Solution Procedures for
3.5 Concluding Remarks
In this chapter, first a dynamic programming formulation to minimize total unweighted earliness and tardiness penalties on a single machine is developed.
This formulation leads to an incomplete dynamic programming and a heuristic based on the incomplete dynamic programming is presented in the following sections. The heuristic also involves a simple interchange procedure and five well known dominance properties.
The heuristic was tested in two parts. In the first part, there are 750 randomly generated problems for problem sizes n = 8,10,12,14,16,18. For this set of problems, the optimal solutions are obtained from the solution of the mixed- integer program for the problem l|dj| ^ ^ (jE'j + Tj). The average relative error of our heuristic is 8.6% and it outperforms all other dispatching rules used in the study. Furthermore, the percentage of time the optimal solutions found by the heuristic is over 25%. For these problems, two problem parameters, namely relative range of due dates and average tardiness factor, are also changed from 0.2 to 1.0 with 0.2 increments. It is seen that our heuristic performs well when the value of average tardiness factor is large for any value of the relative range of due dates.
In the second part of the study, we generated 625 random problems for n = 20,25,30,35,40 for which we could not obtain the optimal solutions. These
Chapter 3. Solution Procedures for l|c?j| Yi{Ej + Tj) 72
set of problems are evaluated with respect to two well known heuristics from the literature. The computational experience shows that our heuristic outperforms the heuristic of Kim and Yano (1987) in all respects. However, the performance of our heuristic is not better than that of Fry et al. (1990) considering the relative errors. But since the latter one is an exponential time algorithm, this is what we expect from the computational experiments.
The development of an exact dynamic programming which is still pseudo
polynomial for this class of scheduling problems remains as a further research topic. Since this class of problems are known to be MV-hard in the ordinary sense, the development of such a dynamic programming may lead to fully-polynomial approximation schemes.
Chapter 4
l \ d j \ T . { E j - \ - T j )
with Special Structures for Distinct Due Dates
Since the problem \\dj\Y^[Ej + Tj) is A/’P-hard in its general form, it may be beneficial to deal with some special cases which will lead to efficient solution procedures for the general case. Considering this fact, we deal with the problems in which we can restrict the structure of the due dates by applying special due date assignment rules. These different due date structures can be viewed as the priority rules for the jobs since they determine the jobs’ urgency in the shop.
Indeed, these rules are studied in the literature but as the due date assignment rules in the multi-machine environments (see for example Baker (1984)). In recent years, some research have been evolved that analyze different due date assignment rules for the single machine scheduling problems in which one of the aims is to determine the optimal due dates, such as Cheng (1984) and Quaddus (1987).
Since the optimality criteria like earliness and tardiness depend on due dates, the assignment rule of the due dates affects the efficiency of the solution procedures developed. Conway et al. (1967) reported four rules to assign due dates in a job
73
shop environment where as Baker (1984) extended these rules to more complex ones.
The importance of the due date assignment rules for the problem that we deal with is that, as Baker and Scudder (1990) states, these type of models may be structurally less complicated than the general model with distinct due dates.
Furthermore, the solution procedures provided for these restricted models may be helpful in the analysis of the general model. Baker and Scudder (1990) have given the following two rules for determining the distinct due dates which relate to the processing times:
• Equal-Slack(SLK): dj = Pj + q
• Total-Work-Content(TWC): dj = kpj
Chapter 4. l\dj\ + Tj) with Special Structures for Distinct Due Dates 74
As its name implies, the SLK rule gives an equal slack for every job, hence their risk to be tardy and early is equal whereas their due dates differ according to their processing times. As it will be seen in Section 4.1, this rule provides a nice structure of the problem in certain cases. The second rule, TW C rule, gives to a job an allowance which is a multiple of its processing time. Hence, a job can wait (k — l ) pj time units before it is processed and to be completed on-time. These rules result in reasonable and attainable due dates since they provide a greater allowance for a job with larger processing time among all the jobs.
In the rest of this chapter, we consider these two rules and the results are presented in the following sections. In Section 4.H , we study the structure of the problem l\dj\Y^[Ej -}- Tj) when due dates are given according to SLK rule. The problem is analyzed in Section 4.1.1. We discuss that this problem can be classified as restricted and unrestricted according to a constraint on q. In Section 4.1.2, the equivalence of the unrestricted case to a polynomially solvable problem is given. We further analyze the restricted case of the problem
tThis section draws heavily on Oğuz and Dinçer (1992)
Chapter 4. \\dj\ Y^{Ej + Tj) with Special Structures for Distinct Due Dates 75
in section 4.1.3 and identify some optimality conditions. Finally, in Section 4.1.4 we state that the restricted case is A/^'P-hard.
In Section 4.2, we consider the second rule. Hence our problem is 1 \dj \
where due dates are given according to the Total-Work-Content rule. In Section 4.2.1, we analyze the problem of l\dj\Yi{Ej Tj) under TW C rule and we present several properties of this problem. In Section 4.2.2, we develop a polynomial time heuristic algorithm with a very good performance. The computational experience regarding this heuristic is given in Section 4.2.3.