Optimal Control of High-Volume Assemble-to-Order Systems written by Eric L. Plambeck Amy R. Ward December 15, 2003 presentation for 15.764 The Theory of Operations Management March 4, 2004 15.764 The Theory of Operations Management Presentation by Ping XuThis summary presentation is based on: Plambeck, Erica, and Amy Ward. "Optimal Control of High-Volume Assemble-to-Order Systems." Stanford University, 2003.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? 15.764 The Theory of Operations Management 1Overview Literature review • Model formulation • – Dynamic control problem – Static formulation • Asymptotic analysis • Delay bound and expediting component option 15.764 The Theory of Operations Management 2Literature • ATO survey by Song and Zipkin (2001) • not FIFO assembly – Agrawal and Cohen (2001), Zhang (1997) • one component and multi-product assembly sequencing — multi-class, single-server queue – Wein (1991) , Duenya (1995) – Maglaras and Van Mieghem (2002), Plambeck, Kumar, and Harrison (2001) fill rate constraints• – Lu, Song, and Yao (2003), Cheng, Ettl, Lin, and Yao (2002) – Glasserman and Wang (1998) 15.764 The Theory of Operations Management 3Model 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 15.764 The Theory of Operations Management 4Model Formulation - notations J components K finished products akj no. of type j components needed by product k pk product price δj component production rate Ok product demand arrival renewal process, rate θk(p) Cj component arrival renewal process, rate δj cj component unit production cost Ak(t) cumulative no. of type k orders assembled up to t u = (pu, δu, Au) admissible policy (prices, production rates, assembly sequence rule) Qu,k(t) Iu,j (t) order queue-length, = Ou,k(t) − Au,k(t) √ 0 inventory levels, = Cu,j (t) − �K k=1 akj Au,k(t) √ 0 15.764 The Theory of Operations Management 5� 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 δθk(p)in pm, m =≤ k. δθk(p) < 0 while √ 0, m = k.δpk δpm ≤Increase in the price of one product cannot lead to an increase in the total rate δθmof demand for all products. −δθk > =k δpk .δpk m∗Revenue rates for each product class, rk(θ) = θkpk(θ) are concave. Renewal processes Ok and Cj started in steady state at time zero. 15.764 The Theory of Operations Management 6� � � � � � � � � � � Model Formulation - profit expression infinite horizon discounted profit: K � � J � � � = pke −�tdAk(t) − cje −�tdCj(t) k=1 oj=1 0 K �� � � � � J � � � = pke −�tdOk(t) − Qk(t)�e −�tdt − j=1 0 cje −�tdCj(t), 0 0k=1 where Qk(t) is the order queue-length e −�tdOk(t) − e −�tdAk(t) = �e −�tQk(t)dt 0 0 0 15.764 The Theory of Operations Management 7� � �Model Formulation - static planning problem if we assume that demand and production flow at the long run average rates continuously and deterministically, K J ¯� = max pkθk(p) − δj cj p�0,α�0 k=1 j=1 K s.t. akj θk(p) ∼ δj , j = 1, ..., J k=1 – optimal solution (p�, δ�) assumed to be unique, positive. the first order condition imply that all constraints are tight (p�, δ�). – � 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�). 15.764 The Theory of Operations Management 8� � � Asymptotic analysis - high demand volume conditions any strictly increasing sequence {n} in [0, →), n tends to infinity. order arrival rate function θn, where θn k (p) = nθk(p), k = 1, ..., K. n¯� upper bounds the expected profit rate in the nth system, �n ��e −�tdt = �−1 n¯∼ 0 n¯plug (p�, nδ�) into the nth system, n−1�(p�,nα�,An) ∀ �−1 ¯� as n ∀ →, given that n−1Qn ∀ 0 a.s., as n ∀ →. 15.764 The Theory of Operations Management 9� � � � � Asymptotic analysis - proposed assembly policy component shortage process: K K Sj(t) = akjOk(t) − Cj(t) = akjQk(t) − Ij(t), j = 1, ..., J k=1 k=1 min. instantaneous cost arrangement of queue-lengths and inventory levels (Q�(S), I�(S)), K min pkQk Q,I�0 k=1 K s.t. Ij = akjQk − Sj √ 0, j = 1, ..., J k=1 15.764 The Theory of Operations Management 10Asymptotic analysis - proposed assembly policy for the nth system, the review period ln = n−�, where � = (4(3 + 2σ1))−1(6 + 5σ1) > 1/2 15.764 The Theory of Operations Management 11Asymptotic analysis - system behavior 15.764 The Theory of Operations Management 12 (See Theorem 1 on page 12 of the Plambeck and Ward paper)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 π2 if X(t) = X(0) + µt + πW (t), �t then E[X(t) − X(0)] = µt, V ar[X(t) − X(0)] = π2t. variance of a Brownian motion increases linearly with the time interval. 15.764 The Theory of Operations Management 13Optimality of Nearly Balanced Systems 15.764 The Theory of Operations Management 14 (See Theorem 2 on page 15 of the Plambeck and Ward paper)System with delay constraints propose a near-optimal discrete review control