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C. E.</a>; <a href="/authors/wu.wen-hsiang" title="Author Profile">Wu, Wen-Hsiang</a>; <a href="/authors/cheng.shuenn-ren" title="Author Profile">Cheng, Shuenn-Ren</a>; <a href="/authors/wu.chin-chia" title="Author Profile">Wu, Chin-Chia</a></div> <h2 class="title"> <strong>Two-agent scheduling with position-based deteriorating jobs and learning effects.</strong> <i>(English)</i> <a class="label nowrap" href="/1231.90182">Zbl 1231.90182</a> </h2> <div class="source"> <a href="/serials/331" title="Journal Profile">Appl. Math. Comput.</a> <a href="/?q=in%3A281099" title="Articles in this Issue">217, No. 21, 8804-8824 (2011)</a>. </div> <div class="abstract">Summary: Scheduling with deteriorating jobs and learning effects has been widely studied. However, multi-agent scheduling with simultaneous considerations of deteriorating jobs and learning effects has hardly been considered until now. In view of this, we consider a two-agent single-machine scheduling problem involving deteriorating jobs and learning effects simultaneously. In the proposed model, given a schedule, we assume that the actual processing time of a job of the first agent is a function of position-based learning while the actual processing time of a job of the second agent is a function of position-based deterioration. The objective is to minimize the total weighted completion time of the jobs of the first agent with the restriction that no tardy job is allowed for the second agent. We develop a branch-and-bound and several simulated annealing algorithms to solve the problem. Computational results show that the proposed algorithms are efficient in producing near-optimal solutions.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A5912686">Cited in <strong>24</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A90B35" title="MSC2020">90B35</a> </td> <td class="space"> Deterministic scheduling theory in operations research </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A90C59" title="MSC2020">90C59</a> </td> <td class="space"> Approximation methods and heuristics in mathematical programming </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Ascheduling">scheduling</a>; <a href="/?q=ut%3Atwo-agent">two-agent</a>; <a href="/?q=ut%3Asimulated+annealing">simulated annealing</a>; <a href="/?q=ut%3Aposition-based+learning">position-based learning</a>; <a href="/?q=ut%3Aposition-based+deteriorating">position-based deteriorating</a></div> <!-- Modal used to show zbmath metadata in different output formats--> <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;"> <!-- alert for connection errors etc --> </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function"> <!-- Button trigger metadata modal --> <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1231.90182" data-ciurl="/ci/05912686" data-biburl="/bibtex/05912686.bib" data-amsurl="/amsrefs/05912686.bib" data-xmlurl="/xml/05912686.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/05912686.pdf" title="Zbl 1231.90182 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1016/j.amc.2011.04.005" aria-label="DOI for “Two-agent scheduling with position-based deteriorating jobs and learning effects”" title="10.1016/j.amc.2011.04.005">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Alidaee, B.; Womer, N. K., Scheduling with time dependent processing times: review and extensions, Journal of the Operational Research Society, 50, 711-720 (1999) · <a href="/1054.90542" class="nowrap">Zbl 1054.90542</a></td> </tr><tr> <td>[2]</td> <td class="space">Agnetis, A.; Mirchandani, P. B.; Pacciarelli, D.; Pacifici, A., Scheduling problems with two competing agents, Operations Research, 52, 229-242 (2004) · <a href="/1165.90446" class="nowrap">Zbl 1165.90446</a></td> </tr><tr> <td>[3]</td> <td class="space">Agnetis, A.; Pacciarelli, D.; Pacifici, A., Multi-agent single machine scheduling, Annals of Operations Research, 150, 3-15 (2007) · <a href="/1144.90375" class="nowrap">Zbl 