Simulated annealing based robust algorithm for routing-scheduling problem with uncertain execution times

Proceedings of the 13th IFAC Symposium on Information Control Problems in Manufacturing Moscow, Russia, June 3-5, 2009

Simulated annealing based robust algorithm for routing-scheduling problem with uncertain execution times Jerzy Józefczyk*, Michał Markowski** Wroclaw University of Technology, Wybrzeze Wyspianskiego 27, 50-370 Wroclaw, Poland *( e-mail: [email protected] ) **( e-mail: [email protected] ) Abstract: A routing-scheduling problem treated as a generalization of a traditional task scheduling problem is investigated in the paper. The generalization deals with the necessity to drive-up of executors, being the performers of tasks, to workstations where the tasks are carried out. An uncertain version of a simple problem with the makespan as the performance index is considered. It is assumed that execution times are not known a priori, but they are elements of given intervals. The uncertain decision-making problem is formulated with the performance index being the absolute regret based on the makespan. A heuristic solution algorithm, which uses a simulated annealing metaheurisics, is proposed. The property of the performance index is shown, which makes easier its calculations. Selected results of numerical experiments are given which evaluate the quality of the uncertain problem in terms of the absolute regret and the time of computation. Keywords: manufacturing systems, scheduling algorithms, routing algorithms, uncertainty, robustness,

decision support systems, artificial intelligence, numerical simulation.

1.

sum of completion times were considered, e.g. (Józefczyk, 1997, 2001b).

INTRODUCTION

An uncertain version of a routing-scheduling problem with a prospective application to manufacturing and logistic systems is investigated in the paper. In general, task scheduling and various problems connected with a movement of different elements in manufacturing and logistic systems, e.g. transportation as well as their connection are widely investigated and reported in the literature. The routingscheduling problem considered in the paper is a generalization of a standard task scheduling problem. It is assumed that executors of tasks move among places where the tasks can be performed. Modern discrete manufacturing systems are the main area of applications for the routingscheduling problem in the cases when the movement of plants being in production is impossible or too expensive and it is better to move executors. For example, it occurs when plants in production (e.g. big engine casings, railway carriage elements) are too big or too heavy to move among executors (machines). Another example may concern maintenance tasks on stationary machines or transportation tasks performed by mobile robots or automated guided vehicles in flexible manufacturing systems. Transportation systems, automated service systems as well as computer systems, where the times of the movement of executors (agents) are not neglected, can be the prospective examples for the applications of the problem under consideration. Till now different cases of the routing-scheduling problems concerning different performance indices have been investigated. The makespan, the maximum lateness and the

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The alike approach referred to as a routing-scheduling with application to flow-shop problems and with travelling machines is presented in (Averbakh and Berman, 1999). The routing-scheduling problem investigated is similar to capacitated vehicle routing problem (CVRP) in terms of the results obtained in the form of routes for executors (e.g. see Fisher, 1995)). The main difference consists in performance indices, i.e. for CVRP the cost is minimized. The problem considered in the paper is also connected with the travelling salesman problem with time windows (TSPTW) which has been stated in (Christofides et al., 1981). Further works develop it. For example, in (Langevin et al., 1993) the makespan problem with time windows (MPTW) is considered (see Mingozzi et al., 1997) for a survey), the additional resources are taken into account which leads to TSP-TWPC. New solution methods are presented, e.g. in (Dumas et al., 1995). In all these works all tasks are performed only by one executor. The problem considered can be also treated as a generalization of the task scheduling with setup times, e.g. (Rajendran and Ziegler, 2003) as well as is connected with hoist scheduling where each task is also composed of two parts, but routes for hoists are much more restricted, e.g. (Che and Chu, 2004; Lei and Wang, 1991). 1.1 Routing-scheduling problem Main notions are now explained to determine more precisely the idea of the routing-scheduling considered. A task is understood like in the scheduling theory, but it has its own

