Environmental safety construction programs optimization and reliability of their implementation

This paper shows a problem of creating the construction programs to ensure the environmental safety with regard to their reliability. The problem is in choosing the right projects for the program to achieve the required effect with minimum costs by restriction either the number of highrisk projects or their funding amount. This paper suggests algorithms of solving the problems using Branch and Bound method and Costeffectiveness analysis.


Introduction
In recent years, the problems of creating safety programs of different types, i.e. reforming of enterprises, regional development, traffic safety, destruction of chemical weapons, safety in emergency situations have developed considerably in terms of applications of mathematical methods.In formal terms the problems, in general, have the following statements.There are a lot of construction projects that are candidates for participation in the program.Each project is characterized by effects -project contribution to a program, costs on implementation and reliability.
Reliability is estimated either by the probability of a project realization or qualitative characteristics: high, medium or low reliability, respectively, there is a high, medium or low risk project.Reliability of a program is defined as either the probability of a successful program implementation or a qualitative characteristic, i.e. a number of projects with high and medium risk or the funding amount of such projects.Accordingly, the problem is to choose projects that ensure the implementation of the program objectives with minimum costs by restriction either the probability of successful implementation of the program or a number of projects with a high and medium risk, or funding amount of such projects.
The aim of this paper is to develop a methodological approach for creating environmental safety programs taking into account the reliability of their implementation.To that end, the authors suggest to use a well-known Branch and Bound method and its modification based on the Cost-effectiveness analysis to simplify the algorithm of achieving the goal.Branch

Problem statement
The program is estimated in m criteria.The status of each criterion is usually scored on the scale: bad -1, satisfactory -2, good -3, excellent -4.In recent years, creating of an integrated assessment program based on the matrix package has become popular.Accordingly, the objective of a program is to achieve either the required value of scores on directions or the required integrated assessment.
There are n projects, i.e. candidates for the participation in a program.Each project is characterized by aij effects (contributions) that it has in one or more directions of the program 1, , where Qi is a set of directions to which an i-type project contributes to.As a rule, it is assumed that effects are added.Each project can be implemented in two optionswith low risk or high risk.Let's denote bi as the costs of implementing a project with low risk and ci as the costs of implementing a project with high riskit is clear that , 1, To define the number of projects that ensures the achievement of objectives in all program directions (or the required values of the integrated assessment) with minimum costs by restriction either the financing amount of high-risk projects or their number.Let's consider the example when each program has its set of projects, i.e. all projects are single-purposeeffect only in one direction.In this case, the problem is solved separately for each direction.

MATEC
In formal statement of the problem let xi = 1 if the i-type project was included in the program with low risk and xi = 0 to the contrary.Accordingly, let yi = 1 if the i-type project was included in the program with high risk and yi = 0 to the contrary.It is obvious, that 1, 1, At given , , 1, , the effect for the appropriate direction of the program will be Assume that the set of boundary values of the effect , 1, 4 the score is equal to j.
To account risk limits denote C as the maximum financing amount of risky projects and p as the maximum number of high-risk projects, then the appropriate limits have the form or The problem is to find , , 1, with delimitations of (2), ( 4) or ( 5) and delimitation of where A is equal to one of Aj values depending on the set objective of the direction.

Branch and Bound method
To apply the Branch and Bound method it is necessary to have means of estimating from below the subsets of solutions [20].Consider the algorithm for obtaining the lower bounds.
To that end, let's first turn to new variables , 1, In the new variables the problem will be to minimize where b c i i i    with delimitations of (4), ( 8) and ( 9): Consider two evaluation problems.
Denote W1 as a value of F1(z) in the optimal solution of the Problem 1 and W2 as a value of F2(y) in the optimal solution of the Problem 2.
Use estimation (12) in the Branch and Bound method.
Lemma 1.If there are optimal solutions z and y of the Problems 1 and 2 so that y < z, ( , 1, ) , the pair (z, y) finds the optimal solution of the problem.The proof is obvious as this pair (z, y) is a feasible solution.

