Production and transportation integration for commit-to-delivery mode with general shipping costs

Feng Li, Zhou Xu, Zhi Long Chen

Research output: Journal article publicationJournal articleAcademic researchpeer-review

9 Citations (Scopus)


We study an integrated production and transportation problem for a make-to-order manufacturing company that operates under the commit-to-delivery mode and uses third-party logistics service providers to deliver products to customers on or before certain committed delivery dates. Such third-party logistics service providers often provide various shipping modes with quantity discounts and different guaranteed shipping times. As a result, the company’s shipping costs need to be represented by general shipping cost functions that are typically nondecreasing, subadditive, and piecewise linear with shipping quantities, and nonincreasing with guaranteed shipping times. To the best of our knowledge, this paper is the first attempt to solve such an integrated production and transportation problem for the commit-to-delivery mode with general shipping costs. We prove that with general shipping costs, the problem is strongly 13-hard when the planning horizon consists of an arbitrary number of days. For the two-day problem, we show that it is ordinarily 13-hard, but is unlikely to have a fully polynomial time approximation scheme (FPTAS) unless 13 = 3. Interestingly, we find that when the unit inventory holding cost is relatively small, which is often true in practice, there exists an FPTAS for the two-day problem, the development of which hinges on a newly discovered property for minimizing the sum of two general piecewise linear functions. For the multiday problem, we develop a heuristic algorithm based on column generation, which novelly uses a dynamic program for a variant of the problem with a single customer. Results from computational experiments demonstrate that the heuristic algorithm can find near-optimal solutions with optimality gaps less than 1% in a short running time.

Original languageEnglish
Pages (from-to)1012-1029
Number of pages18
JournalINFORMS Journal on Computing
Issue number4
Publication statusPublished - Sept 2020


  • Algorithm analysis
  • Column generation
  • Commit-to-delivery mode
  • Computational complexity
  • General shipping cost functions
  • Integer programming
  • Integrated production and transportation

ASJC Scopus subject areas

  • Software
  • Information Systems
  • Computer Science Applications
  • Management Science and Operations Research


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