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meaning by reason of the fact that the movement of executors is taken into account. An executor is the subject performing tasks. Usually, it is a technological device. The executor performs the task at a place called a workstation. Moreover, a depot is distinguished being a workstation where each executor starts and ends its work. At the depot no task is performed. The executor should drive-up to the workstation to accomplish the task. Consequently, each task consists of two parts: a driving–up to the workstation and a performing of an activity at the workstation. Then, the execution time being the main data for every task scheduling problem is the sum of the driving–up time and the activity performing time. This generalization defines a new scheduling problem in which routes of executors should be determined, i.e. the routing-scheduling problem. This paper deals with the generalization of previous investigations, leading to a non-deterministic case in which execution times – the task scheduling problem’s most important parameters – are uncertain. 1.2 Uncertainty in decision-making problems The stochastic approach is the most popular (and widely investigated) way to model uncertainty in decision making problems (see, for instance, survey (Birge and Dempster, 1996)). It is assumed that a certain probability distribution exists over the space of all the possible realizations (scenarios) of all the random parameters of the problem and the objective is to determine a solution that fulfils a selected probabilistic performance index. Main drawbacks of the stochastic approach are discussed in (Kouvelis and Yu,1997). The so-called robust (worst-case) absolute regret performance index, for which the resulting routingscheduling algorithm is referred to as the robust (worst-case) absolute regret routing-scheduling algorithm (Kouvelis and Yu, 1997); Averbakh, 2000), is applied in the present paper. The case is considered in the paper that the uncertainty of parameters is described in the form of a set of their possible values (in particular, in the form of intervals). Nevertheless, a selected form of a substantiation of the criterion evaluating a decision (solution) with respect to uncertain parameters is used in each case, e.g. (Zadeh, 2005; Yager, 2004). This substantiation can be performed directly on criterion values or on terms based on them. One such a term (perhaps the most known), called an absolute regret, was introduced in (Savage, 1951). It is defined as a difference between the cost of a specific decision and the corresponding cost of the optimal decision for any given realization of unknown parameters. A simple case with unrelated executors performing a set of non-preemptive, independent tasks to minimize the makespan is investigated as the traditional task scheduling problem (in fact, an assignment problem because no precedence constraints are considered). It is generalized taking into account the movement of executors, e.g. (Józefczyk, 2001b). Consequently, the routing-scheduling problem introduced in previous subsection is obtained. This paper is organized as follows. The routing-scheduling problem is introduced and formulated in Section 2 as a discrete optimization problem for both deterministic and

uncertain execution times. A heuristic solution algorithm based on a simulated annealing metaheuristics is given in Section 3. Moreover, in this Section, the property of the absolute regret is proved which facilitates the calculation of the performance index values. Selected results of numerical experiments evaluating the algorithm as well as an analytical assessment of the performance index optimal value are presented in Section 4. Final remarks complete the paper. The same uncertain routing–scheduling problem is investigated in (Jozefczyk and Markowski, 2008) where not only simulated annealing based but also tabu search based solution algorithm is presented together with other results of numerical experiments. The present paper evaluates the simulated annealing based heuristic algorithm analytically and empirically more accurately and in a more comprehensive way. 2.

PROBLEM FORMULATION

2.1 Deterministic case Let us formulate the routing-scheduling problem in the deterministic case. Only one task, being a sequence of driving up and performing an activity by the same executor, is carried out at every workstation, or exactly speaking, on a product in manufacture located at the workstation. Sets of tasks and workstations are identically denoted as H = {1, 2, ..., l} where l is the number of tasks or workstations. A depot for executors, being the place where no activity is performed and which is the beginning and the end of each executor’s route, is distinguished among the workstations. A depot is denoted by h = l + 1 . Then H = H ∪ {l + 1} is a set of workstations with the depot. Similarly, R and m are respectively a set of executors and a number of executors. Task execution times p r , g ,h have now a more complex form than for a version

without moving executors, depending not only on the executors but also on the workstations from which the executors set off to perform the tasks. Each execution time ) ( ) pr , g , h is the sum pr , g , h = pr , h + pr , g , h , where pr , h and ( pr , g , h are respectively the time for which activity h ∈ H is performed by executor r ∈ R at workstation h and the time for which the executor drives up to workstation h from ( workstation g . Assumption pr , h , h = +∞ , h = 1,2,..., l + 1 means that it is not possible for executors to return to the same workstation without performing any activity. Times pr , g , h form three-dimensional matrix P= [ p r , g ,h ] r =1, 2, ..., m

