{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,9]],"date-time":"2026-05-09T00:40:20Z","timestamp":1778287220047,"version":"3.51.4"},"reference-count":21,"publisher":"Institute for Operations Research and the Management Sciences (INFORMS)","issue":"5","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Management Science"],"published-print":{"date-parts":[[2021,5]]},"abstract":"<jats:p> In 1964, Kenneth Lane proposed an algorithm to optimize the production schedule of a single-metal, single-processor open pit mine. For this, he proposed a policy based on varying, over time, the so-called \u201ccutoff grade\u201d\u2014or grade threshold used to determine if extracted material should be ore (processed material) or waste (thrown away). Lane\u2019s algorithm had a profound impact on the mining industry. However, though it has been used in multiple commercial software systems and has traditionally been taught to every aspiring mining engineer, it is widely considered a heuristic, and little is known regarding the quality of the solutions it produces. In this paper, we formally study Lane\u2019s problem. We show that Lane\u2019s algorithm can be viewed as an approximate dynamic programming scheme and that Lane\u2019s optimality conditions can be formally derived in two different ways: by considering a variant of the problem where the future value function is linearly approximated or by deriving the optimality conditions of a continuous-time version of the problem. We further show that Lane\u2019s algorithm can naturally be extended to this continuous-time version of the problem and that when this algorithm converges, it converges to an optimal solution. Finally, through a reformulation, we show that Lane\u2019s original problem can be solved using convex mixed-integer programming. Though hypothetical counterexamples can be constructed, computational experiments prove that Lane\u2019s algorithm can produce the optimal solution in every real-world data set tested, thereby lending solid support for its practical application. <\/jats:p><jats:p> This paper was accepted by Chung Piaw Teo, optimization. <\/jats:p>","DOI":"10.1287\/mnsc.2020.3685","type":"journal-article","created":{"date-parts":[[2020,10,2]],"date-time":"2020-10-02T11:50:06Z","timestamp":1601639406000},"page":"3087-3103","source":"Crossref","is-referenced-by-count":9,"title":["Lane\u2019s Algorithm Revisited"],"prefix":"10.1287","volume":"67","author":[{"given":"Marcos","family":"Goycoolea","sequence":"first","affiliation":[{"name":"Business School, Universidad Adolfo Ib\u00e1\u00f1ez, Pe\u00f1al\u00f3len 7941169, Chile;"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Patricio","family":"Lamas","sequence":"additional","affiliation":[{"name":"Business School, Universidad Adolfo Ib\u00e1\u00f1ez, Pe\u00f1al\u00f3len 7941169, Chile;"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/2.zoppoz.workers.dev:443\/https\/orcid.org\/0000-0003-2968-0073","authenticated-orcid":false,"given":"Bernardo K.","family":"Pagnoncelli","sequence":"additional","affiliation":[{"name":"Business School, Universidad Adolfo Ib\u00e1\u00f1ez, Pe\u00f1al\u00f3len 7941169, Chile;"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Adriana","family":"Piazza","sequence":"additional","affiliation":[{"name":"Department of Economics, Faculty of Economics and Business, University of Chile, Santiago 8330015, Chile"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"109","reference":[{"issue":"11","key":"B1","first-page":"741","volume":"111","author":"Asad M","year":"2011","journal-title":"J. South African Institute Mining Metallurgy"},{"key":"B2","doi-asserted-by":"publisher","DOI":"10.1016\/j.resourpol.2016.05.005"},{"issue":"2","key":"B3","first-page":"189","volume":"56","author":"Azimi Y","year":"2011","journal-title":"Arch. Mining Sci."},{"key":"B4","volume-title":"Dynamic Programming","author":"Bellman R","year":"1957"},{"issue":"1","key":"B7","first-page":"39\u201345","volume":"60","author":"Dagdelen K","year":"2008","journal-title":"Mining Engrg."},{"key":"B8","doi-asserted-by":"publisher","DOI":"10.1007\/BF02592064"},{"key":"B9","doi-asserted-by":"publisher","DOI":"10.1007\/s10479-012-1258-3"},{"key":"B10","doi-asserted-by":"publisher","DOI":"10.17159\/2411-9717\/2016\/v116n11a4"},{"issue":"3","key":"B11","first-page":"54","volume":"84","author":"Henning U","year":"1963","journal-title":"Canadian Mining J."},{"key":"B12","volume-title":"Open Pit Mine Planning and Design","author":"Hustrulid W","year":"2013"},{"key":"B13","doi-asserted-by":"publisher","DOI":"10.21236\/AD0672094"},{"key":"B15","doi-asserted-by":"publisher","DOI":"10.1137\/0108053"},{"key":"B16","unstructured":"King B (2001) Optimal mine scheduling policies. Unpublished PhD thesis, Royal School of Mines, Imperial College, London."},{"issue":"4","key":"B17","first-page":"811","volume":"59","author":"Lane K","year":"1964","journal-title":"Colorado School Mines Quart"},{"key":"B18","unstructured":"Lane KF, Hamilton D, Parker JJB (1984) Cutoff grades for two minerals. Proc. 18th Internat. Sympos. Appl. Comput. Oper. Res. Minerals Indust. (Institution of Mining and Metallurgy, London), 485\u2013492."},{"key":"B19","volume-title":"The Economic Definition of Ore: Cut-off Grades in Theory and Practice","author":"Lane KF","year":"2018"},{"key":"B20","doi-asserted-by":"publisher","DOI":"10.1137\/0305004"},{"key":"B21","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-017-1191-y"},{"key":"B22","first-page":"357","volume":"59","author":"Mortimer G","year":"1950","journal-title":"Trans. Inst. Mining Metallurgy"},{"issue":"6","key":"B23","first-page":"578","volume":"13","author":"Vickers E","year":"1961","journal-title":"Mining Engrg."},{"key":"B24","doi-asserted-by":"publisher","DOI":"10.1016\/0098-1354(95)87027-X"}],"container-title":["Management Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/2.zoppoz.workers.dev:443\/https\/pubsonline.informs.org\/doi\/pdf\/10.1287\/mnsc.2020.3685","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,4,2]],"date-time":"2023-04-02T18:22:19Z","timestamp":1680459739000},"score":1,"resource":{"primary":{"URL":"https:\/\/2.zoppoz.workers.dev:443\/https\/pubsonline.informs.org\/doi\/10.1287\/mnsc.2020.3685"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,5]]},"references-count":21,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2021,5]]}},"alternative-id":["10.1287\/mnsc.2020.3685"],"URL":"https:\/\/2.zoppoz.workers.dev:443\/https\/doi.org\/10.1287\/mnsc.2020.3685","relation":{},"ISSN":["0025-1909","1526-5501"],"issn-type":[{"value":"0025-1909","type":"print"},{"value":"1526-5501","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,5]]}}}