1144.90375</a></td> </tr><tr> <td>[4]</td> <td class="space">Bachman, A.; Cheng, T. C.E.; Janiak, A.; Ng, C. T., Scheduling start time dependent jobs to minimize the total weighted completion time, Journal of the Operational Research Society, 53, 6, 688-693 (2002) · <a href="/1059.90063" class="nowrap">Zbl 1059.90063</a></td> </tr><tr> <td>[5]</td> <td class="space">Bachman, A.; Janiak, A., Scheduling jobs with position-dependent processing times, Journal of the Operational Research Society, 55, 257-264 (2004) · <a href="/1095.90033" class="nowrap">Zbl 1095.90033</a></td> </tr><tr> <td>[6]</td> <td class="space">Baker, K. R.; Smith, J. C., A multiple-criterion model for machine scheduling, Journal of Scheduling, 6, 7-16 (2003) · <a href="/1154.90406" class="nowrap">Zbl 1154.90406</a></td> </tr><tr> <td>[7]</td> <td class="space">Ben-Arieh, D.; Maimon, O., Annealing method for PCB assembly scheduling on two sequential machines, International Journal of Computer Integrated Manufacturing, 5, 361-367 (1992)</td> </tr><tr> <td>[8]</td> <td class="space">Biskup, D., Single-machine scheduling with learning considerations, European Journal of Operational Research, 115, 173-178 (1999) · <a href="/0946.90025" class="nowrap">Zbl 0946.90025</a></td> </tr><tr> <td>[9]</td> <td class="space">Biskup, D., A state-of-the-art review on scheduling with learning effect, European Journal of Operational Research, 188, 315-329 (2008) · <a href="/1129.90022" class="nowrap">Zbl 1129.90022</a></td> </tr><tr> <td>[10]</td> <td class="space">Browne, S.; Yechiali, U., Scheduling deteriorating jobs on a single processor, Operations Research, 38, 495-498 (1990) · <a href="/0703.90051" class="nowrap">Zbl 0703.90051</a></td> </tr><tr> <td>[11]</td> <td class="space">Chang, P.-C.; Chen, S.-H.; Mani, V., A note on due-date assignment and single machine scheduling with a learning/aging effect, International Journal of Production Economics, 117, 1, 142-149 (2009)</td> </tr><tr> <td>[12]</td> <td class="space">Chen, Z. L., A note on single-processor scheduling with time-dependent execution times, Operations Research Letters, 17, 127-129 (1995) · <a href="/0841.90072" class="nowrap">Zbl 0841.90072</a></td> </tr><tr> <td>[13]</td> <td class="space">Cheng, T. C.E.; Ding, Q., The complexity of single machine scheduling with release times, Information Processing Letters, 65, 2, 75-79 (1998) · <a href="/1338.68096" class="nowrap">Zbl 1338.68096</a></td> </tr><tr> <td>[14]</td> <td class="space">Cheng, T. C.E.; Wang, G., Single machine scheduling with learning effect considerations, Annals of Operations Research, 98, 273-290 (2000) · <a href="/0967.68019" class="nowrap">Zbl 0967.68019</a></td> </tr><tr> <td>[15]</td> <td class="space">Cheng, T. C.E.; Ding, Q., Single machine scheduling with step-deteriorating processing times, European Journal of Operational Research, 134, 623-630 (2001) · <a href="/0984.90014" class="nowrap">Zbl 0984.90014</a></td> </tr><tr> <td>[16]</td> <td class="space">Cheng, T. C.E.; Ding, Q.; Lin, B. M.T., A concise survey of scheduling with time-dependent processing times, European Journal of Operational Research, 152, 1-13 (2004) · <a href="/1030.90023" class="nowrap">Zbl 1030.90023</a></td> </tr><tr> <td>[17]</td> <td class="space">Cheng, T. C.E.; Ng, C. T.; Yuan, J. J., Multi-agent scheduling on a single machine to minimize total weighted number of tardy jobs, Theoretical Computer Science, 362, 273-281 (2006) · <a href="/1100.68007" class="nowrap">Zbl 1100.68007</a></td> </tr><tr> <td>[18]</td> <td class="space">Cheng, T. C.E.; Ng, C. T.; Yuan, J. J., Multi-agent scheduling on a single machine with max-form criteria, European Journal of Operational Research, 188, 603-609 (2008) · <a href="/1129.90023" class="nowrap">Zbl 1129.90023</a></td> </tr><tr> <td>[19]</td> <td class="space">Della Croce, F.; Narayan, V.; Tadei, R., The two-machine total completion time flow shop problem, European Journal of Operational Research, 90, 227-237 (1996) · <a href="/0916.90148" class="nowrap">Zbl 