. In order to formulate the corresponding

g , h =1, 2 ,..., l +1

optimization problem, let us define a decision variable as matrix C = [cr , g , h ]r =1, 2, ..., m where c r , g ,h = 1(0) if g , h =1, 2, ..., l +1

executor r performs task h after driving-up from workstation g (otherwise). The constraints imposed on decision matrix C ensure the feasibility of solutions (Józefczyk, 2001). They form the set Dc . The makespan

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l +1 l +1

Q( P , C ) =

max

∑ ∑ cr , g ,h pr , g ,h

(1)

r =1, 2,..., m h =1 g =1

is used as the performance index. The routing-scheduling problem under consideration can be formulated as the discrete optimization problem OPT( P ) : Find the minimum of Q ( P , C ) with respect to C ∈ Dc for fixed value of P. Let us also denote ∆

Q* (P) =

min Q( P , C ) .

C ∈D

c

Actually, the solution of the problem consists in the determination of the routes for executors with the beginning and the end at the depot. Let us denote the route for the current executor as a sequence t r = (t r ( j )) j =1, 2, ..., mr where t r ( j ) , mr are the j th element of t r , its length, respectively. Moreover, t r ( 0) = t r ( m r ) = l + 1 and t r ( j ) = h, j = 1, 2, ..., mr , i.e. the task (the workstation) for which cr , t r ( j −1),t r ( j ) = 1 holds is the element of t r . 2.2 Uncertain case The exact values of execution times

p r , g ,h

or their

components are not known in many cases (Józefczyk, 2000, , pr , g , h ] , 2001a, 2007). Let us assume that pr , g , h ∈ [ p r, g ,h

0< p

r, g ,h

≤ pr , g , h where values of p r , g , h , pr , g , h are known

and given. Then, each realization of matrix P comprising execution times p r , g ,h , being a parameter of task scheduling

~ As the result, matrix C (a robust optimal in the sense of (2) and expressing the robust absolute regret schedule) as well as ~ the value of zA (C ) are obtained. Analogously, the optimization problem min z R (C ) = min max

C∈Dc

3.

1.

Set n = 0 , C ' = C 0 and calculate zA (C 0 ) .

2.

Find the best solution C n +1 in the neighbourhood of C n and calculate z A (C n +1 ) ≤ z A (C ′) set C ′ = C n +1 .

ratio (Q(P, C) − Q * (P)) / Q * (P) , respectively. Then, uncertain optimization problems, which correspond to the optimization problem OPT( P ) , can be formulated. The first one, which corresponds to the absolute regret, referred to as UOPTA consists in the minimization with respect to C ∈ Dc of the performance index

~ ∆ min z A (C) = min max[Q(P, C) − Q* (P) = zA (C) . (3)

C∈Dc

C∈Dc P∈Dp

SOLUTION ALGORITHM

V (C n )

Subsection 1.2 is proposed to cope with this uncertainty. The objective is to find matrix C that performs well for all the scenarios in the sense of makespan (1). The robust optimization approach can be used for this purpose. The uncertainty in decisions due to the uncertainty in parameters P can be evaluated in different ways. In further considerations, the approach based on the regret is proposed. Two its forms can be used, i.e. the absolute regret and the relative regret being the difference Q( P , C ) − Q* ( P ) and the

called the robust absolute regret criterion, i.e.

