Benchmarking Non-First-Come-First-Served Component Allocation in an Assemble-To-Order System Kai Huang McMaster University
June 4, 2013
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Table of Contents 1
Introduction
2
Non-First-Come-First-Served Component Allocation Last-Come-First-Served-Within-One-Period (LCFP) Product-Based-Priority-Within-Time-Windows (PTW)
3
Demand Fulfillment Rates Demand Fulfillment Rates of the LCFP Rule Demand Fulfillment Rates of the PTW Rule
4
Inventory Replenishment Policy Base Stock Level Optimization of the LCFP Rule Base Stock Level Optimization of the PTW Rule
5
Benchmark Models
6
Numerical Experiment
7
Conclusions
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Assemble-To-Order System (ATOS) Two levels: Products and components. c o m p o n e n ts
p ro d u c ts
r e p le n is h m e n t
c u s to m e r d e m a n d
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Assemble-To-Order System (ATOS) Two levels: Products and components. c o m p o n e n ts
p ro d u c ts
r e p le n is h m e n t
c u s to m e r d e m a n d
In the middle of single-echelon and two-echelon.
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Assemble-To-Order System (ATOS)
Assumptions: ◮
Periodic review.
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Assemble-To-Order System (ATOS)
Assumptions: ◮
Periodic review.
◮
Independent base stock policy for each component.
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Assemble-To-Order System (ATOS)
Assumptions: ◮
Periodic review.
◮
Independent base stock policy for each component.
◮
Consignment policy: once a unit of component is assigned to an order, it is not available to other orders anymore even if it still stays in the inventory.
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Assemble-To-Order System (ATOS)
Assumptions: ◮
Periodic review.
◮
Independent base stock policy for each component.
◮
Consignment policy: once a unit of component is assigned to an order, it is not available to other orders anymore even if it still stays in the inventory.
Optimization problems: ◮ ◮
Base stock level optimization. Component allocation optimization.
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Last-Come-First-Served-Within-One-Period (LCFP)
In a period, the unfulfilled orders come from t1 , t1 + 1, · · · , t − 1, t: ◮ ◮
FCFS: Fulfill the orders in the sequence t1 , t1 + 1, · · · , t − 1, t. LCFP: Fulfill the orders in the sequence t, t1 , t1 + 1, · · · , t − 1.
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Product-Based-Priority-Within-Time-Windows (PTW)
Each product has a priority j and a time window wj .
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Product-Based-Priority-Within-Time-Windows (PTW)
Each product has a priority j and a time window wj . Product j can only be considered for fulfillment from period t + wj onward.
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Product-Based-Priority-Within-Time-Windows (PTW)
Each product has a priority j and a time window wj . Product j can only be considered for fulfillment from period t + wj onward. The fulfillment follows the priority list.
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Product-Based-Priority-Within-Time-Windows (PTW)
Each product has a priority j and a time window wj . Product j can only be considered for fulfillment from period t + wj onward. The fulfillment follows the priority list. Example: Let w1 = 0, w2 = 1, w3 = 2. Then the sequence of satisfying the demands P1,t , P2,t , P3,t will be P1,t , P2,t−1 , P3,t−2 , P1,t+1 , P2,t , P3,t−1 , P1,t+2 , P2,t+1 , P3,t .
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Demand Fulfillment Rates of the LCFP Rule
The amount of inventory committed to the demand Di,t should be Ei,t = Min{(Si − Di [t − Li − 1, t − 1])+ + Di,t−Li −1 , Di,t }, while in FCFS, this amount is Min{(Si − Di [t − Li , t − 1])+ , Di,t }.
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Demand Fulfillment Rates of the LCFP Rule (Zero Time Window)
Lemma The available on-hand inventory at the end of period t is (Si − Di [t − Li , t])+ under the LCFP rule, which is the same as that under the FCFS rule.
Theorem The demand Di,t will be satisfied exactly in period t if and only if (Si − Di [t − Li − 1, t − 1])+ + Di,t−Li −1 ≥ Di,t under the LCFP rule.
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Demand Fulfillment Rates of the LCFP Rule (Positive Time Window)
Theorem The demand Di,t will be satisfied within a time window w ≥ 1 if and only if (Si − Di [t − Li − 1, t − 1])+ + Di,t−Li −1 ≥ Di,t (i.e. Ei,t = Di,t ), or, + (Si − Di [t − Li − 1, t − 1]) Pw + Di,t−Li −1 < Di,t (i.e. Ei,t < Di,t ) and Si − Di [t − Li + w , t] − s=1 Ei,t+s ≥ 0, under the LCFP rule.
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Demand Fulfillment Rates of the PTW Rule (Zero Time Window)
Theorem When the PTW rule is applied, the net inventory just before satisfying the demand aij Pj,t in period t + wj is: Si P − Di [t − PLi + wj , t − 1] − k:k<j s:s≥t,s+wk ≤t+wj aik Pk,s P P + k:k>j s:sj s:s 0} ≥ αj
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∀j.
