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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.

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Assemble-To-Order System (ATOS)

Two levels: Products and components. In the middle of single-echelon and two-echelon.

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Assemble-To-Order System (ATOS)

Assumptions:

◮ Periodic review. Kai Huang (McMaster University) Fields Institute June 4, 2013 4 / 26

Assemble-To-Order System (ATOS)

Assumptions:

◮ Periodic review. ◮ Independent base stock policy for each component. Kai Huang (McMaster University) Fields Institute June 4, 2013 4 / 26

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.

Kai Huang (McMaster University) Fields Institute June 4, 2013 4 / 26

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. Kai Huang (McMaster University) Fields Institute June 4, 2013 4 / 26

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. Kai Huang (McMaster University) Fields Institute June 4, 2013 5 / 26

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

• nward.

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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

• nward.

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

• nward.

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])+ + Di,t−Li−1 < Di,t (i.e. Ei,t < Di,t) and Si − Di[t − Li + w, t] − w

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 aijPj,t in period t + wj is: Si − Di[t − Li + wj, t − 1] −

k:k<j

• s:s≥t,s+wk≤t+wj aikPk,s

+

k:k>j

• s:s<t,s+wk≥t+wj aikPk,s.

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Demand Fulfillment Rates of the PTW Rule (Positive Time Window)

Theorem

When the PTW rule is applied, the net inventory just before satisfying the demand aijPj,t in period t + wj + δj is: Si − Di[t − Li + wj + δj, t − 1] −

k:k<j

• s:s≥t,s+wk≤t+wj aikPk,s

+

k:k>j

• s:s<t,s+wk≥t+wj aikPk,s.

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Base Stock Level Optimization of the LCFP Rule

Min

• i∈M

ciSi s.t. P{(Si − DLi+1

i

)+ + Di,t−Li−1 ≥ Di,t, ∀i : aij > 0} ≥ αj ∀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 − DLi+1

i

)+ + Di,t−Li−1 ≥ Di,t, ∀i : aij > 0} ≥ αj ∀j}, where S = (Si)i∈M ∈ R|M|

+

is the vector of nonnegative base stock levels. The set X is not necessarily convex.

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Illustration

S1 S2

50 100 150 200 250 50 100 150 200 250

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Base stock Level Optimization of the PTW Rule

Min

• i∈M

ciSi s.t. P{X j

it ≤ Si, ∀i : aij > 0} ≥ αj

∀j. where X j

it

= Di[t − Li + wj, t − 1] +

k:k≤j

• 0≤q≤wj−wk aikPk,t+q

−

k:k>j

• 0<q≤wk−wj aikPk,t−q.

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Base stock Level Optimization of the PTW Rule

Theorem

Assume the PTW 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{X j

it ≤ Si, ∀i : aij > 0} ≥ αj

∀j}, where S = (Si)i∈M ∈ R|M|

+

is the vector of nonnegative base stock levels. The set X is convex.

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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

• f 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 : L+1

k=0 xjk = Pj,t

∀j ∈ N k

µ=0

n

j=1 aijxjµ ≤ Ok i

∀i ∈ M, k < k∗, k ∈ L k

µ=0

n

j=1 aijxjµ = Di,t

∀i ∈ M, k ≥ k∗, k ∈ L xjk ∈ Z+ ∀j ∈ N, k ∈ L }, where Ok

i = Min{(Si − Di[t − Li + k, t − 1])+, Di,t} and

k∗ = Min{k ∈ L : Ok

i = Di,t} and Z+ is the set of nonnegative integers.

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Benchmark for the Demand Fulfillment Rates under FCFS

C1(S, ξ(ω)) = Min f1(S, ξ(ω), x, z) s.t. Pj,t − wj

k=0 xjk ≤ Pj,tzj

∀j ∈ N zj ∈ {0, 1} ∀j ∈ N x ∈ X, where z = (zj)j∈N and f1(S, ξ(ω), x, z) = n

j=1 1 nzj.

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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) = m

i=1 hi[(Si − DLi i )+ − n j=1 aijPj,t]+

+ m

i=1

L+1

k=0 hi(Ok i − k µ=0

n

j=1 aijxjµ)

+ n

j=1

L+1

k=0 bj(Pj,t − k 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

0.5 1 1.5 2 2.5 3 3.5 4 0.4 0.5 0.6 0.7 0.8 0.9 1 FCFS−FS LCFS−FS FCFS−GCF LCFS−GCF FCFS−LFF LCFS−LFF Benchmark

Figure : Comparison of demand fulfillment rates

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Performance Measure of the LCFP Rule

0.5 1 1.5 2 2.5 3 3.5 4 250 300 350 400 450 500 550 600 650 700 750 FCFS−FS LCFS−FS FCFS−GCF LCFS−GCF FCFS−LFF LCFS−LFF Benchmark

Figure : Comparison of operatoinal costs

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Performance Measure of the PTW Rule

0.5 1 1.5 2 2.5 3 3.5 4 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 FCFS−GCF LCFS−GCF FCFS−LFF LCFS−LFF PTW PTW* Benchmark

Figure : Comparison of demand fulfillment rates

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Performance Measure of the PTW Rule

0.5 1 1.5 2 2.5 3 3.5 4 200 300 400 500 600 700 800 900 1000 1100 1200 FCFS−GCF LCFS−GCF FCFS−LFF LCFS−LFF PTW PTW* Benchmark

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

• ptimization.

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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

• ptimization.

The Sample Average Approximation algorithm is viable in solving small to medium instances.

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