A1256656645DLQtF

1
Batching deteriorating items
Mikhail Y. Kovalyov
Belarusian State University, Minsk
Joint work with F. Al-Anzi and A. Allahverdi

1. Introduction.
2. Parallel machine problem.
3. Fractional relaxation and Arithmetic-Geometric
   Mean (Cauchy) inequality.
4. Optimal solution almost equal batch sizes.
5. Single machine work-rework problem.
6. Optimality of Last Come First Served rule.
7. Fractional relaxation and AG Mean inequality.
8. Optimal solution almost equal batch sizes.
9. Conclusion.

2
1. Introduction.
Batching identical items (first part of decision
making)
Batches
N items
3
Introduction.
Batch sequencing on a single machine (second part
of decision making)
Setup time, setup cost
Setup time, setup cost
Setup time, setup cost
Sequential item processing, item availability.
4
Introduction.
Parallel batch processing with machine activation
costs (second part of decision making)
Each batch is assigned its own machine, so 4
batches -gt 4 machines.
Setup time, setup cost
Setup time, setup cost
Setup time, setup cost
Setup time, setup cost
Sequential item processing, item availability.
5
Introduction.

• Surveys on batch scheduling
• A. Allahverdi, J.N.D. Gupta, T. Aldowaisan, A
  review of scheduling research involving setup
  considerations, OMEGA 27 (1999) 219-239.
• 2. C.N. Potts, M.Y. Kovalyov, Scheduling with
  batching a review,
• European Journal of Operational Research 120
  (2000) 228-249.
• Surveys on scheduling with start time dependent
  processing times
• B. Alidaee, N.K. Womer, Scheduling with time
  dependent processing times Review
  and extensions, Journal of the Operational
  Research Society 50 (1999) 711-720.
• 2. T.C.E. Cheng, Q. Ding, B.M.T. Lin, A concise
  survey of scheduling with time-depen- dent
  processing times, European Journal of Operational
  Research 152 (2004) 1-13.

6
2. Parallel machine problem.
Item positions
1 2 3 4 5 6 7
Machine (batch) 1
s0
Machine (batch) 2
s0
Machine (batch) 3
s0
time
Cmax
0
Start time dependent processing times p(t)atb
and costs c(t)ctd.
p1as0b, p2a(s0p1)b(1a)p1, ,
pj(1a)j-1p1 c1cs0d, c2c(s0p1)dc1cp1,
, cjc1cp1((1a)j-1-1)/a
Objective CTCaCmaxßTC
TC total machine activation and job processing
cost.
7
Parallel machine problem.

• Literature on scheduling with machine activation
  costs
• D. Cao, M.Y. Chen, G.H. Wan, Parallel machine
  selection and job scheduling to minimize machine
  cost and job tardiness, Computers and Operations
  Research
• 32 (2005) 1995-2012.
• 2. G. Dosa, Y. He, Better online algorithms for
  scheduling with machine cost,
• SIAM Journal on Computing 33 (2004)
  1035-1051.
• 3. Y. He, S.Y. Cai, Semi-online scheduling with
  machine cost,
• Journal of Computer Science and Technology 17
  (2002) 781-787.
• 4. Y.W. Jiang, Y. He, Preemptive online
  algorithms for scheduling with machine cost,
• Acta Informatica 41 (2005) 315-340.
• 5. S.S. Panwalkar, S.D. Liman, Single operation
  earliness-tardiness scheduling with machine
  activation costs, IIE Transactions 34 (2002)
  509-513.

8
Parallel machine problem.
3. Fractional relaxation and Arithmetic-Geometric
Mean (Cauchy) inequality
k number of batches, x1,,xk batch sizes,
?i1k xiN.
A10, A20 A3 0
A10, A20 A3 0
CTC(x1,,xk)aCmaxßTC
A1max1ik(1a)x_iA2?i1k(1a)x_i A3kA4 .

AG Mean inequality (Auguste Cauchy (1789-1857),
see G. Hardy, H. Littlewood, G. Polya,
Inequalities, Cambridge University Press, 1934)
(y1yk)/k (y1yk)1/k, for any non-negative
y1,,yk.
?
?i1k(1a)x_i
k (1a) (? x_i)/k k(1a)N/k and
max1ik(1a)x_i (?i1k(1a)x_i)/k (1a)N/k

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Parallel machine problem.
? CTC(x1,,xk) A1(1a)N/kA2k(1a)N/k-A3kA4.
? If batch sizes are allowed to be fractional
(rational) numbers ?
k min A1(1a)N/kA2k(1a)N/k-A3k, subject to
k1,,N
? O(N)
or O(log N) by a three-section search if almost
unimodal
??
Optimal fractional solution each of the k
batches contains N/k items.
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Parallel machine problem.
4. Optimal solution almost equal batch sizes.
Given k and a feasible solution (x1,,xk) let
xiN/kzi, i1,,k.
CTC(z1,,zk)B1(1a)N/kmax1ik(1a)z_i

