Applied Soft Computing 10 (2010) 919–925 Contents lists available at ScienceDirect Applied Soft Computing journal homepage: www.elsevier.com/locate/asoc The use of a fuzzy multi-objective linear programming for solving a multi-objective single-machine scheduling problem Reza Tavakkoli-Moghaddam ∗ , Babak Javadi, Fariborz Jolai, Ali Ghodratnama Department of Industrial Engineering, College of Engineering, University of Tehran, Tehran, Iran article info Article history: Received 23 April 2007 Received in revised form 1 March 2009 Accepted 17 October 2009 Available online 25 October 2009 Keywords: Single-machine scheduling Multi-objective linear programming Fuzzy multi-objective linear programming Decision maker abstract This paper develops a fuzzy multi-objective linear programming (FMOLP) model for solving a multi- objective single-machine scheduling problem. The proposed model attempts to minimize the total weighted tardiness and makespan simultaneously. In this problem, a proposed FMOLP method is applied with respect to the overall acceptable degree of the decision maker (DM) satisfaction. A number of numer- ical examples are solved to show the effectiveness of the proposed approach. The related results are compared with the Wang and Liang’s approach. These computational results show that the proposed FMOLP model achieves lower objective functions and higher satisfaction degrees. © 2009 Elsevier B.V. All rights reserved. 1. Introduction Scheduling consists of planning and arranging jobs in an orderly sequence of operations in order to meet customer’s requirements [17]. The scheduling of jobs and the control of their flows through a production process are the most significant elements in any mod- ern manufacturing systems. The single-machine environment is the basis for other types of scheduling problems. In single-machine scheduling, there is only one machine to process all jobs optimizing the system performance measures such as makespan, completion time, tardiness, number of tardy jobs, idle times, sum of the maxi- mum earliness and tardiness [18,19]. In single-machine scheduling, most research is concerned with minimizing a single criterion. However, scheduling problems often involve more than one aspect and therefore may require multiple criteria analyses [14]. Ishi and Tada [11] considered a single- machine scheduling problem minimizing the maximum lateness of jobs with fuzzy precedence relations. A fuzzy precedence relation relaxes the crisp precedence relation and represents a satisfaction level with respect to the precedence between two jobs. Thus, the problem considers an additional objective in order to maximize the minimum satisfaction level that is obtained by the fuzzy prece- dence relations. An algorithm for determining non-dominated solutions is proposed based on a graph representation of the prece- dence relations. ∗ Corresponding author. Tel.: +98 21 82084183; fax: +98 21 88013102. E-mail address: tavakoli@ut.ac.ir (R. Tavakkoli-Moghaddam). Adamopoulos and Pappis [2] presented a fuzzy-linguistic approach to a multi-criteria sequencing problem. They considered a single machine, in which each job is characterized by fuzzy pro- cessing times. The objective was to determine the processing times of jobs and the common due as well as to sequence the jobs on the machine where penalty values are associated with due dates assigned, earliness, and tardiness. Another approach to solve a multi-criteria single-machine scheduling problem is presented by Lee et al. [12]. They used linguistic values to evaluate each criterion (e.g., very poor, poor, fair, good, and very good) and to represent its relative weights (e.g., very unimportant, unimportant, moder- ately important, important, and very important). Also, a tabu search method is used as a stochastic tool to find the near-optimal solution for an aggregated fuzzy objective function. Chanas and Kasperski [6] considered two single-machine scheduling problems with fuzzy processing times and fuzzy due dates. They defined the fuzzy tardiness of a job in a given sequence as a fuzzy maximum of zero and the difference between the fuzzy completion time and the fuzzy due date of this job. In the first problem, they minimized the maximal expected value of a fuzzy tardiness. In the second one, they considered minimizing the expected value of a maximal fuzzy tardiness. Chanas and Kasper- ski [7] considered the single-machine scheduling problem with parameters given in the form of fuzzy numbers. It is assumed that the optimal schedule in such a problem cannot be determined pre- cisely. In their paper, it is shown how to calculate the degrees of possible and necessary optimality of a given schedule in one of the special cases of single-machine scheduling problems. Azizoglu et al. [4] studied the bi-criteria scheduling problem of minimizing the maximum earliness and the number of tardy 1568-4946/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.asoc.2009.10.010