1 Introduction 2 Problems of theory of scheduling

Analyze of some problems of theory of scheduling, industrial planning and nancial mathematics by methods of mathematical programming and discrete optimization. Lazarev Alexander A. Kazan State University, Chair Economic Cybernetics, Kremlevskaya str., 18, Kazan, 420008, Russia e-mail [email protected] http://www.kcn.ru/tat en/university/persons/9508.en.html

1 Introduction Problems of allocating limited resources arise in a large variety of real world applications. They create an interdisciplinary area of research, where discrete optimization, theory of scheduling, nancial mathematics, mathematical programming and design of algorithms. The considered project is a continuation of a long-term research activities of our team involved, in the area concerning deterministic and stochastic problems of task scheduling. Furthermore there was possible use of this apparatus to solve dierent practical problems. There is planned to develop eective algorithms, including approached, for the following problems of theory of scheduling: 1 j rj j Lmax , 1 jj Tj ; industrial planning: on various graph topology of precedence relations and resource restrictions; nancial mathematics - H.Markowitz's problems. It is assume to use methods of mathematical programming and discrete optimization for the speci ed problems.

2 Problems of theory of scheduling Following polynomial resolved private cases are founded for problems of theory of scheduling for single machine to minimize of maximal lateness and total tardiness when given release time ri , processing time pi , deadline terms dj , of job j, j = 1; n, [1]. For problem 1 j rj j Lmax : 1

 d  d  :::  dn , d ; r ; p  d ; r ; p  :::  dn ; rn ; pn, [3];  dj = rj + pj + Const, j = 1; n, [3,8];  r  r  :::  rn, d  d  :::  dn; dk ; pk  dm , k = 1; n ; 1, 1

2

1

1

1

1

2

1

2

1

2

1

2

2

2

2

m = k + 1; n, [10];  r  r  :::  rn, d  d  :::  dn; dk ; pk  dm , k = 1; n ; 1, m = k + 1; n, [6,10]. For the problem 1 j dj = rj + pj j Lmax polynomial algorithm have been constructed (complexity O(n log n)). We proved 0  Lmax (0 ) ; Lmax ( )  , where  and 0 are optimum schedules of two problems 1 j rj j Lmax with parameters of jobs frj ; pj ; dj g and frj ; pj ; d0j g, j 2 N, respectively, and  = max fd ; d0j g ; min fd ; d0j g. Minimum absolute error of approximate j 2N j j 2N j decision has been found P by the linear programming method. For problem 1 jj Tj :  d  d  :::  dn , dj  dj + pj , j = 1; n ; 1, [3];  if pi < pj then pi + di  pj + dj , or if pi + di < pj + dj then pi  pj , [4]. Approached pseudopolynomial P algorithmsPare constructed. For problems 1 j rj j Lmax , 1 j rj j Cj and 1 jj Tj have been found the necessary conditions of optimality [13, 14]. 3

1

2

+1

3 Problems of industrial planning Following problem was considered: to sequence n jobs with resource restrictions and precedence relations was obtained LP formulation with zero-one variables. We introduce variables xij , i = 1; n, j = 0; T (T is planning interval). They are appointed either value 1, if job i is completed at moment j, or value 0, otherwize. Optimization criteria is the minimum of total weighted completion time: F (x) =

X i2N

!iti =

X i2N

were ti is completion time of job i, i 2 N. Follow restrictions are derived: T X

all jobs should be processed.

 =0

T X

!i(

 =0

xi = 1; i 2 N;

2

xi );

T X

where ri is release

xi   ri + pi ; i 2 N;  =0 time, pi is processing time of job i, i 2 N.

L X X

xi qik +

 =0 i2N^k

X i2Nk



xi  pi qik  0; L = 0; T; k 2 Q ( +

)

resource increase should be more than resource consumption at every time of planning interval. Resource k is increased, qik > 0, at job i completion time and is consumed, qik < 0, at job beginning time. Sets N^k and Nk are de ned as: N^k = fi 2 N j qik > 0g; Nk = fi 2 N j qik < 0g: Restrictions ensued from precedence relations: T X  =0

x   00



T X  =0

xi   + pi ; i ! i ; i ; i 2 N: 0

0

00

0

00

00

For solving this task was applied the method of branch and bounds. To calculate lower bounds of tasks we used simplex method. Braching was realized on variable that had inadmissible value and a maximum of objective function coecients. The proposed variants have been tested on a large collection of test problems, a good eect was observed on problems with some graph topology (outtree).

