1 ie&m shanghai jiao tong university supply chain management one's work may be finished...
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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
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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
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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
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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
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ABCABC Analysis (2)Analysis (2)
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%
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Tong University
ABCABC Analysis (3)Analysis (3)
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
ABCABC Analysis (4)Analysis (4)
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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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)
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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
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Multi-Prod (Multi-Prod (QQ,,rr) Systems – Constraint Formulations) Systems – Constraint Formulations
Backorder model
min Inventory investment
subject to:
Average order frequency F
Average backorder level B
Fill rate model (or Stockout model)
min Inventory investment
subject to:
Average order frequency F
Average fill rate S
Two different waysto represent customerservice.
36IE&MShanghai Jiao
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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
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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
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Backorder Constraint FormulationBackorder Constraint Formulation Verbal Formulation:
min Inventory investment
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
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Backorder Cost FormulationBackorder Cost Formulation Verbal Formulation:
min Ordering Cost + Backorder Cost + Holding Cost
Mathematical Formulation:
)},(),(),({ min N
1iiiiiiiiiii rQIhrQbBrQAF
“Coupling” of Q and r makes this hard to solve.
40IE&MShanghai Jiao
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Fill Rate Constraint FormulationFill Rate Constraint Formulation
Verbal Formulation:min Inventory investment
subject to: Average order frequency F
Average fill rate S
Mathematical Formulation:
N
iiiii
N
iiii
iiii
SrQSD
FrQF
rQIc
1tot
1
N
1i
),(D
1
),(N
1 :subject to
),( min
“Coupling” of Q and r makes this hard to solve.
41IE&MShanghai Jiao
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Fill Rate Cost FormulationFill Rate Cost Formulation
Verbal Formulation:
min Ordering Cost + Stockout Cost + Holding Cost
Mathematical Formulation:
)},()),(1(),({ min N
1iiiiiiiiiiii rQIhrQSkDrQAF
Note: a stockout cost penalizeseach order not filled from stockby k regardless of the duration ofthe stockout
“Coupling” of Q and r makes this hard to solve.
42IE&MShanghai Jiao
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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
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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
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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
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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
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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
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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
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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
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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
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Multi-Product Multi-Product (Q,r) (Q,r) ExampleExample
j Qj (units)
kDj/(kDj+hjQj)
(unitless)
rj (units)
Fj (Order Freq.)
Sj (Fill Rate)
Bj (Backo
rder Level)
Ij (Invent
ory Level)
($)
1
2
3
4
36.1
114.1
11.4
36.1
0.666
0.863
0.387
0.666
169.9
92.1
25.9
5.0
27.7
8.8
8.8
2.8
0.922
0.995
0.749
0.988
0.544
0.022
0.918
0.014
2410.6
670.24
512.52
189.33
12.0 0.95 1.497 3782.7
Results of multipart Stockout model (Q,r) calculations
Step 1. Assume a target average order frequency of F=12.
Step 2. Assume a target average fill rate of S=0.95.
Result: k = 7.213
Too low!!
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Conclusions of Multipart Stockout (Q,r) Model
This algorithm produce a very high fill rate (99.5%) for inexpensive, low-demand part 3.
Intuitively, the algorithm is trying to achieve an average fill rate of 95% as cheaply as possible, so it makes service high where it can do so cheaply (i.e., where the unit holding cost is low) and where it has a big impact on the overall average (i.e., where annual demand is high).
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Multi-Product Multi-Product (Q,r) (Q,r) ExampleExample
j Qj (units)
bj/(bj +h)
(unitless)
rj (units) Fj (Order Freq.)
Sj (Fill Rate)
Bj (Backor
der Level)
Ij (Invento
ry Level)
($)
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.
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Multi-Product Multi-Product (Q,r) (Q,r) ExampleExample
j Qj (units) bj/(bj +h)
(unitless)
rj (units) Fj (Order Freq.)
Sj (Fill Rate)
Bj (Backorde
r Level)
Ij (Inventory Level)
($)
1
2
3
4
36.1
114.1
11.4
36.1
0.538
0.921
0.538
0.921
165.6
95.0
27.9
7.0
27.7
8.8
8.8
2.8
0.875
0.997
0.840
0.998
0.974
0.010
0.511
0.002
2024.8
698.76
671.85
209.1
12.0 0.934 1.497 3604.5
Results of multipart Backorder model (Q,r) calculations
Conclusions:
• This algorithm results in low backorder levels for inexpensive parts 2 and 4, but higher backorder levels for expensive parts 1 and 3.
• Higher backorder levels for high-demand parts (i.e., parts 1 and 2) because higher demand produces more backorders when all other
things are equal.
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Multi-Product Multi-Product (Q,r) (Q,r) ExampleExample
j Qj (units) bj/(bj +h)
(unitless)
rj (units) Fj (Order Freq.)
Sj (Fill Rate)
Bj (Backorde
r Level)
Ij (Inventory Level)
($)
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
Results of multipart Backorder model (Q,r) calculations
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.
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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
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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?
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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?
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Types of Multi-Echelon Systems
Level 1 Level 2 Level 3
Stocking Site Inventory Flow
SerialSystem
GeneralArborescent
System
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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.
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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.
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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
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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
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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
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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:
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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
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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
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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
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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
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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.)
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