Branch algorithm description
Step 1. Solve the Problems 1 and 2. Denote M1 as the set of optimal solutions of the Problem 1 and M2 as the set of optimal solutions of the Problem 2. If there is a pair (z, y), , 1 2 z M y M   and y < z, Lemma (z, y) is the optimal solution.Otherwise, let's choose the j-type project so that yj = 1 and zj = 0, and partition the set of all solutions into two subsets.
In the first one yj = 1, zj = 1, and in the second one yj = 0. Next, for these subsets let's solve the evaluation problems; choose the subset with the best evaluation, etc. in accordance with the scheme of the Branch and Bound method.
Example 1.There are 7 projects which details are shown in Table 1.The optimal option corresponds to a cell (170; 145).Get solution: Solve the Problem 2 that is also a knapsack problem.Take the structure of a dichotomic representation from Fig. 1.The optimal solution is determined by the cell (67; 73).Applying the "reverse order" method, define the optimal solution: y1 = y4 = y5 = y6 = 0, y2 = y3 = y7 = 1 with the costs 67 and effect 73. W1 = 170, W2 = 73.Obtain the lower bound that is W1 -W2 = 97.Since y3 = 1 and z3 = 0, this solution is not a feasible one for the initial problem.
Carry out branching of the project 3.In the first subset y3 = 1, z3 = 1, and in the second one y3 = 0.
Estimate the first subset.Since y3 = 1, the solution of the second problem does not change.Exclude all options when z3 = 0 in the first problem.It is obvious that the resulting table will have the same table without the first three lines.The optimal solution is determined by the cell (175; 155) with the costs 175.Let's estimate from below the first subset F(y3 = 1) = 175 -73 = 102.Estimate the second subset.Since y3 = 0, the solution to the first problem does not change.Consider the second problem by setting y3 = 0. Perform the summary Table 4.The optimal solution is determined by the cell (62; 63) with the costs 62 and effect W2(y3 = 0) = 63.Let's estimate from below the second subset W2(y3 = 0) = 170 -63 = 107.Choose the first subset with a lower bound.The corresponding solution has the form of z1 = z2 = z4 = z6 = 0, z3 = z5 = z7 = 1.Comparing with the solution of the second problem y1 = y4 = y5 = y6 = 0, y2 = y3 = y7 = 1, we see that y2 = 1 and z2 = 0. Let's carry out branching of the second project, partitioning it into two subsets (y2 = 1).The solution of the second problem does not change.

Cost-effect method
The described method of obtaining the lower bounds requires at each step of branching solving a knapsack problem.There is an efficient approximate algorithm to solve the knapsack problem that is called the Cost-effect method; its brief description was given in Introduction.Apply this method to obtain the approximate lower bounds in the method of Branch and Bound.To that end, let's define the effectiveness of projects contribution to the Arrange the projects in descending order of the efficiencies for Problems 1 and 2 respectivelysee Tables 6 and 7.Under the Cost-effect method the projects are chosen in accordance with the indicated orderings.Apply this method to the solution of Example 1. Carry out branching of the project 2.
The result can be improved if in the last step of selection to look through all the projects the inclusion of which provides the desired effect, and then choose among them a project with minimal costs.This technique insignificantly increases the volume of calculation but allows in some cases to improve the result.So, if in the last step to take project 4 instead of project 5, the costs will decrease by 20.

Example 2 .
Solve the Problem 1.The solution contains projects 7, 3, and 5 with the costs 175.Solve the Problem 2. The solution contains projects 2, 3, 1, 4 with the effect 78. Obtain the lower bound that is W = 175-78 = 97.

Table 1 .
Data on projects.Structure of the network representation of the problem.
Take A = 140 and C = 70.Solve the Problem 1.This is a knapsack problem.Let's solve it by the dichotomic programming method.The structure of the network representation of the problem is shown in Fig.1.

Table 5 .
Projects efficiency for the directions development.