4 Problems of nancial mathematics The next problem we consider is the problem of optimal capital control under conditions of risk existence. It means that investor doesn't know the future pro t de nitely and it may be considered as a random variable. Let  ui is a decision concerning capital control chosen at the moment i, i = 1; 2; :::;  Ui is a set of all possible decisions concerning capital control at the moment i;  Ai (u ; u ; :::; ui; ) is a numeric value of assets at the moment i, it depends of decisions u ; u ; :::; ui; chosen at previous moments; 1

2

1

1

2

1

3

 ri (u ; u ; :::; ui) is the relative gain of capital for period of time between 1

2

moments i and i + 1: ri (u ; u ; :::; ui) = Ai (u ; u ;


; ui)=Ai(u ; u ; :::; ui; ): 1

2

+1

1

2

1

2

1

 Rm (u ; u ; :::; um) is the relative gain of capital for period between mo1

2

ments 1 and m + 1:

Rm (u ; u ; :::; um) = 1

2

m Y i=1

ri (u ; u ;


; ui) 1

2

The basis supposition of our model is that distribution of random variables ri(u ; u ;


; ui) does not depend of u ; u ; :::; ui; , and it is possible to think that ri (u ; u ;


; ui) = ri (ui). Let's consider the following characteristic of a random variable r: Tr = exp(M ln r) We oer to use that characteristic as the estimation of expected capital gain and as a way of comparison of dierent decisions concerning capital control. The expediency of using it may be shown with the following armation: Armation 1. Let fvi g; fwig are two sequences of decisions and the following conditions are realized: a) For every m and every vector (u ; u ;


; um ) random variables r (u ); r (u );


; rm (um ) are independent. b) For existing numbers K; V and for every i the following restrictions are realized: max(M j lnri (vi )j; M j lnri(wi )j)  K max(D ln ri (vi ); D lnri (wi ))  V c) For existing  > 0 and for every i the following inequalities are ful lled: Tri (vi )  (1 + )
T ri(wi ) Then mlim !1 p (Rm (w ;


; wm) > Rm (v ;


; vm )) = 0 Armation 2. For every m the following con rmation is realized: max TRm (u ;


; um ) = TRm (v ;


; vm ); u ;


;u 1

2

1

1

1

2

1

2

1

2

1

2

2

1

(

1

1

m)

1

1

where Tri(vi ) = max Tri(ui ). ui Using the oered approach there was constructed the following optimization model for the problem of portfolio control: m P n Q rij
xj i j 8 P < n xj = 1 : jx  0; j

max

=1

=1

=1

4

where xj is a share of securities number j in the portfolio, rij is the relative gain of security number j for the moment number i. In future it is planned to improve the optimization model and to approach it to real stock market. At the same time it is planned to carry out experiments for comparison our model with another existent models.

5 General formulation The researched problems can be formulated in following formulation:

fextrf(x) j Ax  0g where f(x) is linear, piecewise linear or quadratic function on area integer or real numbers. For the considered problem it is supposed to use methods of mathematical programming, theory of scheduling, discrete optimization. For each of the listed above three classes of problems will be formulated statements and solution in the terms of other classes.

6 Experimental research for problem 1 rj Lmax j

j

We created the software for running various experiments. It includes realization of some algorithms for polynomially solvable cases of the problem and the branch&bound method for it. Visual C++ 6.0 and Delphi 3 programming environments were used for software creation. Branch&bound method are used for researching the problem. Branching scheme: for statement to a current partial schedule only that jobs j are claimed, for which rj  tr , where tr = jmin fmaxfCmax (part ; rj g + pj g and 2N jobs are included in order nondecreasing dj . P Lower valuations: Cmax (part ) + pj ; jmax d and 2N j j 2N maxfLmax (part ); max fmaxfCmax (part ); rj g + pj ; dj gg, where N 0 - set of j 2N nonsequencing jobs. Research of experimental value of minimumabsolute error () was perfomed. Data values were simulated according to various distributions: Uniform (U, rj 2 [0; br ]; pj 2 [1; bp]; dj 2 [10; bd]), Binomial (B, n = b; p = 0:5), Poisson (P,  = b ), Exponential (E,  = b ), Normal (N, N( b ; b )), Chi-square (n; n = b), Cauchy (C, a = b ; c = b ), where br = 100, bp = 20, bd = 150. For each distribution and for n = 5; :::; 14 experiments had more 2000 launches. Results are shown in table below. There are following denotements accepted: av = prac:  100%; m- number of launches our polynomial algorithm constructed optimal schedule in % from all launches; prac: - experimental value of absolute 0

