A note on scheduling problems with competing agents and earliness minimization objectives

European Journal of Operational Research 245 (2015) 875–876

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Short Communication

A note on scheduling problems with competing agents and earliness minimization objectives Christos Koulamas∗ College of Business, Florida International University, Miami, FL 33199, USA

a r t i c l e

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a b s t r a c t

Article history: Received 9 January 2015 Accepted 6 April 2015 Available online 11 April 2015

We show that a number of scheduling problems with competing agents and earliness minimization objectives are equivalent to the corresponding problems with tardiness minimization objectives. © 2015 Elsevier B.V. All rights reserved.

Keyword: Scheduling

Mor and Mosheiov (2010) (to be called M–M in the sequel) analyzed scheduling problems with two competing agents and earliness minimization objectives. The purpose of this note is to point out that by invoking the well-known equivalence between earliness and tardiness problems on a single machine (see Du & Leung, 1990), we can derive the results in M–M from known results for the corresponding problems with tardiness minimization objectives. We also list additional competing agent problems with earliness minimization objectives (not considered by M–M) that can be analyzed by invoking the earliness–tardiness equivalence. Using the notation in M–M knowledge of which is assumed, for each two-agent earliness problem PRE , we can define an equivalent two-agent tardiness problem PRT in the [0, D] interval with p Aj = pAj , d Aj = D − dAj + pAj for all j = 1, . . . , nA and p Bj = pBj , d Bj = D − dBj + pBj for all j = 1, . . . , nB . Let SE be an optimal sequence of JA ∪ JB for PRE ; then, we can construct an optimal sequence ST for PRT so that the job in position j in SE is sequenced in position nA + nB − j + 1 in ST . As  where [j] denotes shown in Du and Leung (1990), E[j] = T[n +n −j+1] A

B

the job in position j in a given sequence; this equivalence is also true when E[j] is replaced by fj (E[j] ) where fj (x) is a non-decreasing function of x with fj (x) = 0 for x ≤ 0. All jobs are available at time zero, with no precedence relationships among them; the machine is continuously available without any inserted idle time in the [0, D] interval. In a single-agent setting, a number of known results for tardiness-related objectives can be extended to the corresponding earliness-related objectives. The focus of this note is to utilize the earliness–tardiness equivalence for single-machine competing agent problems; our findings are summa-

rized next in Table 1 where A ⇔ B denotes that problems A, B are equivalent. Problems 1–5 were considered by M–M. Problem 1 is equivalent A B to the 1//fmax : fmax problem solved in O[(nA )2 + nB log nB ] time by Agnetis, Mirchandani, Pacciarelli, and Pacifici (2004). Problem 2 was solved by Agnetis et al. (2004) in O(nA log nA + nB log nB ) time. Prob B problem because of lem 3 is equivalent to the 1// wAj CjA : Cmax  A A B problem is ordithe common job due date; the 1// wj Cj : Cmax nary NP-hard (Agnetis et al., 2004). Problem 4 is equivalent to the  B problem which is strongly NP-hard (Leung, Pinedo, 1// wAj CjA : fmax  B & Wan, 2010). Problem 5 is a special case of the 1// CjA : fmax problem which was solved in O(nA log nA + nB log nB ) time by Agnetis et al. (2004). Problems 6–7 were not considered by M–M. Problem 6 was shown   to be ordinary NP-hard and solvable in O[(nA )4 ( j∈JA pAj + j∈JB pBj ) + nB log nB ] time by Leung et al. (2010). Problem 7 is a generalization of problem 1 to k > 2 agents and was solved by Agnetis, Pacciarelli, and Pacifici (2007) in O(n log n) time where n is the total number of jobs for all k agents.

Table 1 List of equivalent earliness–tardiness problems. 1 3 4

1/dAj = d/

Tel.: +1 305 348 3309; fax: +1 305 348 4126. E-mail address: koulamas@fiu.edu

http://dx.doi.org/10.1016/j.ejor.2015.04.006 0377-2217/© 2015 Elsevier B.V. All rights reserved.

 j∈JA

j∈JA

wAj EjA : max{EjB } ⇔ 1/dAj = d/

 j∈JA

j∈JA

wAj TjA : max{TjB }

6

 A  A 1/dAj = d/ Ej : max{EjB } ⇔ 1/dAj = d/ Tj : max{TjB } A j∈JA  A j∈JB  A B 1// Ej : fmax ⇔ 1// Tj : fmax

7

1 2 k 1 2 k 1//Emax , Emax , . . . , Emax ⇔ 1//Tmax , Tmax , . . . , Tmax

5

∗

1// max{fjA (EjA )} : max{fjB (EjB )} ⇔ 1// max{fjA (TjA )} : max{fjB (TjB )}

1// max{EjA } : max{EjB } ⇔ 1// max{TjA } : max{TjB }  A A  A A 1/dAj = dBj = d/ wj Ej : max{EjB } ⇔ 1/dAj = dBj = d/ wj Tj : max{TjB }

2

j∈JA

j∈JA

876

C. Koulamas / European Journal of Operational Research 245 (2015) 875–876

Acknowledgment We would like to thank the referees for their insightful comments which helped us improve an earlier version of this note. References Agnetis, A., Mirchandani, P. B., Pacciarelli, D., & Pacifici, A. (2004). Scheduling problems with two competing agents. Operations Research, 52, 229–242 .

Agnetis, A., Pacciarelli, D., & Pacifici, A. (2007). Multi-agent single machine scheduling. Annals of Operations Research, 150, 3–15. Du, J., & Leung, J. Y.-T. (1990). Minimizing total tardiness on one machine is NP-hard. Mathematics of Operations Research, 15, 483–495. Leung, J. Y.-T., Pinedo, M., & Wan, G. (2010). Competitive two-agent scheduling and its applications. Operations Research, 58, 458–469. Mor, B., & Mosheiov, G. (2010). Scheduling problems with two competing agents to minimize minmax and minsum earliness measures. European Journal of Operational Research, 206, 540–546.