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Batching deteriorating items

Batching deteriorating items. Mikhail Y. Kovalyov. Belarusian State University, Minsk. Joint work with F. Al-Anzi and A. Allahverdi. Introduction. Parallel machine problem . Fractional relaxation and Arithmetic-Geometric Mean (Cauchy) inequality. Optimal solution: almost equal batch sizes.

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Batching deteriorating items

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  1. Batching deteriorating items Mikhail Y. Kovalyov Belarusian State University, Minsk Joint work with F. Al-Anzi and A. Allahverdi • Introduction. • Parallel machine problem. • Fractional relaxation and Arithmetic-Geometric Mean (Cauchy) inequality. • Optimal solution: almost equal batch sizes. • Single machine work-rework problem. • Optimality of “Last Come First Served” rule. • Fractional relaxation and AG Mean inequality. • Optimal solution: almost equal batch sizes. • 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 -> 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)=at+b and costsc(t)=ct+d. p1=as0+b, p2=a(s0+p1)+b=(1+a)p1, …,pj=(1+a)j-1p1 c1=cs0+d,c2=c(s0+p1)+d=c1+cp1, …,cj=c1+cp1((1+a)j-1-1)/a Objective: CTC=αCmax+βTC 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 machinecost and job tardiness, Computers and Operations Research • 32(2005) 1995-2012. • 2. G. Dosa, Y. He, Better online algorithms for scheduling withmachine cost, • SIAM Journal on Computing 33 (2004) 1035-1051. • 3. Y. He, S.Y. Cai, Semi-online scheduling with machinecost, • Journal of Computer Science and Technology 17 (2002) 781-787. • 4. Y.W. Jiang, Y. He, Preemptive online algorithms forscheduling with machine cost, • Acta Informatica 41 (2005) 315-340. • 5. S.S. Panwalkar, S.D. Liman, Single operation earliness-tardinessscheduling 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, ∑i=1k xi=N. A1≥0, A2≥0 A3 ≥0 A1≥0, A2≥0 A3 ≥0 CTC(x1,…,xk)=αCmax+βTC= =A1max1≤i≤k{(1+a)x_i}+A2∑i=1k(1+a)x_i –A3k+A4 . AG Mean inequality (Auguste Cauchy (1789-1857), see G. Hardy, H. Littlewood, G. Polya,Inequalities, Cambridge University Press, 1934): (y1+…+yk)/k ≥ (y1·…·yk)1/k, for any non-negative y1,…,yk. ▼ ∑i=1k(1+a)x_i ≥ k (1+a) (∑ x_i)/k =k(1+a)N/k and max1≤i≤k{(1+a)x_i} ≥ (∑i=1k(1+a)x_i)/k ≥ (1+a)N/k

  9. Parallel machine problem. ► CTC(x1,…,xk) ≥ A1(1+a)N/k+A2k(1+a)N/k-A3k+A4. ▼ If batch sizes are allowed to be fractional (rational) numbers ▼ k*: min A1(1+a)N/k+A2k(1+a)N/k-A3k, subject to k€{1,…,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.

  10. Parallel machine problem. 4. Optimal solution: almost equal batch sizes. Given k and a feasible solution (x1,…,xk); let xi=└N/k┘+zi, i=1,…,k. CTC(z1,…,zk)=B1(1+a)└N/k┘max1≤i≤k{(1+a)z_i}+ B2(1+a)└N/k┘∑i=1k(1+a)z_i –A3k+A4 . Denote r=N-k└N/k┘. We have r € {0,1,…,k-1}. Problem P1:Minimize max1≤i≤k{(1+a)z_i}, s.t. ∑i=1kzi=r, zi €Z. Problem P2:Minimize ∑i=1k (1+a)z_i, s.t. ∑i=1kzi=r, zi €Z. Theorem 1There exists an optimal solution for both problems P1 and P2 such that zi €{0,1}, i=1,…,k. CorollaryIf N/k is not integer then x*i =└N/k┘, i=1,…,r, and x*i=┌N/k┐, i=r+1,…,k.

  11. Parallel machine problem. Optimal number of batches: k*: minF(k) = C1(1+a)┌N/k┐+C2((N-k└N/k┘)(1+a)┌N/k┐+ k└N/k┘(1+a)└N/k┘)-C3k, subject tok€{1,…,N} C1≥0, C2≥0 C3 ≥0 ▼ O(N) or O(log N) by a three-section search if almost unimodal: ??

  12. 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, N=vn, n – number of defective items (x – batch size). Waiting time dependent rework times: p(t)=at+b and costsc(t)=ct+d. Objective: CRC=αCmax+βRC RC – total setup and rework cost.

  13. 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-University 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.

  14. Single machine work-rework problem 6. Optimality of “Last Come First Served” rule. Work operations Rework operations 1 2 … v … i1 i2 … xv ix s0 … s1 … ▼ 1st def. ▼ xth def. Will be reworked first waiting time for rework Solution: the number of batchesk, batch sizes x1,…,xk and the processing orderofdefective items in each rework sub-batch. Lemma 1It is optimal to rework defective items in the reversed order of theirprocessingat the work stage. Proof: By the interchange technique.

  15. Single machine work-rework problem 7. Fractional relaxation and AG Mean inequality k – number of batches, x1,…,xk – batch sizes, ∑i=1k xi=n. CRC(x1,…,xk)=αCmax+βRC= D1k+D2∑i=1k(1+a)x_i+D3. AG Mean inequality D1 € R, D2 ≥ 0 ▼ CRC(x1,…,xk) ≥ D1k+D2k(1+a)n/k+D3. ▼ If batch sizes are allowed to be fractional (rational) numbers ▼ k*: min D1k+D2k(1+a)n/k, subject to k€{1,…,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”.

  16. Single machine work-rework problem. 8. Optimal solution: almost equal batch sizes. Given k, denote q=n-k└n/k┘. We have q € {0,1,…,k-1}. StatementIf n/k is not integer, then x*i =└n/k┘, i=1,…,q, and x*i=┌n/k┐, i=r+1,…,k. E1 € R, E2 ≥ 0 Optimal number of batches: k*: minG(k) = E1k+E2((n-k└n/k┘)(1+a)┌n/k┐+ k└n/k┘(1+a)└n/k┘), subject tok€{1,…,n} ▼ O(n) or O(log n) by a three-section search if almost unimodal: ??

  17. 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 considerationof other processing environments. • Open questions: • Are functions F(k) and G(k) almost unimodal? • Are they almost unimodal in some interesting special cases?

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