0916.90148</a></td> </tr><tr> <td>[20]</td> <td class="space">Eren, T.; Guner, E., A bicriteria flowshop scheduling with a learning effect, Applied Mathematical Modelling, 32, 1719-1733 (2008) · <a href="/1145.90382" class="nowrap">Zbl 1145.90382</a></td> </tr><tr> <td>[21]</td> <td class="space">Gupta, J. N.D.; Gupta, S. K., Single facility scheduling with nonlinear processing times, Computers and Industrial Engineering, 14, 387-393 (1988)</td> </tr><tr> <td>[22]</td> <td class="space">Hall, N. G.; Posner, M. E., Generating experimental data for computation testing with machine scheduling applications, Operations Research, 854-865 (2001) · <a href="/1163.90490" class="nowrap">Zbl 1163.90490</a></td> </tr><tr> <td>[23]</td> <td class="space">Janiak, A.; Rudek, R., A new approach to the learning effect: Beyond the learning curve restrictions, Computers and Operations Research, 35, 3727-3736 (2008) · <a href="/1170.90392" class="nowrap">Zbl 1170.90392</a></td> </tr><tr> <td>[24]</td> <td class="space">Kirkpatrick, S.; Gelatt, C.; Vecchi, M., Optimization by simulated annealing, Science, 220, 671-680 (1983) · <a href="/1225.90162" class="nowrap">Zbl 1225.90162</a></td> </tr><tr> <td>[25]</td> <td class="space">Lee, W. C.; Wu, C. C., Minimizing total completion time in a two-machine flowshop with a learning effect, International Journal of Production Economics, 88, 85-93 (2004)</td> </tr><tr> <td>[26]</td> <td class="space">Liu, P.; Tang, L., Two-agent scheduling with linear deteriorating jobs on a single machine, Lecture Notes in Computer Science, 5092, 642-650 (2008) · <a href="/1148.90324" class="nowrap">Zbl 1148.90324</a></td> </tr><tr> <td>[27]</td> <td class="space">Liu, P.; Zhou, X. Y.; Tang, L. X., Two-agent single-machine scheduling with position-dependent processing times, International Journal of Advanced Manufacturing Technology, 48, 325-331 (2010)</td> </tr><tr> <td>[28]</td> <td class="space">Lee, W. C.; Wang, W. J.; Shiau, Y. R.; Wu, C. C., A single-machine scheduling problem with two-agent and deteriorating jobs, Applied Mathematical Modelling, 34, 10, 3098-3107 (2010) · <a href="/1201.90080" class="nowrap">Zbl 1201.90080</a></td> </tr><tr> <td>[29]</td> <td class="space">Lee, W. C.; Chen, S. K.; Wu, C. C., Branch-and-bound and simulated annealing algorithms for a two-agent scheduling problem, Expert Systems with Applications, 37, 6594-6601 (2010)</td> </tr><tr> <td>[30]</td> <td class="space">Lee, W. C.; Chen, S. K.; Chen, C. W.; Wu, C. C., A two-machine flowshop problem with two agents, Computers and Operations Research, 38, 98-104 (2011) · <a href="/1231.90204" class="nowrap">Zbl 1231.90204</a></td> </tr><tr> <td>[31]</td> <td class="space">Nawaz, M.; Enscore, E. E.; Ham, I., A heuristic algorithm for the m-machine, n-job flow-shop sequencing problem, Omega, 11, 91-95 (1983)</td> </tr><tr> <td>[32]</td> <td class="space">Ng, C. T.; Cheng, T. C.E.; Yuan, J. J., A note on the complexity of the problem of two-agent scheduling on a single machine, Journal of Combinatorial Optimization, 12, 387-394 (2006) · <a href="/1126.90027" class="nowrap">Zbl 1126.90027</a></td> </tr><tr> <td>[33]</td> <td class="space">Sundararaghavan, P. S.; Kunnathur, A., Single machine scheduling with start time dependent processing time: some solvable cases, European Journal of Operational Research, 78, 3, 394-403 (1994) · <a href="/0816.90088" class="nowrap">Zbl 0816.90088</a></td> </tr><tr> <td>[34]</td> <td class="space">Sun, L., Single-machine scheduling problems with deteriorating jobs and learning effects, Computers and Industrial Engineering, 57, 843-846 (2009)</td> </tr><tr> <td>[35]</td> <td class="space">Wang, J. B.; Xia, Z. Q., Flow-shop scheduling with a learning effect, Journal of the Operational Research Society, 56, 1325-1330 (2005) · <a href="/1082.90041" class="nowrap">Zbl 1082.90041</a></td> </tr><tr> <td>[36]</td> <td class="space">Wang, J. B., A note on scheduling problems with learning effect and deteriorating jobs, International Journal of Systems Science, 37, 