Using mixed graph coloring to minimize total completion time in job shop scheduling

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Abstract

The problem of scheduling a set of jobs with unit operation times in a job shop to minimize total completion time is addressed. It is shown that this problem can be modeled as finding the optimal coloring of a special mixed graph. Subgraph of such a mixed graph without edges represents union of paths, and subgraph without arcs represents union of cliques. Finding the optimal coloring of the mixed graph with the criterion of minimizing the sum of maximal colors (used for the paths) is shown to determine a schedule for minimizing total completion time for processing jobs in a job shop. Since the problem is NP-hard, we develop a branch and bound algorithm for obtaining the optimal coloring of such a mixed graph. We develop a color-based branching scheme, dominance of the vertices in the solution tree, and three lower bounds of the objective function. The computational experiments indicate that the branch and bound algorithm performs well for randomly generated mixed graphs of order up to 200, if all the paths in the directed subgraph (job routes) have the same length, and of order up to 750 otherwise. Since in the considered job shop problem, machine repetition and absence of some machines in the job route are allowed, the developed methodology can be used to minimize total completion time in the job shop with arbitrary integer operation durations.

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 (vL(i)+1,vL(i)+2,,vL(i)+ri) where L(i)=k=0i-1rk and r0 = 0. For a job Jk, let i = L(k) + l where machine Mki has to process operation vi. In a technological route of (MkL(k)+1,MkL(k)+2,,MkL(k)+rk) 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 Φ(C1,C2,,Cn)=i=1nCi where Ci means completion time of job Ji: Ci=cL(i)+ri. 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 J|pi=1|Ci when jobs may have different technological routes (a job shop problem), and by the notation F|pi=1|Ci 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 F|pi{0,1}|Ci 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 MkiM does not process job Jk  J at all. The problem J|pi=1|Ci is NP-hard since the problem F|pi{0,1}|Ci is a special case of the problem J|pi=1|Ci.

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 Jn = 2∣Φ and Jn = 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 J3|n=3|Ci (with three machines and three jobs) and problem J2|n=3,Pr|Ci (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 J2|n=3,Pr|Ci with integer operation durations implies NP-hardness of the problem J2|n=3,pi=1|Ci due to their equivalence. Thus, to obtain exact solution to the problem Jm|n=no,pi=1|Ci 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 Jpi = 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 J|pi=1|Ci 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 J|pi=1|Ci is proposed in Section 3. In Section 4, three lower bounds on the objective function Ci 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 J|pi=1|Ci 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 inequalityϕ(vi)<ϕ(vj),and edge inclusion [vi, vj]  E impliesϕ(vi)ϕ(vj).Similar 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 J|pi=1|Ci 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

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