A comparison of simulated annealing algorithms in the scheduling of multiproduct serial batch plants

A comparison of simulated annealing algorithms in the scheduling of multiproduct serial batch plants

V. Ruiz Ahon, F.W. Tavares^1* and M. Castier²

Escola de Química, Universidade Federal do Rio de Janeiro,

C.P. 68542, 21949-900, Rio de Janeiro - RJ, Brazil

E-mail: [e-mail]

(Received: September 20, 1999 ; Accepted: March 30, 2000)

Abstract - In this work, several simulated annealing algorithms presented in the literature as solutions to different problems were compared in the scheduling of multiproduct serial batch plants. A systematic performance study of the original algorithm, based on the Boltzmann statistics, and a recently proposed algorithm, based on the Tsallis statistics, were analyzed for three different annealing schemes. It was observed that performance (defined in terms of the number of configurations) is more sensitive to the annealing scheme than to the statistics used. We also observed that the overall performance of the simulated annealing procedure is very dependent on the parameters of the algorithm and on the kind and size of the problems. For large problems, algorithms based on the Tsallis statistics converged to the global minimum more frequently than those based on the Boltzmann statistics, although they needed a larger number of configurations to obtain the results.

Keywords: simulated annealing, scheduling of multiproduct, batch plants.

INTRODUCTION

A large number of specialty and fine chemicals is produced in relatively small amounts in batch plants. Since several processes are often used in these plants, the optimal scheduling of operations is important, but leads to a combinatorial problem. If we restrict our attention to a multiproduct serial batch plant (Das et al., 1990), where the number of chemicals to be produced is equal to I and the same production sequence is adopted in all units, there will be I! possible sequences. The number of alternatives may be so large that, instead of trying to find the best production sequence, finding a near-optimal solution to the scheduling problem is frequently considered to be satisfactory. With this objective, the simulated annealing (SA) algorithm (Kirkpatrick et al., 1983) was used by several authors (Das et al., 1990; Ku and Karimi, 1991; Castier, 1996; Batista et al., 1998).

The most frequently used version of the SA algorithm was inspired by the Boltzmann formulation of statistical mechanics. However, Tsallis (1988) proposed a generalization of the Boltzmann statistics that has been successfully used in many areas. It is of particular interest that an SA algorithm inspired by the generalized Tsallis statistics was proposed by Tsallis and Stariolo (1996) and used in complex optimization problems such as the determination of protein conformations (Hansmann and Okamoto, 1999) and the experimental planning of phase equilibrium measurements (Dariva et al., 1998). The success of this new version of SA, reported in the literature, motivated us to evaluate the method in the scheduling of batch plant operations. However, the performance of an SA algorithm depends not only on the statistics, but also on the annealing scheme, i.e., the strategy used to reduce the temperature-analogue parameter during the iterations. Therefore, the objective of this paper is to present a systematic study of the computational performance of the combined use of the original algorithm of Kirkpatrick et al. (1983), based on the Boltzmann statistics, and a new algorithm based on the generalized statistics of Tsallis (Tsallis and Stariolo, 1996) with three different annealing schemes. Several test problems were solved, but for the sake of conciseness we only present two cases of multiproduct plants with an unlimited number of intermediate storage (UIS) tanks.

RECURRENCE FORMULA FOR A UIS MULTIPRODUCT SERIAL PLANT

Multiproduct plant operation is assumed to be nonpreemptive, i.e., once a unit starts processing a given product, it is not interrupted until the product is ready to leave. Defining and as the processing time and the completion time in unit j of product k that occupies position i in the sequence, the following expression can be recursively used to determine completion times in a UIS batch plant (Mah, 1989):

(1)

The initial conditions represent the availability of all units at the start of the plant operation:

(2)

(3)

where I and J represent the number of products and the number of units, respectively.

The scheduling problem is formulated as the minimization of the makespan (), i.e., the time that elapses between the moment the first product in the sequence enters the process and the moment the last product is ready to leave.

SIMULATED ANNEALING

In the SA method, a chain of system configurations is generated. In the scheduling problem, each configuration corresponds to a production sequence. Assuming that a current sequence (C) is available, a new sequence (N) is randomly generated. The acceptance probability of the new sequence (P_C®N ) is

(4)

where p is a real number whose evaluation depends on the type of statistics that is adopted. A new sequence is accepted if a uniformly distributed random number, r, belonging to the interval [0,1], satisfies r £ P_C®N; otherwise, it is rejected and the current sequence is retained as the starting point for the generation of another trial sequence.

In SA algorithms that use the Boltzmann statistics, p is given by

(5)

where DF represents the difference between the objective function for the current (F_C) and new (F_N) configurations, as follows

(6)

In the scheduling problem, F represents the makespan, recursively calculated using eq. 1. The value of b is

(7)

where G is a parameter of the SA algorithm, which is analogous to the product kT of statistical thermodynamics (k and T are the Boltzmann constant and the absolute temperature, respectively).

In the SA algorithm, the value of G decreases throughout the iterations, thereby reducing the probability of accepting a new sequence, and contributing towards "freezing" the system in its final configuration. For given statistics, such as the Boltzmann statistics discussed previously, SA algorithms differ in their annealing scheme, i.e., the initial value of G and how often and how fast G is reduced.

