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Managing Inventory, Backorders, and Quality in Stochastic Systems

Explore joint inventory, backorder, and quality management in manufacturing systems. Analyze control policies and optimization strategies for maximizing profit. Utilize analytical models and simulations to enhance decision-making.

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Managing Inventory, Backorders, and Quality in Stochastic Systems

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  1. 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 FIFTH International Conference on ``Analysis of Manufacturing Systems -- Production Management'‘ Zakynthos, Greece, 2005

  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 • Production control: when to produce and when to stop producing? Stock is costly, but so are stockouts. • 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. • 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  1 2 Assumptions Processing times in factory Mi are exponential rv’s with rates i Demand is Poisson with rate  Demand during stockout periods in buffer B1 or B2 is satisfied immediately by purchasing from subcontractors Factory Mi produces until stock ni reaches base stock level bi, i = 1, 2

  5. Parameters • p1price at which factory M1 sells a component to M2 • p2selling price of the final product • ci unit production cost at factory Mi • si cost of purchasing one unit from subcontractor i • hi unit holding cost rate in buffer Bi

  6. Equilibrium probabilitiesP(n1, n2) The system is Markovian. State: number of components and products in stock: (n1, n2). Define P(n2) =  [P(0, n2) P(1, n2) … P(b1, n2)]. Chapman-Kolmogorov equations: P(0)A0P(1)C0 P(n2)AP(n21)B + P(n2+1)C , n2 1, …, b21 P(b2)A1P(b21)B1 where A0, A1, A, C0, C, B1, and B are matrices that describe the transition rates among the various states (n1, n2). 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 (n1, n2) = 1.

  7. Mean profit rate of the systemJ(b1, b2) J(b1, b2)  p2  [production costs in M1 and M2]  [costs of purchasing from subcontractors in M1 and M2]  [inventory costs in M1 and M2]  a function of the equilibrium probabilities • Coordination • FULL: Perform exhaustive search to track down values for b1 and b2that jointly maximize the mean profit rate of the system. • PARTIAL: • Factory M2 determines a base stock b2 which maximizes the mean profit rate by considering its own costs and profits. Factory M2 is an M/M/1/b2 queue. • Factory M1 uses the individually optimal value b2 to estimate its demand rate and to compute a base stock b1 which, again, maximizes its own profit rate.

  8. Numerical comparisons Standard parameters:  5, 12 6.25, p1 70, p2 100, c1 50, c2 10, s1 60, s2 90, h1 3, h2 8

  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 n0 = nF+ (c - nB)

  10. Relationship between the original and closed systems

  11. Parameters p unit profit h unit holding cost rate b unit backlog cost rate iC inspection cost per outgoing item rCrework/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 Mean quality cost per outgoing item Q = iC + rC (1 q) + k q E[(Y t)2, given that Y is acceptable] Mean profit rate J(, m, s) = pTH hH  bB QTH/q TH = throughput , H = average inventory, B = average backlog ms + c base stock + base backlog

  12. Assumption: The equivalent system is of the Jackson type Let (n0, n1, …, nN) be the vector whose entries are the items in each machine. Then ms + c base stock + base backlog U [U0U1 … UN ], UUΠ, Π[pij] matrix of part-routing probabilities Then: HE [nH] sP(n0c) + (s+1)P(n0c1) +…+ mP(n0 0) BE [nB]  1P(n0c1) + 2P(n0c2) +…+ cP(n0 0)

  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 sis 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 G0 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. Numerical comparisons • 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.

  16. Conclusions • 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.

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