0

0

0

5

2

2 3

2

2

2

30

5

6

error. n

av

5

U

m

av

3.0

92.6

6

4.0

7

B

m

av

38.6

95.9

87.5

41.2

5.2

80.2

8

7.0

9

P

m

av

32.0

84.1

97.8

36.9

42.5

98.9

71.3

42.8

9.1

61.9

10

11.6

11

E

m

av

1.8

95.4

86.5

2.3

41.4

91.6

99.3

43.3

44.0

99.6

50.7

44.4

14.6

42.3

12

18.1

13 14

N

m

av

9.3

87.7

92.9

12.5

2.8

90.4

95.5

3.5

45.3

97.5

99.7

46.0

44.9

99.8

35.5

45.2

22.8

30.5

26.8

26.9

2n

C

m

av

29.1

79.9

2.85

81.8

67.9

36.0

81.0

4.57

80.4

16.4

57.8

40.5

86.6

4.56

79.7

87.0

21.2

49.7

43.9

91.8

5.23

80.2

4.3

83.9

26.6

46.1

45.3

95.6

5.85

80.2

98.6

5.2

81.2

31.8

47.7

46.1

97.5

0.50

81.6

46.4

99.0

6.6

75.7

36.6

53.4

47.0

98.5

2.82

80.9

99.8

46.8

99.3

8.0

73.9

40.9

59.4

47.6

99.0

6.13

83.1

45.5

99.9

47.6

99.4

10.0

68.3

44.2

67.4

48.4

99.2

4.59

83.1

45.8

99.9

47.5

99.6

11.3

68.8

47.2

77.8

49.0

99.5

3.80

80.3

We are going to continue to research this problem in following ways:  To increase the size of problem  To realize experiments with other distribution's parameters  To study common regularity of absolute error  and m for other distribution's parameters Described results was presented on next conferences:  In book of Abstracts [16] on Fourth International Congress on Industrial and Applied Mathematics, ICIAM 99, July 5-9, 1999, Edinburgh, Scotland, UK (WWW page: http://www.ma.hw.ac.uk/iciam99/).  19th IFIP TC7 Conference on System Modelling and Optimization, July 12 - 16, 1999, Cambridge, England, UK (WWW page: http://www.damtp.cam.ac.uk/user/na/tc7con/).  In Book of Abstracts [12, 17] on Workshop on the Complexity of Multivariate Problems, October 4-8, 1999, Hong Kong, China (WWW page: http://www.math.hkbu.edu.hk/complexity99/).  In theses [9,10,11] on 9-th Belgian-French-German Conference on Optimization, September, 7-11, 1998, Namur, Belgium (WWW page: http://www.fundp.ac.be/ bfgconf9/).  In theses [6,7] on 11-th Baikal International School-Seminar "Optimization Methods and their Applications", July, 5-12, 1998, Baikal, Irkytsk, Russia.  In theses [5] on III-rd International Conference "Mathematical Methods and Computers in Economics", May, 28-29, 1998, Penza, Russia. The potential results obtained during a realization of the proposal would be joint published in international journals of a good standard (the Journal of Complexity, European Journal of Operational Research, Mathematics and Economics) and the monograph would appear in a world known publishing house. As a result of the project proceedings of well reputed international conferences are expected: 6

m

 IME 2000, Fourth International Congress on Insurance: Mathematics

and Economics, July 24-26, 2000, Barcelona, Spain, UK (WWW page: http://www.ub.es/congres/ime2000/).  3rd European Congress of Mathematics, July 10 - 14, 2000, Barcelona, Spain (WWW page: http://www.iec.es/3ecm/).  DAOR-2000, Siberian Conference on Discrete Analysis and Operational Research, 26 June- 1July, 2000, Novosibirsk, Russia (WWW page: http://www.math.nsc.ru/conference/DAOR/daor.html).  Franch-German-Italian Conference on Optimization, 4-8 September, 2000, Montpellier, France (WWW page: http://www.math.univ-montp2.fr/fgi2000.html).  Symposium on Risk Management in the Global Economy: Measurement, Management and Macroeconomic Implications, 21-23 September, 2000, Chicago, USA.  9th Tor Vergata Financial Conference "Risk and Regulation in the Global Financial Market", 15-17 November, 2000, Rome, Italy. It is also planned to defend one habilitation.