A new expression for p in the acceptance criterion of the SA algorithm (eq. 4) was recently proposed by Tsallis and Stariolo (1996), based on the generalized statistics of Tsallis (1988), as follows

(8)

where parameter q_a is a real number that is kept constant. Since the value of p is calculated in eq. 8, the remaining operations to either accept or reject a new configuration are the same as described previously.

For the implementation of the SA algorithms, the initial value of G (G₀) was calculated as

(9)

where DF^* is the difference between the largest and the smallest values of F in a small number of attempted sequences (I sequences were used) (Das et al., 1990). For each value of G , I(I-1) sequences are tried. The average value and the standard deviation (s ) of the accepted configurations for this value of G are computed. The random generation of attempted configurations was done by the removal-insertion technique (Das et al., 1990; Castier, 1996).

The following annealing schemes to generate a new value of G (G_N) from its current value (G_C) were tested:

(a) Exponential procedure (Kirkpatrick et al., 1983) (Exp):

(10)

(b) Aarts and van Laarhoven procedure (Das et al., 1990) (AvL):

(11)

Tsallis and Stariolo (1996) procedure (TS):

(12)

where q_v is a parameter of the annealing scheme and N_cfg is the number of configurations generated up to the current value of G (G_C).

RESULTS AND DISCUSSSION

The performance of the SA algorithms was compared in several problems. Each SA algorithm is the combination of an acceptance criterion and an annealing scheme. For conciseness, we only present results of makespan minimization in two UIS plants. To ensure a consistent comparison between the algorithms, each of them was used to solve each problem 100 times with randomly different initial scheduling sequences and the same termination criterion (G <10^-30 or no new configuration was accepted at a given value of G).

In the first problem, there are six products and ten processing units. The processing time for each product in each unit is shown in Table 1. In this relatively small problem, the minimum makespan (equal to 88) was always obtained by all SA algorithms, except for the combination of the Boltzmann statistics and the TS annealing scheme with q_v = 1.5, for which this value was obtained in 99% of the cases (in the remaining 1%, a value of 91 was obtained). Figures 1A-1L show the frequency (in %) of the number of iterations required by the different SA algorithms for convergence. The fastest algorithm for this problem was the combination of the Boltzmann statistics and the TS annealing scheme with q_v = 1.7 shown in Figure 1D.

In the second problem, there are 12 products and six processing units. The processing time of each product in each unit is shown in Table 2. Figures 2A-2L show the frequency (in %) of the converged makespan values. In terms of frequency of convergence to the minimum makespan value (equal to 86), the combination of the Tsallis statistics with q_a = 3 and the TS annealing scheme with q_v = 1.5, shown in Figure 2K, had the best performance, converging to the minimum value in more than 70% of the cases. For the Boltzmann statistics (Figures 2A-2D) and the Tsallis statistics with q_a = 2 (Figures 2E-2H), the TS annealing scheme with q_v = 1.5 also had the largest frequency of convergence to the minimum makespan. It can be observed in each set of Figures (2A-2D, 2E-2H, 2I-2L) that the TS annealing scheme with q_v = 1.5 required the largest number of iterations for the corresponding statistics. Therefore, the best performance in terms of frequency of convergence to the minimum makespan value was obtained at the cost of a larger number of configurations for the given statistics. It was also observed that the performance of SA algorithms that use the Tsallis statistics is strongly dependent on the annealing scheme and on the values of parameters q_a and q_v, and it may be difficult to tune them for their best performance. We also noted that, in terms of number of iterations, SA algorithms based on the Tsallis statistics were not faster than those based on the Boltzmann statistics, as shown in Figures 3A-3L.

CONCLUSIONS

Several simulated annealing algorithms were compared in the solution of the scheduling problem in multiproduct serial batch plants. In this type of problem, it was observed that performance is very sensitive to the scheme used to reduce the annealing parameter. In large problems, algorithms based on the Tsallis statistics converged to the global minimum more frequently than did those based on the Boltzmann statistics, although they did so at the cost of visiting a larger number of configurations and, therefore, being slower. Provided that the number of configurations in the application of an algorithm that uses the Tsallis statistics is not unfeasibly large, these algorithms can be useful for the scheduling of batch plants. A difficulty, however, with these algorithms is to tune the best values of q_a and q_v parameters.

ACKNOWLEDGMENTS

The authors thank Prof. C. Tsallis for his helpful comments. M.C. thanks Prof. José Tojo Suárez for his hospitality at the University of Vigo (Spain), where part of this work was done. V.R.A. acknowledges a scholarship received from CAPES (Ministry of Education, Brazil). M.C. and F.W.T. are grateful for the financial support of CNPq/Brazil and PRONEX (Grant no. 124/96).

NOMENCLATURE

completion time of product k occupying position i in unit j makespan F objective function I number of products J number of units N_cfg number of configurations generated P_C®N acceptance probability of a new configuration q_a parameter of the acceptance criterion in the Tsallis statistics q_v parameter of the Tsallis annealing scheme r random number T_k,j processing time of product k occupying position i in unit j

Greek letters

D average value of the accepted configurations s standard deviation of the accepted configurations G annealing parameter

Subscripts

C current configuration N new configuration

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*

To Whom correspondence should be addressed

Publication Dates

• Publication in this collection
  06 July 2000
• Date of issue
  June 2000

History

• Accepted
  30 Mar 2000
• Received
  20 Sept 1999