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Information about Optimal Control Of High-Volume Assemble To Order Systems Ping Xu

Optimal Control Of High-Volume Assemble To Order Systems Ping Xu

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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 1

Overview • Literature review • Model formulation – Dynamic control problem – Static formulation • Asymptotic analysis • Delay bound and expediting component option 15.764 The Theory of Operations Management 2

Literature • 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 3

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 15.764 The Theory of Operations Management 4

Model 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) order queue-length, = Ou,k (t) − Au,k (t) √ 0 �K Iu,j (t) inventory levels, = Cu,j (t) − 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 in pm, m = k. δθk (p) < 0 while δθkm √ 0, m = k. (p) ≤ ≤ δp δp k Increase in the price of one product � cannot lead to an increase in the total rate of demand for all products. −δθk > m∗=k δθm . 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. 15.764 The Theory of Operations Management 6

Model Formulation - proﬁt expression inﬁnite horizon discounted proﬁt: K J � � �� �� −�t cj e−�tdCj (t) �= pk e dAk (t) − 0 o j=1 k=1 � K J � � � � �� � � �� −�t −�t cj e−�tdCj (t), �= pk e dOk (t) − Qk (t)�e dt − 0 0 0 j=1 k=1 where Qk (t) is the order queue-length � � � � � � −�t −�t �e−�tQk (t)dt e dOk (t) − e dAk (t) = 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 pk θk (p) − δj cj p�0,α�0 j=1 k=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(p) = nθk (p), k = 1, ..., K. k n¯ upper bounds the expected proﬁt rate in the nth system, � � � �n ∼ n¯ −�tdt = � −1n¯ ��e � 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 ∀ →. 15.764 The Theory of Operations Management 9

Asymptotic analysis - proposed assembly policy component shortage process: K K � � Sj (t) = akj Ok (t) − Cj (t) = akj Qk (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 � p� Q k min k Q,I�0 k=1 K � s.t. Ij = akj Qk − Sj √ 0, j = 1, ..., J k=1 15.764 The Theory of Operations Management 10

Asymptotic 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 11

Asymptotic analysis - system behavior (See Theorem 1 on page 12 of the Plambeck and Ward paper) 15.764 The Theory of Operations Management 12

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. 15.764 The Theory of Operations Management 13

Optimality of Nearly Balanced Systems (See Theorem 2 on page 15 of the Plambeck and Ward paper) 15.764 The Theory of Operations Management 14

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. 15.764 The Theory of Operations Management 15

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