Alexander A. Pavlov

Planning Automation in Discrete Systems with a Given Structure of Technological Processes

In this paper, we consider mathematical models and algorithms for efficient planning process automation in discrete systems of a wide class (innovative software development, small-scale production). An effective solution of the proposed models is based on earlier results of M.Z. Zgurovsky, A.A. Pavlov, E.B. Misura, and E.A. Khalus in the field of intractable single stage single machine scheduling problems.

NP-Hard Scheduling Problems in Planning Process Automation in Discrete Systems of Certain Classes

In this paper, we consider an intractable problem of total tardiness of tasks minimization on single machine. The problem has a broad applications solutions during planning process automation in systems in various spheres of human activity. We investigate the solutions obtained by the exact algorithm for this problem earlier developed by M.Z. Zgurovsky and A.A. Pavlov. We propose an efficient approximation algorithm with ( Oleft( {n^{2} } right) ) complexity with estimate of the maximum possible deviation from optimum. We calculate the estimate separately for each problem instance. Based on this estimate, we construct an efficient estimate of the deviation from the optimum for solutions obtained by any heuristic algorithms. Our statistical studies have revealed the conditions under which our approximation algorithm statistically significantly yields a solution within 1–2% deviation from the optimum, presumably for any problem size. This makes possible obtaining efficient approximate solutions for real practical size problems that cannot be solved with known exact methods.

The Total Weighted Tardiness of Tasks Minimization on a Single Machine

We solve the problem of constructing a schedule for a single machine with various due dates and a fixed start time of the machine that minimizes the sum of weighted tardiness of tasks in relation to their due dates. The problem is NP-hard in the strong sense and is one of the most known intractable combinatorial optimization problems. Unlike other PSC-algorithms in this monograph, in this chapter we present an efficient PSC-algorithm which, in addition to the first and second polynomial components (the first one contains twelve sufficient signs of optimality of a feasible schedule) includes exact subalgorithm for its solving. We have obtained the sufficient conditions that are constructively verified in the process of its execution. If the conditions are true, the exact subalgorithm becomes polynomial. We give statistical studies of the developed algorithm and show the solving of well-known examples of the problem. We present an approximation algorithm (the second polynomial component) based on the exact algorithm. Average statistical estimate of deviation of an approximate solution from the optimum does not exceed 5% of it.

The Total Earliness/Tardiness Minimization on a Single Machine with Arbitrary Due Dates

We solve a single machine problem of constructing a schedule of tasks with arbitrary due dates on a single machine that minimizes the total earliness/tardiness in relation to their due dates. This problem is solved in three different formulations: (1) the start time of the machine is fixed. In this case the problem is NP-hard; (2) the start time of the machine belongs to a specified time segment. The problem is intractable because there is no exact polynomial algorithm for its solving; (3) the start time of the machine is arbitrary. The problem is intractable because there is no exact polynomial algorithm for its solving. For the first two problems we give PSC-algorithms, each of them contains sufficient signs of a feasible solution optimality and is based on the optimal solution for the single machine problem to minimize the total tardiness of tasks in relation to their various due dates (with equal weights). We show that the PSC-algorithm of its solving is a simplified modification of the PSC-algorithm presented in Chap. 4. For the third problem solving we build an efficient approximation algorithm.

The Total Tardiness of Tasks Minimization on Identical Parallel Machines with Arbitrary Fixed Times of Their Start and a Common Due Date

We solve a NP-hard problem of constructing a schedule for identical parallel machines that minimizes the total tardiness of tasks in relation to a common due date in case when the start times of machines are fixed at arbitrary time points less than the due date. We present an efficient PSC-algorithm of its solving which is a generalization of our previously developed results: for the problem with equal start times of machines we have derived two sufficient signs of optimality of a feasible solution and constructed two PSC-algorithms. Each of the algorithms checks one of these signs. In this chapter we propose a generalized PSC-algorithm for equal start times of machines that combines the best properties of both PSC-algorithms. We have obtained a modification of the generalized PSC-algorithm for the case of arbitrary start times of machines, its complexity is determined by O(n2m) function. The first polynomial component of the PSC-algorithm coincides with its second polynomial component. We obtain an efficient estimate of the deviation from an optimal solution for an approximation algorithm of the problem solving. We also present the statistical studies of the PSC-algorithm that showed its high efficiency (efficient exact solutions have been obtained for problems with tens of thousands of variables which is unique for NP-hard combinatorial optimization problems).