B2(1a)N/k?i1k(1a)z_i
A3kA4 .
Denote rN-kN/k. We have r 0,1,,k-1.
Problem P1 Minimize max1ik(1a)z_i, s.t.
?i1kzir, zi Z.
Problem P2 Minimize ?i1k (1a)z_i, s.t.
?i1kzir, zi Z.
Theorem 1 There exists an optimal solution for
both problems P1 and P2 such that zi 0,1,
i1,,k.
Corollary If N/k is not integer then xi N/k,
i1,,r, and xiN/k, ir1,,k.
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Parallel machine problem.
Optimal number of batches
k min F(k) C1(1a)N/kC2((N-kN/k)(1a)N/k
kN/k(1a)N/k)-C3
k, subject to k1,,N
C10, C20 C3 0
? O(N)
or O(log N) by a three-section search if almost
unimodal
??
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5. Single machine work-rework problem.
Work operations
Rework operations
1 2 v
i1 i2
xv
ix
s0
s1


? 1st def.
? xth def.
waiting time t for rework
A batch with x defective items reworked in the
order i1,,ix.
N total number of items, Nvn, n number of
defective items (x batch size).
Waiting time dependent rework times p(t)atb
and costs c(t)ctd.
Objective CRCaCmaxßRC
RC total setup and rework cost.
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Single machine work-rework problem.

• Literature on scheduling work and rework
  processes
• M. de Brito, R. Dekker, Reverse logistics a
  framework. In R. Dekker, M. Fleischmann, K.
  Inderfurth and L. N. van Wassenhove (eds.),
  Reverse Logistics - Quantitative Models for
  Closed-Loop Supply Chains, Springer, 2004, 3-27.
• S.D.P. Flapper, J.C. Fransoo, R.A.C.M.
  Broekmeulen, K. Inderfurth, Planning and control
  of rework in the process industries a review,
  Production Planning Control 1 (2002) 26-34.
• K. Inderfurth, A. Janiak, M.Y. Kovalyov, F.
  Werner, Batching work and rework pro-cesses with
  limited deterioration of reworkables, Computers
  and Operations Research 33 (2006) 1595-1605.
• K. Inderfurth, M.Y. Kovalyov, C.T. Ng, F. Werner,
  Cost minimizing scheduling of work and rework
  processes on a single facility under
  deterioration of reworkables, Interna-tional
  Journal of Production Economics 2006, to appear.
• K. Inderfurth, G. Lindner, N.P. Rahaniotis,
  Lotsizing in a production system with rework and
  product deterioration, Preprint 1/2003, Faculty
  of Economics and Manage-ment, Otto-von-Guericke-Un
  iversity Magdeburg, Germany, 2003.
• K. Inderfurth, R.H. Teunter, Production planning
  and control of closed-loop supply chains. In
  V.D.R. Guide Jr. and L.N. van Wassenhove (eds.),
  Business perspectives on closed-loop supply
  chains, Carnegie Mellon University Press, 2003,
  149-173.

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Single machine work-rework problem
6. Optimality of Last Come First Served rule.
Work operations
Rework operations
1 2 v
xv
i1 i2
ix
s0
s1


? 1st def.
? xth def.
Will be reworked first
waiting time for rework
Solution the number of batches k, batch sizes
x1,,xk and the processing order of defective
items in each rework sub-batch.
Lemma 1 It is optimal to rework defective items
in the reversed order of their processing at the
work stage. Proof By the interchange technique.
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Single machine work-rework problem
7. Fractional relaxation and AG Mean inequality
k number of batches, x1,,xk batch sizes,
?i1k xin.
CRC(x1,,xk)aCmaxßRC
D1kD2?i1k(1a)x_iD3.
AG Mean inequality
D1 R, D2 0
?
CRC(x1,,xk) D1kD2k(1a)n/kD3.
? If batch sizes are allowed to be fractional
(rational) numbers ?
k min D1kD2k(1a)n/k, subject to k1,,n
?
O(n)
or O(log n) by a three-section search if almost
unimodal
??
Optimal fractional solution each of the k
batches contains n/k defective items reworked
according to the rule Last Come First Served.
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Single machine work-rework problem.
8. Optimal solution almost equal batch sizes.
Given k, denote qn-kn/k. We have q
0,1,,k-1.
Statement If n/k is not integer, then xi n/k,
i1,,q, and xin/k, ir1,,k.
E1 R, E2 0
Optimal number of batches
k min G(k) E1kE2((n-kn/k)(1a)n/k
kn/k(1a)n/k),
subject to k1,,n
? O(n)
or O(log n) by a three-section search if almost
unimodal
??
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9. Conclusions.

• The AG Mean inequality was used to reduce
  parallel machine problem and single machine
  work-rework problem to minimizing a function of
  one variable k being the number of batches.
• In an optimal solution of either problem, there
  are at most two batch sizes obtained by rounding
  down and rounding up the total number of
  (defective) items divided by the optimal number
  of batches k.
• The study of more complex objective functions
  that include, for example, inventory holding and
  shortage costs, is an interesting topic for
  future research, as well as the consideration of
  other processing environments.
• Open questions
• Are functions F(k) and G(k) almost unimodal?
• Are they almost unimodal in some interesting
  special cases?