= z R (C ′′) (4)

algorithm given in the form of matrix C , C ' – the best solution determined by the algorithm. The values of parameters N, θ min , θ max as well as randomly generated initial solution C 0 are the data for the solution algorithm that can be presented in seven steps:

r , g ,h

(2)

Q* (P)

The optimization problem UOPTA is obviously the NP– hard one because its deterministic counterpart is NP–hard. Therefore, the simulated annealing algorithm (SA) is proposed to solve the task scheduling problem UOPTA . Let us introduce the basic notation: N – number of iterations of the algorithm which is used also as the stop condition, n, n = 1, 2, ..., N – index of the current iteration of the algorithm, θ , θ ∈ [θ min , θ max ] – parameter of the algorithm called temperature, C n – solution in the nth step of the

scenarios is the Cartesian product of all intervals [p , pr , g , h ] . The robust approach introduced in

P ∈D p

C∈Dc P∈Dp

called UOPTR should be solved for the relative regret. As the result, matrix C ′′ as well as the value of zR (C ′′) are obtained (see Józefczyk, 2007) for the solution algorithm).

problem OPT( P ) , is called a scenario. Set Dp of all the

z A (C ) = max [Q ( P , C ) − Q* ( P )]

Q( P , C ) − Q * ( P )

z A (C n + 1 ) .

If

3.

If zA (C n +1 ) ≤ zA (C n ) go to Step 6, otherwise go to the next step.

4.

Generate randomly the number d from the interval [0, 1] according to the rectangular distribution.

5.

If

exp

zA (C n ) − z A (C n +1 )

θ

>d

set

C n +1 = C n ,

otherwise go to the next step.

θ − θ min θ where λ = max . Nθ maxθ min 1 + λθ

6.

Set θ =

7.

If n < N set n = n + 1 and go to Step 2, otherwise stop the algorithm with the solution C ' and z A (C ′) .

The following procedure has been applied to generate the neighbourhood. Some number of tasks denoted as H N are taken randomly from the longest working executor, and they are inserted into all possible positions of the shortest working executor.

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It is easy to see that the deterministic NP–hard optimization problem OPT( P ) must be solved for any scenario P to calculate each value of the performance index zA (C n ) . The set Dp of all scenarios as the Cartesian product of the intervals is of the power of the continuum. In the consequence, the computational complexity of the optimization problem UOPTA is enormous even using heuristic solution algorithms. Therefore, it is crucial to limit the set Dp . The following Lemma shows that it can be done

Hence, it is enough to fix p j , g ,h = p j , g ,h and p j * , g , h = p

j, g ,h

to maximize the difference in (8). So, it is sufficient to consider the bounds of the intervals instead of all intervals. Let Q r ( Pr , C ) denote the time executor r is engaged, i.e. Q( P , C ) =

max Q r ( Pr , C )

r =1, 2, ..., m

where Pr is a part of matrix P with the execution times

and the set of all possible scenarios Dp can be substantially

which correspond to executor r . Additionally, P is matrix P with all elements equal to p r , g , h , and, analogously, P is

limited.

composed only of elements p

Lemma (sufficient condition)

max [Q ( P , C ) − Q * ( P )] = max [Q( P , C ) − Q * ( P )] P∈Dp′

1.

(5)

where Dp′ is the Cartesian product of all 2 element sets

{p

r , g ,h

Calculate Q r ( Pr , C ) for r = 1, 2, ..., m and find the index of the executor that works the longest time, i.e. j = arg

, pr , g , h } .

Let C * denote the optimal solution of OPT( P ) for fixed matrix P . Then

2.

l +1 l +1

∑∑ p

z A (C ) = max [ max P ∈D p

r =1, 2 , ..., m

max

r =1, 2 , 2, ..., m

∑∑ p

(6)

r , g , h c r , g , h ].

3.

