1. 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

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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)

8. 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)

9. 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)

10. 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.

11. 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

12. 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.

13. 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)

14. 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.

15. 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

16. 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

17. 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

18. 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}

19. 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

20. 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!