2, 827-833 (2006) · <a href="/1126.90347" class="nowrap">Zbl 1126.90347</a></td> </tr><tr> <td>[37]</td> <td class="space">Wang, J. B., Single-machine scheduling problems with the effects of learning and deterioration, Omega, 35, 397-402 (2007)</td> </tr><tr> <td>[38]</td> <td class="space">Wang, J. B.; Cheng, T. C.E., Scheduling problems with the effects of deterioration and learning, Asia-Pacific Journal of Operational Research, 24, 2, 1-17 (2007) · <a href="/1121.90066" class="nowrap">Zbl 1121.90066</a></td> </tr><tr> <td>[39]</td> <td class="space">Wang, X.; Cheng, T. C.E., Single-machine scheduling with deteriorating jobs and learning effects to minimize the makespan, European Journal of Operational Research, 178, 57-70 (2007) · <a href="/1110.90045" class="nowrap">Zbl 1110.90045</a></td> </tr><tr> <td>[40]</td> <td class="space">Wang, J. B., Single-machine scheduling with general learning functions, Computers and Mathematics with Applications, 56, 1941-1947 (2008) · <a href="/1165.90470" class="nowrap">Zbl 1165.90470</a></td> </tr><tr> <td>[41]</td> <td class="space">Wang, J. B., Single machine scheduling with past-sequence-dependent setup times and time-dependent learning effect, Computers and Industrial Engineering, 55, 3, 584-591 (2008)</td> </tr><tr> <td>[42]</td> <td class="space">Wang, J. B., Single machine scheduling with a time-dependent learning effect and deteriorating jobs, Journal of the Operational Research Society, 60, 583-586 (2009) · <a href="/1163.90515" class="nowrap">Zbl 1163.90515</a></td> </tr><tr> <td>[43]</td> <td class="space">Wang, J. B., Single-machine scheduling with a sum-of-actual-processing-time-based learning effect, Journal of the Operational Research Society, 61, 172-177 (2010) · <a href="/1193.90115" class="nowrap">Zbl 1193.90115</a></td> </tr><tr> <td>[44]</td> <td class="space">Wang, J. B.; Guo, Q., A due-date assignment problem with learning effect and deteriorating jobs, Applied Mathematical Modelling, 34, 309-313 (2010) · <a href="/1185.90099" class="nowrap">Zbl 1185.90099</a></td> </tr><tr> <td>[45]</td> <td class="space">Wang, J. B.; Liu, L. L., Two-machine flow shop problem with effects of deterioration and learning, Computers and Industrial Engineering, 57, 1114-1121 (2009)</td> </tr><tr> <td>[46]</td> <td class="space">Wang, J. B.; Sun, L.; Sun, L., Single machine scheduling with exponential sum-of-logarithm-processing-times based learning effect, Applied Mathematical Modelling, 34, 2813-2819 (2010) · <a href="/1201.90088" class="nowrap">Zbl 1201.90088</a></td> </tr><tr> <td>[47]</td> <td class="space">Xu, D.; Yin, Y.; Li, H., Comments on a note on minimizing maximum lateness in an m-machine scheduling problem with a learning effect, Applied Mathematics and Computation, 217, 2, 939-943 (2010) · <a href="/1197.90238" class="nowrap">Zbl 1197.90238</a></td> </tr><tr> <td>[48]</td> <td class="space">Yang, S.-J.; Yang, D.-L., Minimizing the makespan on single-machine scheduling with aging effect and variable maintenance activities, Omega, 38, 6, 528-533 (2010)</td> </tr><tr> <td>[49]</td> <td class="space">Yang, S.-J.; Yang, D.-L., Single-machine scheduling problems with aging/deteriorating effect under an optional maintenance activity consideration, INFOR, 48, 171-179 (2010) · <a href="/1548.90235" class="nowrap">Zbl 1548.90235</a></td> </tr><tr> <td>[50]</td> <td class="space">Yang, S.-J.; Yang, D.-L., Minimizing total completion time in single-machine scheduling with aging/deteriorating effects and deteriorating maintenance activities, Computers and Mathematics with Applications, 60, 2161-2169 (2010) · <a href="/1205.90141" class="nowrap">Zbl 1205.90141</a></td> </tr><tr> <td>[51]</td> <td class="space">Yang, S. J., Single-machine scheduling problems with both start-time dependent learning and position dependent aging effect under deteriorating maintenance consideration, Applied Mathematics and Computation, 217, 7, 3321-3329 (2010) · <a href="/1202.90149" class="nowrap">Zbl 1202.90149</a></td> </tr><tr> <td>[52]</td> <td