Batching Work and Rework Processes on Dedicated Facilities to Minimize the Makes Irina V. Gribkovskaia Molde University College, Norway Sergey Kovalev Ecole des Mines des Saint Etienne, France Frank Werner Otto-von-Guericke-University Magdeburg, Germany EURO XXIV, Lisbon, July 11 – 14, 2010

Content of the talk Introduction Related Literature Properties of an Optimal Solution An LP Formulation An O(log K) Solution Approach A per Time Unit Cost Minimization Problem

1. Introduction - imperfect production of a single product - product consists of a sufficiently large number of units (continuously divisible) - two facilties: a main facility (machine 1): original production a re-manufacturing facility (machine 2): re-manufacturing of defective items pi - processing time of one unit of the product on machine i, i = 1,2 s - inspection time of any batch t - transportation time of the batch to the re-manufacturing facility

Two types of inspections: Assumptions: - fraction defective (1/v) is the same for any batch (see e.g. Gitlow et al. (1989) ) - re-manufacturing process is perfect Two types of inspections: on-line: no unit can be proceesed on machine 1 while the inspection is in progress off-line: machine 1 is available for production during the inspection off-line inspections are a special case of on-line inspections: set t:=t+s and s:=0

Given: Problem P-Time: - demand for N good quality units - upper bound K on the number of batches Problem P-Time: Find a partition of the set of N units into at most K batches, a sequence of these batches on machine 1 and a sequence of the corresponding batches of defective units on machine 2 such that the makespan is minimized assumption: s, t, p1, p2, N, v, K integer motivation: make-to-order toy and kitchenware production

2) Related Literature 3 categories of problems of batching work and rework processes: 1) scheduling with batching and lot-sizing or lot-streaming 2) optimal lot-sizing for imperfect production systems 3) reverse logistics category 1: mainly deterministic and discrete problems categories 2 and 3: mainly stochastic and continuous problems

Problems of type 1: Potts and van Wassenhove, JORS (1992) Potts and Kovalyov, EJOR (2000) SURVEYS Allahverdi, Ng, Cheng, Kovalyov, EJOR (2008) Tang and Wang, Omega (2008) APPLICATIONS Problems of type 2: Rosenblatt and Lee, IEEE Transactions (1986) Groenevelt, Pintelon and Seidman, Management Sci. (1992) Liu and Yang, EJOR (1996) Dolgui, Levin, Louly, IJCIM (2005) Chiu and Chang, Omega (2005) Buscher and Lindner, C&OR (2007)

Problems of type 3: Fleischmann, Krikke, Dekker, Flapper, Omega (2000) Flapper, Fransoo, Broekmeulen, Inderfurth, Production Planning and Control (2002) mostly: work and rework processes on the same facility see e.g. Lindner, Buscher and Flapper, OvGU (2001) Teuntner and Flapper, OR Spectrum (2003) Inderfurth, Janiak, Kovalyov and Werner, C&OR (2006) Inderfurth, Kovalyov, Ng and Werner, IJPE (2007)

dedicated facilities: Teunter, Kaparis and Tang, EJOR (2008) - two independent flows of product units for manufacturing and re-manufacturing - objective: minimization of total setup and inventory holding costs P-Time is a generalization of the 2-machine flow-shop lot-streaming problem considered by Potts and Baker, OR Letters (1989) (s = t = 0)

3) Properties of an Optimal Solution Statement 1: There exists an optimal solution with the following properties: 1) machine 1 starts at time 0 and has no idle time until the completion of the last unit; 2) machine 2 has no idle time (since it starts processing until the completion of the last product unit) 3) batches of defective items are processed in the same order on machine 2 as the corresponding original batches on machine 1.

Comment: The problem with on-line inspections, whose time is a linear function of the batch size, can be reduced to the same problem with a constant inspection time. let: s(B) = s + |B| a re-set: p1 = p1 + a and s(B) = s

Parameters: s – the quality inspection time of a batch; t – the transportation time of the defective units of the same batch from machine 1 to machine 2; pi – the processing time for any unit of a product on machine i, i = 1,2; N – the demand for good quality units; 1/v – the fraction of defective units in any batch; n – the total number of defective units, n = N/v; K – the upper bound on the number of batches.

Decision variables: k – the number of batches; xi – the size of a batch sequenced i-th, i = 1,…,k. Objective function to be minimized: T(k)(x) – the makespan of a solution x = (x1,…,xk).  problem P-Time reduces to k problems LP(k)

Problem LP(k): yk --> min! subject to Mj(x) = t + np2 + js + (vp1 – p2)∑j xi+ vp1xj ≤ yk, j = 1, ..., k ∑i xi = n xj ≥ 0, j = 1,…,k.

K problems LP(k) with k+1 variables and k+1 constraints LP(k) can be solved in O(k3.5) time (see Kamarkar, Combinatorica (1984)) -> P-Time can be solved in O(K4.5) time

5. On O(log K) Solution Approach Theorem 1: For any optimal solution x = (x1, …, xk), k ε {1,..,K} of problem P-Time, the following equations are satisfied: Mj(x) = t + np2 + js + (vp1 – p2) ∑ij-1 xi + vp1xj = T*, j = 1, ..., k.  Every manufacturing and re-manufacturing is critical for problem P-Time

Introduce variables T(k) and x(k) For given k consider t + np2 + js + (vp1 – p2) ∑1j-1 xi + vp1xj = yk, j = 1, ..., k xi = n Case vp1 = p2: k* computable in O(1) time each value xj(k*) computable in O(1) time  case solvable in O(1) time

Statement 1: Problem P-Time is solvable in O(log K). Case vp1 = p2: k* computable in O(log K) time each value xj(k*) computable in O(1) time  case solvable in O(log K) time Statement 1: Problem P-Time is solvable in O(log K). Input of P-Time: N, v, s, t, K, p1, p2  polynomial algorithm must be polynomial in log max {N,v,s,t,K,p1,p2}

6. A per Time Unit Cost Minimization Problem ci – cost associated with the production of one unit of the product on machine i, i = 1,2 c3 – batch handling cost Cp.t.u. := [ c1 (v-1) n + c2 n + c3 k ] / T(k) problem P-Cost Case: p2 = vp1 can be similarly solved in O(log K) time

detailed version of this talk: Gribkovskaia, I.; Kovalev, S.; Werner, F.: Batching for Work and Rework Processes on Dedicated Facilities to Minimize the Makespan, Omega, Vol. 38, December 2010, 522 – 527. Thank you for attention!