao

Tong University

Facility Lead Times (mean)Facility Lead Times (mean)

Delay due to backordering:

Effective lead time for part i to facility m:

i

iiii D

rQBW

),(

][

][

imimim

iimim

LED

WLE

by Little’s law

use this in place of in base stock modelfor facilities

Wait is analogous to cycle time, backorder level is analogous to WIP, and demand rate is analogous to throughput.

ngbackorderi todue seat warehou waitspart for order an timeexpected :W ii

66IE&MShanghai Jiao

Tong University

Facility Lead Times (std dev)Facility Lead Times (std dev)

If y=delay for an order that encounters stockout, then:

Variance of Lim:

)1(

][

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

j

j

jjmjm

jjmjmjjmjjm

S

Wy

WLE

ySySSLE

)()(][

1)(

1)1(][][)(

))(1(][

222

2222

222

mmimimim

ii

iim

ii

iiiimimim

imiimiim

LdDLE

WS

SL

WS

SySSLELELVar

ySSLE

Note: Si=Si(Qi,ri) this just picks y to match mean, which we already know

we can use this in place of in normal base stockmodel for facilities

Suppose there are 2 possibilities: Either order sees no delay and lead time is ljm, or it encounters a stockout delay and has lead time ljm+y (y is a deterministic delay), since the probability of stockout is 1-Sj, we have

67IE&MShanghai Jiao

Tong University

Two Echelon (Single Product) ExampleTwo Echelon (Single Product) ExampleD = 14 units per year (Poisson demand) at warehouse

l = 45 days

Q = 5

r = 3

Dm = 7 units per year at a facility

lm = 1 day (warehouse to facility)

B(Q,r) = 0.0114

S(Q,r) = 0.9721

W = 365B(Q,r)/D = 365(0.0114/14) = 0.296 days

E[Lm] = 1 + 0.296 = 1.296 days

m= DmE[Lm] = (7/365)(1.296) = 0.0249 units

from previous example

single facility that accounts for half of annual demand

68IE&MShanghai Jiao

Tong University

Two Echelon Example (cont.)Two Echelon Example (cont.)

Standard deviation of demand during replenishment lead time:

Backorder level:

994.0)2(

9225.0)1(

1612.0)7474.1)(0192.0()0192.0(296.1

)()(][

7474.1296.09721.01

9721.0

1)(

22

222

B

B

LdDLE

WS

SL

mmm

m

computed from basestock

model using m and m

Conclusion: base stock level of 2 probably reasonable for facility.

69IE&MShanghai Jiao

Tong University

Observations on Multi-Echelon SystemsObservations on Multi-Echelon Systems Service matters at locations that interface with customers:

– fill rate (fraction of demands filled from stock)– average delay (expected wait for a part)

Multi-echelon systems are hard to model/solve exactly, so we try to “decouple” levels.

– Example: set fill rate at at DC and compute expected delay at facilities, then search over DC service to minimize system cost.

Structural changes are an option – (e.g., change number of DC's or facilities, allow cross-sharing, have

suppliers deliver directly to outlets, etc.)

1 IE&M Shanghai Jiao Tong University Supply Chain Management One's work may be finished some day,...

Documents

Transcript of 1 IE&M Shanghai Jiao Tong University Supply Chain Management One's work may be finished some day,...