Int J Adv Manuf Technol (2014) 71:377–380 DOI 10.1007/s00170-013-5466-z ORIGINAL ARTICLE A counterexample to a proposed dynamic programming algorithm for optimal bid construction in an auction-based fully distributed manufacturing system Eiji Mizutani Received: 22 January 2013 / Accepted: 28 October 2013 / Published online: 27 November 2013 © Springer-Verlag London 2013 Abstract The paper by Veeramani and Wang (Int J Adv Manuf Technol 28:541–550, 2006) published in this journal offers a general bid construction scheme for minimizing the job flow time in auction-based manufacturing control and claims the optimality of the procedure. The purpose of this note is to provide a small example, in which their proposed new method fails to produce an optimal solution. We also address efficient correct algorithms so as to protect the read- ers from mistakenly believing that their new procedure is a better way of computing solutions. Keywords Bid construction scheme · Forward dynamic programming · Setup (or changeover) times · Job class scheduling 1 A counterexample and incorrect and correct methods Section 4 of the paper of Veeramani and Wang “Bid con- struction scheme for job flow time reduction in auction- based fully-distributed manufacturing systems” appeared in this journal [1] proposes a new forward dynamic pro- gramming (DP) algorithm for optimal bid construction. In conforming to the scheduling theory, it is a procedure for solving 1|s fg | ∑ C j , a single-machine scheduling prob- lem [4] that involves changeover (or setup) times between job classes (or families) for minimizing the total job com- pletion time (or flow time). In the problem 1|s fg | ∑ C j ,a given set of N jobs are partitioned into disjoint F families: E. Mizutani () Department of Industrial Management, National Taiwan University of Science and Technology, Taiwan, 106 Taipei, Republic of China e-mail: eiji@mail.ntust.edu.tw N= ∑ F f =1 N f , where N f is the number of jobs in family f , and those N f jobs within each family f are ordered accord- ing to the shortest processing time (SPT) first rule (e.g., see [6, p. 267]; [5, p. 800]). To denote the ordered jobs in each family, we use the notation f j for the j th job in fam- ily f . The problem data include p(f j ), the processing time of job f j , and s(f, g), the changeover (or setup) time from family f to family g. 1.1 Example for N = 4 and F = 2 As a counterexample, we consider an instance of only two job families (F = 2), families a and b; family a has only one job (N a = 1), whereas family b has three jobs (N b = 3), hence four jobs in total (N = 4). The data of process- ing time p(f j ) and setup time s(f, g) are summarized in Table 1. The posed example is small enough to list all four admissible job sequences, as shown in Table 2, where ∑ C j is evaluated for each, and Seq 1 therein is the optimal sequence: a 1 -b 1 -b 2 -b 3 with ∑ C j (Seq 1 ) = 59. 1.2 An incorrect procedure As a general solver to 1|s fg | ∑ C j , Veeramani and Wang propose a new forward DP, defining C(n 1 ,n 2 , ..., n F ,f) 1 as the optimal cost-so-far value function when the n i th job is completed for each family i (i = 1, ..., F ), and f is the family of the last job. For our later demonstration, we repro- duce their forward DP formulation below for the two-family case (F = 2) using slightly different notations: 1 There is a typographical error in Eq. (2) on p. 545 in [1], where the arguments of the optimal value function C(.) on the right-hand side must be n ′ i (rather than n i ) as C(n ′ 1 ,n ′ 2 , ..., n ′ F , a).