7 References: 1. Graham R.L., Lawler E.L., Lenstra J.K., Rinnooy Kan A.H.G. Optimization and Approximation in Deterministic Sequencing and Scheduling: a Survey. Ann. Discrete Math. 5 (1979), 287-326. 2. Markowitz H.M. Portfolio Selection, J.Wiley & Sons Inc., New York, 1959. 3. Lazarev A.A. Ecient Algorithms of Decisions some Problems of Theory of Scheduling for Single Machine with Deadline Terms of Service Jobs.// Dissertation, 1989. 108 p. 4. Berkyta O.N., Lazarev A.A. Algorithm of Diculty O(n ) Deciding of Problem to Minimization of Total Tardiness.// Investigations on Applied Mathematics, Vol. 22, Kazan, 1995, 67-78. 5. Lazarev A.A., Siraev R.R. Solving the Problem of Calendar Planning by the Branch and Bound Method.// Proceedings of III-rd International Conference "Mathematical Methods and Computers in Economics", May, 2829, 1998, Part I, Penza, Russia, 27-28. 6. Lazarev A.A., Siraev R.R. Solving the Problem of Calendar Planning by the Branch and Bound Method.// Proceedings of 11-th Baikal International School-Seminar "Optimization Methods and their Applications", 3

7

7.

8. 9. 10.

11.

12. 13. 14. 15.

16.

July, 5-12, 1998, Section 1: Mathematical Programming, Irkytsk, Russia, 159-162. Lazarev A.A., Shul'gina O.N. A Pseudopolinomial Algorithm for Solving NP-hard Problem of Maximal Time Shift Minimization.// Proceedings of 11-th Baikal International School-Seminar "Optimization Methods and their Applications", July, 5-12, 1998, Section 1: Mathematical Programming, Irkytsk, Russia, 163-167. Lazarev A.A. Scheduling to Minimize Maximum Lateness for Single Machine: New Approach of Investigation.// 9-th Belgian-French-German Conference on Optimization, September, 7-11, 1998, Namur, Belgium. Lazarev A.A., Siraev R. R. Scheduling to Minimize Total Weighted Completion Time: Branch and Bound Method. // 9th Belgian-French-German Conference on Optimization, September, 7-11, 1998, Namur, Belgium. Lazarev A.A., Shul'gina O.N. Problem Minimizing Maximum Lateness for Single Machine: Properties, Procedures, Algorithms.// 9-th BelgianFrench-German Conference on Optimization, September, 7-11, 1998, Namur, Belgium. Lazarev A.A. Scheduling to Minimize Maximum Lateness and to Minimize Maximum Completion for Single Machine: Minimum Absolute Error, 1 j dj = rj + pj j Lmax , 1 j dj = rj + pj ; Lmax  y j max , 1 j dj = rj + pj j Lmax ; Cmax.// European Journal of Operational Research (in print). Lazarev A.A. Minimum absolute error for NP-hard scheduling problem for single machine - minimizing maximum lateness//the Journal of Comlexity (in print). Lazarev A.A. Scheduling algorithms based on necessary optimality//Journal of Soviet Mathematics, V 44, N 5 (1989), 635-642. Lazarev A.A. Algorithms in the schedule theory based on necessary conditions of optimality (Russian)//Issledovania Prikladnoy Matematike, N 10 (1984), 102-110. Lazarev A.A. Analize of structure of optimal schedule the problem minimizing maximum lateness for single machine// the Fourth International Congress on Industrial and Applied Mathematics, 5-9 July 1999, Edinburgh, Scotland, p 283. Lazarev A.A. Minimum absolute error for NP-hard scheduling problem for single machine - minimizing maximum lateness// Workshop on the Complexity of Multivariate Problems, October 4-8, 1999, Hong Kong, China, p. 13. 8