class="space">Yang, S. H.; Wang, J. B., Minimizing total weighted completion time in a two-machine flow shop scheduling under simple linear deterioration, Applied Mathematics and Computation, 217, 9, 4819-4826 (2011) · <a href="/1230.90104" class="nowrap">Zbl 1230.90104</a></td> </tr><tr> <td>[53]</td> <td class="space">Yin, Y.; Xu, D.; Sun, K.; Li, H., Some scheduling problems with general position-dependent and time-dependent learning effects, Information Sciences, 179, 2416-2425 (2009) · <a href="/1166.90342" class="nowrap">Zbl 1166.90342</a></td> </tr><tr> <td>[54]</td> <td class="space">Yuan, J. J.; Shang, W. P.; Feng, Q., A note on the scheduling with two families of jobs, Journal of Scheduling, 8, 537-542 (2005) · <a href="/1123.90040" class="nowrap">Zbl 1123.90040</a></td> </tr><tr> <td>[55]</td> <td class="space">Zhao, C.-L.; Tang, H.-Y., Single machine scheduling with general job-dependent aging effect and maintenance activities to minimize makespan, Applied Mathematical Modelling, 34, 837-841 (2010) · <a href="/1185.90106" class="nowrap">Zbl 1185.90106</a></td> </tr><tr> <td>[56]</td> <td class="space">Zhang, X.; Yan, G., Single-machine group scheduling problems with deteriorated and learning effect, Applied Mathematics and Computation, 216, 4, 1259-1266 (2010) · <a href="/1187.90147" class="nowrap">Zbl 1187.90147</a></td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching. </div> </div></article> </div></div> </div> </div> <div class="clearfix"></div> </div> </div> <div id="foot"><div class="copyright"> © 2025 <a target="fiz" href="https://www.fiz-karlsruhe.de/en">FIZ Karlsruhe GmbH</a> <a href="/privacy-policy/">Privacy Policy</a> <a href="/legal-notices/">Legal Notices</a> <a href="/terms-conditions/">Terms & Conditions</a> <div class="info"> <ul class="nav"> <li class="mastodon"> <a href="https://mathstodon.xyz/@zbMATH" target="_blank" class="no-new-tab-icon"> <img src="/static/mastodon.png" title="zbMATH at Mathstodon (opens in new tab)" alt="Mastodon logo"> </a> </li> </ul> </div> </div> <div class="clearfix" style="height: 0px;"></div> </div> </div> <script src="https://static.zbmath.org/contrib/jquery/1.9.1/jquery.min.js"></script> <script src="https://static.zbmath.org/contrib/jquery-caret/1.5.2/jquery.caret.min.js"></script> <script src="/static/js/jquery-ui-1.10.1.custom.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap/v3.3.7zb1/js/bootstrap.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-lightbox/v0.7.0/bootstrap-lightbox.min.js"></script> <script src="https://static.zbmath.org/contrib/retina/unknown/retina.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-select/v1.13.14/js/bootstrap-select.min.js"></script> <script> var SCRIPT_ROOT = ""; </script> <script src="/static/scripts.js?v=20240926"> </script> <script src="https://static.zbmath.org/contrib/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script> <script type="text/x-mathjax-config"> MathJax.Hub.Config({ "HTML-CSS": { preferredFont: "TeX", availableFonts: [ "STIX", "TeX" ], linebreaks: { automatic: true }, EqnChunk: (MathJax.Hub.Browser.isMobile ? 10 : 50) }, tex2jax: { processEscapes: true, ignoreClass: "tex2jax_ignore|dno" }, TeX: { Macros: { Aut: "\\operatorname{Aut}", Hom: "\\operatorname{Hom}" }, noUndefined: { attributes: { mathcolor: "#039", //"red", mathbackground: "white", //"#FFEEEE", mathsize: "90%" } } }, messageStyle: "none" }); </script> <script type="text/javascript"> $(document).ready(function() { $("#MathInput").stop(true, true).keyup(function() { $.ajax({ url: "/mwsq/", type: "POST", data: { query : $("#MathInput").val() }, dataType: "text" }) .done(function(xml) { $("#MathPreview").html(xml); $(window).resize(); }); }); var press = jQuery.Event("keyup"); press.ctrlKey = false; press.which = 40; $("#MathInput").trigger(press); }); </script> <div id="new_tab_icon" style="display: none"> <span class="glyphicon glyphicon-new-window" aria-hidden="true"></span><span class="sr-only">(opens in new tab)</span></div> </body> </html>