Let j and j * be indices of executors for which the sums in (6) are the maximum, i.e. max

r =1, 2, ..., m

l +1

h =1

g

∑ ∑p l +1

max

r =1, 2, ..., m

4. r , g ,h

cr , g ,h and

h =1

r , g ,h

g

∑∑ p

j , g ,hc j , g , h

−

∑∑ p

j* , g ,h

c

*

j * , g ,h ]

(7)

h =1 g =1

and in the consequence as l +1 l +1

z A (C ) = max

P ∈ Dp

∑∑ ( p

NUMERICAL EXPERIMENTS

r , g ,h

l +1 l +1

h =1 g =1

*

Calculate z A (C ) = Q j ( P j , C ) − Q *, j ( Pˆ ) .

randomly generated according to the rectangular distribution. Namely, left and right bounds, i.e. p and pr , g , h take

l +1 l +1

P∈Dp

* Obtain C * , Q*, j ( Pˆ ) as the solution of problem OPT( Pˆ ) and determine j * .

Numerical experiments were performed to evaluate the solution algorithm. The interval execution times pr , g , h were

c * r , g ,h

Thus (6) can be rewritten as zA (C ) = max [

p r , g ,h cr , g ,h .

h=1 g =1

Obtain the deterministic matrix Pˆ with the following elements

4.

l +1

∑ ∑p

r =1, 2, ..., m

∑∑

*

h =1 g =1

l +1

max

 pr , g ,h , for r = j and cr , g , h = 1 , pˆ r , g ,h =   p r , g , h , otherwise .

r , g , h cr , g , h

h =1 g =1

l +1 l +1

j * = arg

Q r ( Pr , C ) = n +1 n+1

Proof

j = arg

max

r =1, 2, ..., m

arg

−

.

The following procedure, which calculates zA (C ) , results from Lemma.

If P ∈ Dp′ for any C ∈ Dc then P∈Dp

r , g ,h

j, g ,hc j, g ,h

− p j * , g , h c* j * , g , h ).

(8)

h =1 g =1

Let us notice that values of times p j , g , h and p j * , g ,h are only important with respect to calculation of zA (C ) . The values of the other times can be set as arbitrary, for example they can obtain values of the bounds of corresponding intervals.

values from intervals [ 40, 70] and [70, 90] , respectively. At first, the parameters of the solution algorithm have been tuned taking into account the value z A (C ′) for l = 26 and m = 2 . As the result, their best values are as follows: N = 1000 , θ max = 100000 , θ min = 1 , H N = 4 . The dependence of z A on N is presented in Fig. 1 as the example of the tuning. It turned out that increasing the number of iterations beyond 1000 does not substantially improve the quality of the algorithm. So, this value has been used for other experiments taking into account that then the time of computation is the least. In the second part of the

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Table 1 Results of numerical experiments for solution algorithm SA

l 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40

Fig. 1. Dependence of z A on N . experiments, the number of tasks l were changed. Apart from the values of performance index z A (C ′) and the time of ~ computation T , the following evaluation of z A (C ) defined in (3) were used for assessing the solution algorithm for different l ~ ~ z A (C ) = max[Q( P , C ) − min Q( P , C )] P∈Dp′

C∈Dc

C∈Dc

C∈Dc

~ ~ ≤ Q ( P , C ) − min Q( P , C ) = Q( P , C ) − Q * ( P ) C∈Dc

where P * is feasible matrix P minimizing the absolute regret. In such a way, the optimal value of the performance index is evaluated by values of the corresponding criteria for the deterministic problem. It is obvious that for z A (C ′) generated by the heuristic SA algorithm, which is not less ~ ~ ~ than z A (C ) , the inequality z A (C ) ≤ Q( P , C ) − Q * ( P ) ~ ∆ = Qˆ (C ) given in (9) may not be valid. However, this inequality holds for all instances of l as can be observed from Table 1 and Table 2. This property can be found as the rough evidence for the usefulness of the SA based solution algorithm. Table 1 and Table 2 comprise other results of the experiments for m = 2 and two versions of the solution algorithm. The difference between the versions consists in the way the solutions’ evaluation is performed in Step 2 of the algorithm. For the first, basic version called SA (Table 1), the values * z (C ) = Q j ( P , C ) − Q *, j (Pˆ) are calculated where n +1

j

n +1

*

Q*, j ( Pˆ ) is the result of the internal optimization problem. For the second version denoted as SA(A) (Table 2), the * approximation of Q*, j ( Pˆ ) is used. So, the values zA (Cn +1 ) = Q j ( Pj , Cn +1 ) −