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Base Stock Level Optimization of the LCFP Rule
Observation Assume the LCFP rule is applied, and the demands in the same period follow a multi-variate normal distribution, and the demands from different periods are i.i.d. Let X be defined as: {S : P{(Si − DiLi +1 )+ + Di,t−Li −1 ≥ Di,t , ∀i : aij > 0} ≥ αj
∀j},
|M|
where S = (Si )i∈M ∈ R+ is the vector of nonnegative base stock levels. The set X is not necessarily convex.
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Illustration 250
200
S2
150
100
50
0
0
50
100
150
200
250
S1 Kai Huang (McMaster University)
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Base stock Level Optimization of the PTW Rule
Min
X
c i Si
i∈M
s.t. P{Xitj ≤ Si , ∀i : aij > 0} ≥ αj
∀j.
where Xitj
= Di P [t − Li + Pwj , t − 1] + k:k≤j 0≤q≤wj −wk aik Pk,t+q P P − k:k>j 0 0} ≥ αj |M|
where S = (Si )i∈M ∈ R+ The set X is convex.
Kai Huang (McMaster University)
∀j},
is the vector of nonnegative base stock levels.
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Solution Strategies
Use the Sample Average Approximation algorithm to solve the base stock level optimization of the LCFP rule.
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Solution Strategies
Use the Sample Average Approximation algorithm to solve the base stock level optimization of the LCFP rule. Use a line search algorithm to solve the base stock level optimization of the PTW rule.
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Observation of Component Allocation Optimizaiton under FCFS Theorem For a periodic review ATO system with component base stock policy and FCFS allocation, let xjk be the number of product j assembled in period t + k for the demand Pj,t . Then the set of feasible component allocation decisions x = (xjk )j,k is characterized by:
X = {(xjk )j,k :
PL+1 xjk = Pj,t Pk=0 Pn k a x ≤ Oik µ=0 Pk Pnj=1 ij jµ µ=0 j=1 aij xjµ = Di,t xjk ∈ Z+
∀j ∀i ∀i ∀j
∈N ∈ M, k < k ∗ , k ∈ L }, ∈ M, k ≥ k ∗ , k ∈ L ∈ N,k ∈ L
where Oik = Min{(Si − Di [t − Li + k, t − 1])+ , Di,t } and k ∗ = Min{k ∈ L : Oik = Di,t } and Z+ is the set of nonnegative integers. Kai Huang (McMaster University)
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Benchmark for the Demand Fulfillment Rates under FCFS
C1 (S, ξ(ω)) = Min f1 (S, ξ(ω), x, z) Pw j s.t. Pj,t − k=0 xjk ≤ Pj,t zj zj ∈ {0, 1} x ∈ X, where z = (zj )j∈N and f1 (S, ξ(ω), x, z) =
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∀j ∈ N ∀j ∈ N
Pn
1 j=1 n zj .
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Benchmark for the Operational Costs under FCFS
C3 (S, ξ(ω)) = Min f3 (S, ξ(ω), x) s.t. x ∈ X , where f3 (S, ξ(ω), x) =
Kai Huang (McMaster University)
P Pm hi [(Si − DiLi )+ − nj=1 aij Pj,t ]+ i=1 P PL+1 P P + m hi (Oik − kµ=0 nj=1 aij xjµ ) i=1 k=0 P P Pk + nj=1 L+1 k=0 bj (Pj,t − µ=0 xjµ )
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Instances
Agrawal and Cohen (2001)
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Instances
Agrawal and Cohen (2001) Zhang (1997)
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Instances
Agrawal and Cohen (2001) Zhang (1997) Cheng et al. (2002)
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Performance Measure of the LCFP Rule 1
0.9
0.8
0.7
0.6 FCFS−FS LCFS−FS FCFS−GCF LCFS−GCF FCFS−LFF LCFS−LFF Benchmark
0.5
0.4
0
0.5
1
1.5
2
2.5
3
3.5
4
Figure : Comparison of demand fulfillment rates
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Performance Measure of the LCFP Rule 750 700 650 600 550 500 450 FCFS−FS LCFS−FS FCFS−GCF LCFS−GCF FCFS−LFF LCFS−LFF Benchmark
400 350 300 250
0
0.5
1
1.5
2
2.5
3
3.5
4
Figure : Comparison of operatoinal costs
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Performance Measure of the PTW Rule 1
0.95
0.9
0.85
0.8 FCFS−GCF LCFS−GCF FCFS−LFF LCFS−LFF PTW
0.75
0.7
0.65
PTW* Benchmark 0
0.5
1
1.5
2
2.5
3
3.5
4
Figure : Comparison of demand fulfillment rates
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Performance Measure of the PTW Rule 1200 FCFS−GCF LCFS−GCF FCFS−LFF LCFS−LFF PTW
1100 1000
PTW* Benchmark
900 800 700 600 500 400 300 200
0
0.5
1
1.5
2
2.5
3
3.5
4
Figure : Comparison of operational costs
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Conclusions
The consignment property is the key in the analysis of the non-FCFS component allocation policies.
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Conclusions
The consignment property is the key in the analysis of the non-FCFS component allocation policies. Chance-constrained programs naturally arise from ATO system optimization.
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Conclusions
The consignment property is the key in the analysis of the non-FCFS component allocation policies. Chance-constrained programs naturally arise from ATO system optimization. The Sample Average Approximation algorithm is viable in solving small to medium instances.
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