Benchmarking Non-First-Come-First-Served Component Allocation in an Assemble-To-Order System

Transcription

1 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, / 26

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

3 Assemble-To-Order System (ATOS) Two levels: Products and components.? F A J I F K? J I H A F A E I D A J? K I J A A Kai Huang (McMaster University) Fields Institute June 4, / 26

4 Assemble-To-Order System (ATOS) Two levels: Products and components.? F A J I F K? J I H A F A E I D A J In the middle of single-echelon and two-echelon.? K I J A A Kai Huang (McMaster University) Fields Institute June 4, / 26

5 Assemble-To-Order System (ATOS) Assumptions: Periodic review. Kai Huang (McMaster University) Fields Institute June 4, / 26

6 Assemble-To-Order System (ATOS) Assumptions: Periodic review. Independent base stock policy for each component. Kai Huang (McMaster University) Fields Institute June 4, / 26

7 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, / 26

8 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, / 26

9 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 t1,t 1 +1,,t 1,t. LCFP: Fulfill the orders in the sequence t,t1,t 1 +1,,t 1. Kai Huang (McMaster University) Fields Institute June 4, / 26

10 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, / 26

11 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, / 26

12 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, / 26

13 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, / 26

14 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 Li 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, / 26

15 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 Li 1 D i,t under the LCFP rule. Kai Huang (McMaster University) Fields Institute June 4, / 26

16 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 Li 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 Li 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, / 26

17 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] k:k<j 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. Kai Huang (McMaster University) Fields Institute June 4, / 26

18 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] k:k<j 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. Kai Huang (McMaster University) Fields Institute June 4, / 26

19 Base Stock Level Optimization of the LCFP Rule Min c i S i i M s.t. P{(S i D L i+1 i ) + +D i,t Li 1 D i,t, i : a ij > 0} α j j. Kai Huang (McMaster University) Fields Institute June 4, / 26

20 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{(S i D L i+1 i ) + +D i,t Li 1 D i,t, 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 not necessarily convex. Kai Huang (McMaster University) Fields Institute June 4, / 26

21 Illustration S S 1 Kai Huang (McMaster University) Fields Institute June 4, / 26

22 Base stock Level Optimization of the PTW Rule where Min c i S i i M s.t. P{X j it S i, i : a ij > 0} α j j. X j it = D i [t L i +w j,t 1] + k:k j 0 q w j w k a ik P k,t+q k:k>j 0<q w k wj a ikp k,t q. Kai Huang (McMaster University) Fields Institute June 4, / 26

23 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 + The set X is convex. is the vector of nonnegative base stock levels. Kai Huang (McMaster University) Fields Institute June 4, / 26

24 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, / 26

25 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, / 26

26 Observation of Component Allocation Optimizaiton under FCFS Theorem For a periodic review ATO system with component base stock policy and FCFS allocation, let x jk be the number of product j assembled in period t +k for the demand P j,t. Then the set of feasible component allocation decisions x = (x jk ) j,k is characterized by: X = {(x jk ) j,k : L+1 k µ=0 k µ=0 k=0 x jk = P j,t j N n j=1 a ijx jµ Oi k i M,k < k,k L n j=1 a ijx jµ = D i,t i M,k k,k L }, x jk Z + j N,k L where O k i = Min{(S i D i [t L i +k,t 1]) +,D i,t } and k = Min{k L : O k i = D i,t } and Z + is the set of nonnegative integers. Kai Huang (McMaster University) Fields Institute June 4, / 26

27 Benchmark for the Demand Fulfillment Rates under FCFS C 1 (S,ξ(ω)) = Min f 1 (S,ξ(ω),x,z) s.t. P j,t w j k=0 x jk P j,t z j j N z j {0,1} j N x X, where z = (z j ) j N and f 1 (S,ξ(ω),x,z) = n j=1 1 n z j. Kai Huang (McMaster University) Fields Institute June 4, / 26

28 Benchmark for the Operational Costs under FCFS where C 3 (S,ξ(ω)) = Min f 3 (S,ξ(ω),x) s.t. x X, f 3 (S,ξ(ω),x) = m i=1 h i[(s i D L i i ) + n j=1 a ijp j,t ] + + m L+1 n i=1 µ=0 j=1 a ijx jµ ) + n j=1 k=0 h i(oi k k L+1 k=0 b j(p j,t k µ=0 x jµ) Kai Huang (McMaster University) Fields Institute June 4, / 26

29 Instances Agrawal and Cohen (2001) Kai Huang (McMaster University) Fields Institute June 4, / 26

30 Instances Agrawal and Cohen (2001) Zhang (1997) Kai Huang (McMaster University) Fields Institute June 4, / 26

31 Instances Agrawal and Cohen (2001) Zhang (1997) Cheng et al. (2002) Kai Huang (McMaster University) Fields Institute June 4, / 26

32 Performance Measure of the LCFP Rule FCFS FS LCFS FS FCFS GCF LCFS GCF FCFS LFF LCFS LFF Benchmark Figure : Comparison of demand fulfillment rates Kai Huang (McMaster University) Fields Institute June 4, / 26

33 Performance Measure of the LCFP Rule FCFS FS 400 LCFS FS FCFS GCF 350 LCFS GCF FCFS LFF 300 LCFS LFF Benchmark Figure : Comparison of operatoinal costs Kai Huang (McMaster University) Fields Institute June 4, / 26

34 Performance Measure of the PTW Rule FCFS GCF 0.75 LCFS GCF FCFS LFF LCFS LFF 0.7 PTW PTW * Benchmark Figure : Comparison of demand fulfillment rates Kai Huang (McMaster University) Fields Institute June 4, / 26

35 Performance Measure of the PTW Rule FCFS GCF LCFS GCF FCFS LFF LCFS LFF PTW PTW * Benchmark Figure : Comparison of operational costs Kai Huang (McMaster University) Fields Institute June 4, / 26

36 Conclusions The consignment property is the key in the analysis of the non-fcfs component allocation policies. Kai Huang (McMaster University) Fields Institute June 4, / 26

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

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