Using mixed graph coloring to minimize total completion time in job shop scheduling
Introduction
We consider the following job shop problem. A set of jobs J = {J1, J2, … , Jn} has to be processed by a set of machines M = {M1, M2, … , Mm}. At any time, each machine can process no more than one job, and each job can be processed by no more than one machine. Job Ji consists of ri ordered operations where and r0 = 0. For a job Jk, let i = L(k) + l where machine has to process operation vi. In a technological route of that is used for processing job Jk ∈ J, a given machine may be used more than once or not used at all. Duration of each operation vi, i ∈ {1, 2, … , L(n + 1)}, is equal to one unit, and preemption of each operation is not allowed. As a result, for each operation vi the following equality must hold: ci = si + 1 where si and ci denote starting and completion times of operation vi, respectively.
Let where Ci means completion time of job Ji: . The scheduling problem under consideration is to construct an optimal schedule for processing the jobs in the set J by machines in the set M to minimize Φ(C1, C2, … , Cn). This problem can be described by the notation when jobs may have different technological routes (a job shop problem), and by the notation when technological routes of all the jobs are identical and each machine Mk ∈ M is used in the technological route exactly once (a flow shop problem).
It is known that the problem is NP-hard, see [4]. The notation pi ∈ {0, 1} means the operation duration can be equal to either zero or one unit, provided that zero duration of operation vi, i = L(k) + l, means that machine does not process job Jk ∈ J at all. The problem is NP-hard since the problem is a special case of the problem .
Kubiak and Timkovsky [7] considered the unit time job shop scheduling problem to minimize total completion time. They considered the case of two machines and proposed an efficient algorithm to solve this special case of the problem. Timkovsky [13] considered the same problem with a general case of m machines. He discussed the complexities of the problem with respect to different criteria including total completion time. Both problems of J∣n = 2∣Φ and J∣n = 2, Pr∣Φ with two jobs are polynomially solvable for any non-decreasing objective function Φ = Φ(C1, C2, … , Cn), see [9]. Hereafter, the symbol Pr means operation preemption is allowed. Sotskov and Shakhlevich [11] and Brucker et al. [2] proved that both problem (with three machines and three jobs) and problem (with two machines and three jobs) are binary NP-hard. The above results on NP-hardness of scheduling problems have been proven with the usual assumption that all the operation durations are integers. Therefore, NP-hardness proof of the problem with integer operation durations implies NP-hardness of the problem due to their equivalence. Thus, to obtain exact solution to the problem with large numbers m ⩾ 2 or/and no ⩾ 3 we need to use some kind of an implicit enumeration technique, e.g., a branch and bound algorithm.
In this paper, we show that job shop scheduling problem of unit time operations with total completion time criterion can be modeled as finding the optimal coloring of a mixed graph. The problem of coloring of a mixed graph has been studied by Hansen et al. [5] and Sotskov et al. [10], [12]. The problem J∣pi = 1∣Cmax with makespan minimization Cmax = max{Ci : Ji ∈ J} and unit operation times may be reduced to the problem of coloring vertices of a graph or vertices of a mixed graph in minimum number of colors, i.e., with criterion MINMAX, see [8], [3], [10], [12].
The rest of this paper is organized as follows. In Section 2, we show that the problem can be represented as the problem of constructing optimal coloring of a special mixed graph with criterion MINSUM. A branch and bound algorithm for solving the problem is proposed in Section 3. In Section 4, three lower bounds on the objective function are developed. Computational analysis for randomly generated problems is conducted in Section 5. Concluding remarks are made in Section 6.
Section snippets
Reduction of the problem to mixed graph coloring
Let G = (V, A, E) be a mixed graph with an non-empty set of vertices V, a set of arcs A, and a set of edges E. N denotes the set of natural numbers. The function ϕ : V → N is called coloring of mixed graph G, if it assigns a number ϕ(vi) ∈ N called a color to each vertex vi ∈ V in such a way that arc inclusion (vi, vj) ∈ A implies inequalityand edge inclusion [vi, vj] ∈ E impliesSimilar to the case of coloring of a graph (V, Ø, E), in the mixed graph coloring problem studied by Hansen et
Branch and bound algorithm
In this section, we develop a branch and bound algorithm, called MINSUM, for solving the problem via optimal coloring of a mixed graph G = (V, A, E) provided that both properties (3), (4) hold. The algorithm MINSUM is based on the lower bounds of value (5), dominance rule, and the color-based branching procedure for resolution conflicts arising when the same color tends to be assigned to adjacent vertices in the graph (V, Ø, E). A solution tree has to be constructed in order to implicitly
Lower bounds on the objective function
In the algorithm MINSUM, three lower bounds on the value of the objective function (5) have been tested: one local lower bound, represented as LB0 (which uses only local data available at the current vertex of the solution tree), and two global lower bounds represented by LB1 and LB2. Calculation of the latter bounds needs “global data”, i.e., data located outside the current vertex of the solution tree.
If coloring process is in the state determined by vector w(j), then for coloring all the
Computational results
The algorithm MINSUM was coded in C++ and tested on PC Pentium III (600 MHz) for coloring (pseudo) random mixed graphs G of orders 70, 90, 100, 120 and 200 with equal lengths of paths (job routes), and on PC Pentium IV (2000 MHz) for coloring random mixed graphs G of orders 300, 400, 500, 600 and 750 with different lengths of paths (provided that both conditions (4), (5) hold for each randomly generated mixed graph).
Table 1, Table 2, Table 3, Table 4, Table 5 show the results of computational
Concluding remarks
The job shop scheduling problem with unit processing times is addressed with respect to total completion time criterion. It is shown that the problem can be modeled as finding the optimal coloring of a special mixed graph. The results obtained may be presented using only graph terminology which is widely used in OR literature, e.g., one can use terms vertex, path, clique, color, coloring, etc. instead of terms operation, technological route, machine, completion time, schedule, etc. Such an
References (14)
An introduction to timetabling
European Journal of Operational Research
(1985)- et al.
Total completion time minimization in two-machine job shops with unit-time operations
European Journal of Operational Research
(1996) The complexity of shop-scheduling problems with two or three jobs
European Journal of Operational Research
(1991)- et al.
NP-hardness of shop-scheduling problems with three jobs
Discrete Applied Mathematics
(1995) Is a unit-time job shop not easier than identical parallel machines?
Discrete Applied Mathematics
(1998)An efficient algorithm for the job-shop problem with two jobs
Computing
(1988)- et al.
Preemptive job-shop scheduling problems with a fixed number of jobs
Mathematical Methods of Operations Research
(1999)
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