Benchmarking Non-First-Come-First-Served Component Allocation in an Assemble-To-Order System Kai Huang McMaster University June 4, 2013 Kai Huang (McMaster University) Fields Institute June 4, 2013 1 / 26
Table of Contents Introduction 1 Non-First-Come-First-Served Component Allocation 2 Last-Come-First-Served-Within-One-Period (LCFP) Product-Based-Priority-Within-Time-Windows (PTW) Demand Fulfillment Rates 3 Demand Fulfillment Rates of the LCFP Rule Demand Fulfillment Rates of the PTW Rule Inventory Replenishment Policy 4 Base Stock Level Optimization of the LCFP Rule Base Stock Level Optimization of the PTW Rule Benchmark Models 5 Numerical Experiment 6 Conclusions 7 Kai Huang (McMaster University) Fields Institute June 4, 2013 2 / 26
Assemble-To-Order System (ATOS) Two levels: Products and components. � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � Kai Huang (McMaster University) Fields Institute June 4, 2013 3 / 26
Assemble-To-Order System (ATOS) Two levels: Products and components. � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � In the middle of single-echelon and two-echelon. Kai Huang (McMaster University) Fields Institute June 4, 2013 3 / 26
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 t 1 , t 1 + 1 , · · · , t − 1 , t : ◮ FCFS: Fulfill the orders in the sequence t 1 , t 1 + 1 , · · · , t − 1 , t . ◮ LCFP: Fulfill the orders in the sequence t , t 1 , t 1 + 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 w j . Kai Huang (McMaster University) Fields Institute June 4, 2013 6 / 26
Product-Based-Priority-Within-Time-Windows (PTW) Each product has a priority j and a time window w j . Product j can only be considered for fulfillment from period t + w j onward. Kai Huang (McMaster University) Fields Institute June 4, 2013 6 / 26
Product-Based-Priority-Within-Time-Windows (PTW) Each product has a priority j and a time window w j . Product j can only be considered for fulfillment from period t + w j onward. The fulfillment follows the priority list. Kai Huang (McMaster University) Fields Institute June 4, 2013 6 / 26
Product-Based-Priority-Within-Time-Windows (PTW) Each product has a priority j and a time window w j . Product j can only be considered for fulfillment from period t + w j onward. The fulfillment follows the priority list. Example: Let w 1 = 0 , w 2 = 1 , w 3 = 2. Then the sequence of satisfying the demands P 1 , t , P 2 , t , P 3 , t will be P 1 , t , P 2 , t − 1 , P 3 , t − 2 , P 1 , t +1 , P 2 , t , P 3 , t − 1 , P 1 , t +2 , P 2 , t +1 , P 3 , t . Kai Huang (McMaster University) Fields Institute June 4, 2013 6 / 26
Demand Fulfillment Rates of the LCFP Rule The amount of inventory committed to the demand D i , t should be E i , t = Min { ( S i − D i [ t − L i − 1 , t − 1]) + + D i , t − L i − 1 , D i , t } , while in FCFS, this amount is Min { ( S i − D i [ t − L i , t − 1]) + , D i , t } . Kai Huang (McMaster University) Fields Institute June 4, 2013 7 / 26
Demand Fulfillment Rates of the LCFP Rule (Zero Time Window) Lemma The available on-hand inventory at the end of period t is ( S i − D i [ t − L i , t ]) + under the LCFP rule, which is the same as that under the FCFS rule. Theorem The demand D i , t will be satisfied exactly in period t if and only if ( S i − D i [ t − L i − 1 , t − 1]) + + D i , t − L i − 1 ≥ D i , t under the LCFP rule. Kai Huang (McMaster University) Fields Institute June 4, 2013 8 / 26
Demand Fulfillment Rates of the LCFP Rule (Positive Time Window) Theorem The demand D i , t will be satisfied within a time window w ≥ 1 if and only if ( S i − D i [ t − L i − 1 , t − 1]) + + D i , t − L i − 1 ≥ D i , t (i.e. E i , t = D i , t ), or, ( S i − D i [ t − L i − 1 , t − 1]) + + D i , t − L i − 1 < D i , t (i.e. E i , t < D i , t ) and S i − D i [ t − L i + w , t ] − � w s =1 E i , t + s ≥ 0, under the LCFP rule. Kai Huang (McMaster University) Fields Institute June 4, 2013 9 / 26
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 a ij P j , t in period t + w j is: S i − D i [ t − L i + w j , t − 1] − � � s : s ≥ t , s + w k ≤ t + w j a ik P k , s k : k < j + � � s : s < t , s + w k ≥ t + w j a ik P k , s . k : k > j Kai Huang (McMaster University) Fields Institute June 4, 2013 10 / 26
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 a ij P j , t in period t + w j + δ j is: S i − D i [ t − L i + w j + δ j , t − 1] − � � s : s ≥ t , s + w k ≤ t + w j a ik P k , s k : k < j + � � s : s < t , s + w k ≥ t + w j a ik P k , s . k : k > j Kai Huang (McMaster University) Fields Institute June 4, 2013 11 / 26
Base Stock Level Optimization of the LCFP Rule � Min c i S i i ∈M ) + + D i , t − L i − 1 ≥ D i , t , ∀ i : a ij > 0 } ≥ α j s . t . P { ( S i − D L i +1 ∀ j . i Kai Huang (McMaster University) Fields Institute June 4, 2013 12 / 26
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: ) + + D i , t − L i − 1 ≥ D i , t , ∀ i : a ij > 0 } ≥ α j { S : P { ( S i − D L i +1 ∀ j } , i where S = ( S i ) i ∈M ∈ R |M| is the vector of nonnegative base stock levels. + The set X is not necessarily convex. Kai Huang (McMaster University) Fields Institute June 4, 2013 13 / 26
Illustration 250 200 150 S 2 100 50 0 0 50 100 150 200 250 S 1 Kai Huang (McMaster University) Fields Institute June 4, 2013 14 / 26
Base stock Level Optimization of the PTW Rule � Min c i S i i ∈M s . t . P { X j it ≤ S i , ∀ i : a ij > 0 } ≥ α j ∀ j . where X j = D i [ t − L i + w j , t − 1] it + � � 0 ≤ q ≤ w j − w k a ik P k , t + q k : k ≤ j − � � 0 < q ≤ w k − wj a ik P k , t − q . k : k > j Kai Huang (McMaster University) Fields Institute June 4, 2013 15 / 26
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 ≤ S i , ∀ i : a ij > 0 } ≥ α j ∀ j } , where S = ( S i ) i ∈M ∈ R |M| is the vector of nonnegative base stock levels. + The set X is convex. Kai Huang (McMaster University) Fields Institute June 4, 2013 16 / 26
Solution Strategies Use the Sample Average Approximation algorithm to solve the base stock level optimization of the LCFP rule. Kai Huang (McMaster University) Fields Institute June 4, 2013 17 / 26
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. Kai Huang (McMaster University) Fields Institute June 4, 2013 17 / 26
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