The Four-Level Model of Planning and Decision Making

We provide an overview of known models, methods and software for scheduling and operational planning for objects with a network representation of technological processes and limited resources. The problem is an operational plan construction to produce a potential order portfolio which is the best in terms of criteria defined by the customer. We make the conclusion that an immediate solution of this problem (multi-stage network scheduling problem) is inefficient. The result of analysis is the four-level model of planning (including operational) and decision making, in which we formalize formal procedures both for obtaining an operational schedule and for its operative adjustment. The four-level model includes the combinatorial optimization problems presented in Chaps. 2– 7 as well as the Decision Making Unit, a subsystem that performs decision making functions in case if various events appear during planning. In the Decision Making Unit we use our modified Analytic Hierarchy Process which is based on the research of empirical pairwise comparisons matrices with the help of combinatorial optimization models with weighted components of the additive functional.

Algorithms and Software of the Four-Level Model of Planning and Decision Making

We give an interrelated description for all algorithms which implement a procedure of scheduling and operative planning on the basis of formal procedures and expert solutions. We have achieved this result sequentially solving the following problems: (1) we formalize the network representation of a technological process; develop a language which implements the dynamics of the process functioning, define a set of practical optimality criteria for the operational plan; (2) we obtain a formal representation of the technological process, adapt the general procedures of its two-level aggregation; (3) we substantiate the scheduling problem reduction to the approximating problem of the total weighted completion times minimization on a single machine with precedence relations between the tasks; (4) we formalize the procedure of the coordinated planning which defines the processing order of products (or product series) corresponding to a single basic criterion; (5) we formalize the process of an operational plan construction based on results of the coordinated planning; (6) we develop two algorithms for an operational plan adjustment; (7) we substantiate a procedure for a decision maker for the expert evaluation and adjustment both of a potential portfolio of orders and the due dates of the products (or product series) included in it, for the approval of the final operational plan on the basis of analysis of plans obtained by the formal methods. The last section describes an informational software system implementing the hierarchical model in a specific practical area: for solving the problems of scheduling and planning for small-series productions.

Optimal Scheduling for Vector or Scalar Criterion on Parallel Machines with Arbitrary Due Dates of Tasks

We examine the problem of constructing a feasible schedule for parallel machines of equal or various productivities with various due dates of tasks and arbitrary start times of machines. We consider two optimality criteria: maximizing the earliest start time of machines (scalar criterion) or obtaining optimal start times of machines subject to a direct lexicographical order of their assignment (vector criterion). In fact, in this chapter we examine four intractable combinatorial optimization problems (for each one there are no efficient polynomial algorithms for its solving). Each of four presented PSC-algorithms contains separate sufficient signs of optimality of a feasible solution, the first polynomial component that checks the sufficient signs of optimality, and an approximation algorithm. Since the formulated problems are quite complex, each component of the PSC-algorithms contain many subalgorithms, each of which implements a separate original heuristic. We give examples of the problems solving.

The Total Weighted Completion Time of Tasks Minimization with Precedence Relations on a Single Machine

We consider the problem of constructing a schedule for a single machine that minimizes the total weighted completion time of tasks when the restrictions on their processing order are given by an arbitrary oriented acyclic graph. The problem is NP-hard in the strong sense. Efficient polynomial algorithms for its solving are known only for cases when the oriented acyclic graph is a tree or a series-parallel graph. We give a new efficient PSC-algorithm of its solving. It is based on our earlier theoretical and practical results and solves the problem with precedence relations specified by an oriented acyclic graph of the general form. The first polynomial component of the PSC-algorithm contains sixteen sufficient signs of optimality. One of them will be statistically significantly satisfied at each iteration of the algorithm when solving randomly generated problem instances. In case when the sufficient signs of optimality fail, the PSC-algorithm is an efficient approximation algorithm. If the sufficient signs of optimality are satisfied at each iteration then the algorithm becomes exact. We present the empirical properties of the PSC-algorithm on the basis of statistical studies.

Optimal Scheduling for Two Criteria for a Single Machine with Arbitrary Due Dates of Tasks

We consider the problem of constructing a feasible (in which all tasks complete before their due dates) schedule for a single machine with arbitrary due dates and maximum start time of the machine or minimum total earliness of the tasks completion times in relation to their due dates. It is shown that for the criterion of maximum start time of the machine the problem is polynomially solvable, we give a polynomial algorithm for its solving. With a fixed start time of the machine the problem is polynomially solvable for the criterion of minimizing the total earliness of the completion times of the tasks if qualitatively proven and statistically significant properties of an optimal solution (Heuristics 1 and 2) are met. The problem with an arbitrary start time of the machine to construct an optimal schedule minimizing the total earliness of the tasks completion times is intractable: an exact polynomial algorithm for its solving is not known. For the case if the Heuristics 1 and 2 are true for an arbitrary start time of the machine, we develop an efficient PSC-algorithm. For the opposite case, this PSC-algorithm is an efficient approximation algorithm for the problem solving.