1 FIFTH International Conference on ``Analysis of Manufacturing Systems -- Production Management'‘ Zakynthos, Greece, 2005 QUEUEING MODELS FOR MANAGING INVENTORIES, BACKORDERS, AND QUALITY JOINTLY IN STOCHASTIC MANUFACTURING SYSTEMS Vassilis S. Kouikoglou, Technical Univ. of Crete, Greece Stratos Ioannidis, University of the Aegean, Greece Georgios Saharidis, Ecole Centrale de Paris, France

2 Basic components of a production system:  Production facilities: processing units (machines) intermediate storage and transferring of parts (buffers) quality control (inspect, rework, scrap)  Sales department

3 Production control objective Maximize profit from sales less quality, inventory, backlog, etc., costs. Subproblems a)Production control: when to produce and when to stop producing? Stock is costly, but so are stockouts. b)Quality control: accept, rework, or reject a finished item, based on deviations of product characteristics from target values. Rework and scrapping of parts are costly and cause delays in production. c)Admission control: During a stockout period should we backorder incoming orders or reject them? Any better practices other than Lost Sales or Complete Backordering? Remarks Criteria (a)-(c) are in conflict. Space of admissible control policies is vast. Analytical models are not always accurate and simulation is often time consuming. We examine a restricted set of controls which are optimal only for simple systems (base stock, Kanban).

4 Example 1: Base stock levels in a two-stage supply chain Assumptions Processing times in factory M i are exponential rv’s with rates  i Demand is Poisson with rate Demand during stockout periods in buffer B 1 or B 2 is satisfied immediately by purchasing from subcontractors Factory M i produces until stock n i reaches base stock level b i, i = 1, 2 11 22

5 Parameters p 1 price at which factory M 1 sells a component to M 2 p 2 selling price of the final product c i unit production cost at factory M i s i cost of purchasing one unit from subcontractor i h i unit holding cost rate in buffer B i

6 Equilibrium probabilities P(n 1, n 2 ) The system is Markovian. State: number of components and products in stock: (n 1, n 2 ). Define P(n 2 ) = [P(0, n 2 ) P(1, n 2 ) … P(b 1, n 2 )]. Chapman-Kolmogorov equations: P(0)A 0  P(1)C 0 P(n 2 )A  P(n 2  1)B + P(n 2 +1)C, n 2  1, …, b 2  1 P(b 2 )A 1  P(b 2  1)B 1 where A 0, A 1, A, C 0, C, B 1, and B are matrices that describe the transition rates among the various states (n 1, n 2 ). We solve these equations recursively expressing P(1), P(2), … as functions of P(0). The latter is computed from the last equation and the normalization equation  P (n 1, n 2 ) = 1.

7 Mean profit rate of the system J(b 1, b 2 ) J(b 1, b 2 )  p 2  [production costs in M 1 and M 2 ]  [costs of purchasing from subcontractors in M 1 and M 2 ]  [inventory costs in M 1 and M 2 ]  a function of the equilibrium probabilities Coordination FULL: Perform exhaustive search to track down values for b 1 and b 2 that jointly maximize the mean profit rate of the system. PARTIAL: 1)Factory M 2 determines a base stock b 2 which maximizes the mean profit rate by considering its own costs and profits. Factory M 2 is an M/M/1/b 2 queue. 2)Factory M 1 uses the individually optimal value b 2 to estimate its demand rate and to compute a base stock b 1 which, again, maximizes its own profit rate.

8 Standard parameters:  5,  1   2  6.25, p 1  70, p 2  100, c 1  50, c 2  10, s 1  60, s 2  90, h 1  3, h 2  8 Numerical comparisons

9 Example 2: Single-product system with a base stock s, a base backlog c, and quality control Equivalent closed queueing network: #jobs is m = s + c n 0 = n F + (c - n B )

10 Relationship between the original and closed systems When n F is and n B is Then the total number n H of parts in the original system is and n 0 in the equivalent closed system is 00sc 10sc + 1 ………… s0ss + c = m 01s + 1c - 1 ………… 0cs + c = m0

11 punit profit hunit holding cost rate bunit backlog cost rate i C inspection cost per outgoing item r C rework/rejection cost per nonconforming item Y value of quality characteristic of each outgoing item; random variable t target value of Y q probability that Y is in an acceptable region [t  , t +  ] kquality loss coefficient; we assume a quadratic loss function k(Y  t) 2 Parameters Mean quality cost per outgoing item Q = i C + r C (1  q) + k q E[(Y  t) 2, given that Y is acceptable] Mean profit rate J( , m, s) = pTH  hH  bB  Q TH/q TH = throughput, H = average inventory, B = average backlog m  s + c  base stock + base backlog

12 H  E [n H ]  sP(n 0  c) + (s+1)P(n 0  c  1) +…+ mP(n 0  0) B  E [n B ]  1P(n 0  c  1) + 2P(n 0  c  2) +…+ cP(n 0  0) m  s + c  base stock + base backlog U  [U 0 U 1 … U N ], U  UΠ, Π  [p ij ]  matrix of part-routing probabilities Then: Assumption: The equivalent system is of the Jackson type Let (n 0, n 1, …, n N ) be the vector whose entries are the items in each machine. Then

13 Theorem 1 (a) The function J( , m, s) is concave in s for any fixed ( , m) and assumes its maximum value at the point s which satisfies the following condition (b) If s is optimal for m, then the optimal base stock for m + 1 is either s or s + 1. Theorem 2 For any fixed , the profit rate J is a unimodal function of m for all m  k, where k is the smallest nonnegative integer such that G 0 is the normalizing constant of the closed queueing network with node 0 removed.

14 Optimization For   [  min,  max]; for m = 0, 1, …, m , where m  = local maximizer satisfying condition of Th 2; compute optimal s for ( , m) by applying Th 1, find (  *, m*, s*) which maximize mean profit J We perform exhaustive search for  and m, but m  is finite.z

15 Admission control policies PLS: partly lost sales (proposed policy) CB: complete backlog LS: lost sales Coordination FULL: this strategy seeks values for , m, and s that jointly maximize the mean profit rate of the system (proposed strategy). PARTIAL: the quality control department computes the value , which minimizes the mean quality cost Q per outgoing item. Using this value, the production department computes the probability q of a conforming item. Then, m and s are determined so as to maximize the quantity pTH  hH  bB which is the total profit without quality costs. NO: similar to PARTIAL except that the production department assumes that q = 1, ignoring the possibility of rework or scrap. Test case: A six-machine production line. Numerical comparisons

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17  Managing inventory levels, sales, and quality tolerances jointly achieves higher profit than independently determined control policies.  Key to the computational efficiency of the optimization algorithms is the adoption of simple control policies and the use of analytical models.  In all the numerical experiments we have performed, the objective functions appear to be quasiconcave (unimodal). We have supported this by a few theoretical results.  Establishing this property is important in order to speed up search for the optimal control parameters, as it will be safe to stop when a locally optimal solution is found. Conclusions