Optimal Control of High-Volume Assemble-to-Order Systems

Optimal Control of High-Volume Assemble-to-Order Systems Motivation • Assembly-to-Order – hold component inventories – rapid assembly of many products – Dell - grown by 40% per year in recent years. PC industry - grown by less than 20% per year. – GE, American Standard, BMW, Timbuk2, National Bicycle. • Challenges of ATO – product prices? – production capacity for component (supply contract)? – dynamically ration scarce components to customer orders? Overview • Literature review • Model formulation – Dynamic control problem – Static formulation • Asymptotic analysis • Delay bound and expediting component option Model Formulation Sequence of events: 1. set product prices, component production rates – remain fixed throughout time horizon 2. dynamically sequence assembly of outstanding product orders Objective: minimize infinite horizon discounted expected profit Trade-off: inventory vs. customer service (assembly delay, cash flow) Operational Assumptions: – assembly is instantaneous given necessary components – customer order for each product are filled FIFO Model Formulation - notations J K akj pk δj Ok Cj cj Ak (t) components finished products no. of type j components needed by product k product price component production rate product demand arrival renewal process, rate θk (p) component arrival renewal process, rate δj component unit production cost cumulative no. of type k orders assembled up to t u = (pu, δu, Au) admissible policy (prices, production rates, assembly sequence rule) order queue-length, = Ou,k (t) − Au,k (t) √ 0 �K inventory levels, = Cu,j (t) − k=1 akj Au,k (t) √ 0 Qu,k (t) Iu,j (t) Model Formulation - technical assumptions θ(p) is continuous, differentiable, and the Jacobian matrix is invertible. guarantees p(θ) is unique, continuous, and differentiable. Customer demand for product k is strictly decreasing in pk , but may be increasing k (p) in pm, m = ≤ k. δθδpk (p) < 0 while δθ ≤ k. δpm √ 0, m = k Increase in the price of one product � cannot lead to an increase in the total rate δθm k of demand for all products. −δθ > m∗=k δp . δp k k Revenue rates for each product class, rk (θ) = θk pk (θ) are concave. Renewal processes Ok and Cj started in steady state at time zero. Model Formulation - profit expression infinite horizon discounted profit: � = K � � � = pk e −�t o k=1 � � K �� � k=1 dAk (t) − J � � pk e 0 cj e−�tdCj (t) j=1 0 � � � −�t � dOk (t) − Qk (t)�e −�t 0 � dt − J � � j=1 � cj e−�tdCj (t), 0 where Qk (t) is the order queue-length � � e 0 −�t dOk (t) − � � e 0 −�t dAk (t) = � � �e−�tQk (t)dt 0 Model Formulation - static planning problem if we assume that demand and production flow at the long run average rates continuously and deterministically, � ¯ = max p�0,α�0 s.t. K � k=1 K � k=1 pk θk (p) − J � δj cj j=1 akj θk (p) ∼ δj , j = 1, ..., J – optimal solution (p�, δ �) assumed to be unique, positive. condition imply that all constraints are tight (p�, δ �). the first order ¯ is an upper bound on the expected profit rate. –� want to show that under high volume conditions, the optimal prices and production rates are close to (p�, δ �). Asymptotic analysis - high demand volume conditions any strictly increasing sequence {n} in [0, →), n tends to infinity. order arrival rate function θn, where θnk (p) = nθk (p), k = 1, ..., K. n¯ � upper bounds the expected profit rate in the nth system, �n ∼ � � n¯ ��e−�tdt = � −1n¯ � 0 plug (p�, nδ �) into the nth system, n−1�(p�,nα �,An) ∀ � −1� ¯ as n ∀ →, given that n−1Qn ∀ 0 a.s., as n ∀ →. Asymptotic analysis - proposed assembly policy component shortage process: Sj (t) = K � k=1 akj Ok (t) − Cj (t) = K � k=1 akj Qk (t) − Ij (t), j = 1, ..., J min. instantaneous cost arrangement of queue-lengths and inventory levels (Q�(S), I �(S)), min Q,I�0 s.t. K � p�k Qk k=1 Ij = K � k=1 akj Qk − Sj √ 0, j = 1, ..., J Asymptotic analysis - proposed assembly policy for the nth system, the review period ln = n−�, where � = (4(3 + 2σ1))−1(6 + 5σ1) > 1/2 Review on Brownian Motion A standard Brownian Motion (Wiener process) is a stochastic process W having 1. continuous sample paths 2. stationary independent increments 3. W (t) � N (0, t) A stochastic process X is a Brownian motion with drift µ and variance π 2 if X(t) = X(0) + µt + πW (t), �t then E[X(t) − X(0)] = µt, V ar[X(t) − X(0)] = π 2t. variance of a Brownian motion increases linearly with the time interval. System with delay constraints propose a near-optimal discrete review control policies, which both sequences customer orders for assembly and expedites component production in an ATO system with delay constraints.