1 m

l +1

∑ g =1

min

min

r =1, 2,..., m h =1, 2,...,l +1

pˆ r , g ,h

are

only

T [ s]

12,55 13,29 13,61 14,20 13,42 14,05 13,95 13,92 13,54 13,16 13,07 13,38 13,38 13,53 13,34 13,57

1020 1422 1828 2462 3185 3462 4221 4822 5144 5604 6312 6800 7021 7094 7219 7426

l

z A (C ′)

Qˆ (C ′)

δ

T [ s]

10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40

126.00 159.00 185,00 218,00 248,25 278,00 299,25 315,00 349,00 371,25 393,25 422,00 450,50 483,00 502,00 526,50

154.35 196.23 197.18 255.90 281.78 312.95 345.05 338.33 401.58 417.98 441,80 455,20 493,33 530,08 539,33 579,65

12.60 13,25 13,21 13,63 13,79 13,90 13,60 13,13 13,42 13,26 13,11 13,19 13,25 13,42 13,21 13,16

65 90 114 146 187 215 244 278 308 347 386 430 475 529 587 633

the quality of SA(A) is only slightly worse than SA. Therefore, SA(A) can be recommended for further considerations. The increase of the time of computation T is consistent with the computational complexity of SA based algorithm. It is obvious that the value of zA is dependent on the number of tasks l . The index δ = zA / l , being the average value of zA with respect to l , is also presented in Table 1 and Table 2. It is the measure of the uncertainty normalized with respect to the number of tasks (the values of the uncertainty for one task). The values of δ are of the same range and differ each other no more than 14%. 5.

determined and it is not necessary to solve the optimization problem like for SA. It results in substantially reduction of the time of computation T (Tables 1, 2). On the other hand,

δ

Table 2 Results of numerical experiments for solution algorithm SA(A)

~ ~ = Q( P * , C ) − min Q( P * , C ) ≤ Q( P , C ) − min Q( P * , C ) (9)

A

Qˆ (C ′) 155,85 196,23 200,43 262,90 283,28 318,95 353,80 403,83 404,33 412,23 464,80 456,70 532,58 529,58 543,33 597,40

z A (C ′) 125,50 159,50 190,50 227,25 241,50 281,00 307,00 334,00 352,00 368,50 392,00 428,00 455,00 487,00 507,00 542,75

FINAL REMARKS

The uncertain version of the routing-scheduling problem is investigated in the paper. The uncertainty is understood in terms of interval execution times. The simple task scheduling

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13th IFAC INCOM (INCOM'09) Moscow, Russia, June 3-5, 2009

(assignment) problem is addressed. The robust approach based on the absolute regret was used as the tool to manage the uncertainty. Because of the NP–hardness of the basic deterministic problem, the heuristic solution algorithm, using the simulated annealing metaheuristic, has been employed for solving the uncertain version. The solution algorithm uses the procedure which makes easier the calculation of the performance index values. The approach can be easily applied to other routing-scheduling problems. The more effective procedure of determining the values of performance index zA is now under consideration. Further researches will deal also with different ways of the performance index determinization with respect to the uncertain (interval) parameters. 6.

Józefczyk, J., 2001b. Scheduling tasks on moving executors to minimise the maximum lateness. European Journal of Operational Research, 131, 17–187. Józefczyk, J., 2007. Robust algorithm for scheduling of manufacturing tasks with interval execution times. Proc. of IFAC Workshop on Intelligent Manufacturing Systems IMS07, Alicante, Spain [cd-rom]. Józefczyk, J., 2008. Worst-case allocation algorithms in a complex of operations with interval parameters. Kybernetes, vol. 37, s. 652–676. Józefczyk, J., Markowski, M., 2008. Heuristic algorithms for solving uncertain routing-scheduling problem. Artificial Intelligence and Soft Computing – ICAISC 2008, 5097, 10521063.

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