INT J COMPUT COMMUN, ISSN 1841-9836 9(2):151-159, April, 2014. Algorithm of Maximizing the Set of Common Solutions for Several MCDM Problems and it’s Application for Security Personnel Scheduling S. Dadelo, A. Krylovas, N. Kosareva, E.K. Zavadskas, R. Dadeliene Stanislav Dadelo, Aleksandras Krylovas, Natalja Kosareva*, Edmundas Kazimieras Zavadskas Vilnius Gediminas Technical University Sauletekio al. 11, LT-10223 Vilnius, Lithuania E-mail: stanislav.dadelo@vgtu.lt, aleksandras.krylovas@vgtu.lt, natalja.kosareva@vgtu.lt, edmundas.zavadskas@vgtu.lt *Corresponding author: natalja.kosareva@vgtu.lt Ruta Dadeliene Lithuanian University of Educational Sciences Studentu st. 39, LT-08106 Vilnius, Lithuania E-mail: ruta.dadeliene@leu.lt Abstract: The article deals with the task of elite selection of private security per- sonnel on the basis of objective and subjective criteria. One of the possible solutions of this multiple criteria decision making (MCDM) problem is creation of heuristics allowing to minimize discrepancy of ranks calculated for the objective and subjec- tive criteria on the basis of the best security staff. The proposed heuristic combines interval points re-selection and random points generation methods. Two optimizing algorithms are proposed. It is shown how this method is applied for solving specific task of elite selection from security personnel. Keywords: preference feature, multiple criteria decision making, heuristics, optimiz- ing algorithm. 1 Introduction EU private security sector employs millions of people and this figure is constantly growing [1]. Economic trends provide further development of the private security. It is predicted that over the next 10 years, the U.S. private security sector job demand will increase by 15% and this is much higher growth rate compared to the overall job growth around the U.S. economy [2]. Safety business formed specific personnel management challenges. Personnel selection and placement to the necessary positions (ranking) is seen as the most important factor affecting the organizations security, stability and development [3] and for private security this process takes on a deeper meaning. Personnel selection process focuses on measurement and evaluation of specific potential, skills and personal characteristics of candidates. Security personnel evaluation process requires identification of specific characteristics (criteria) for the occupation of the post and their weights determination [4]. It is necessary to develop universal algorithms for personnel selection [5]. Developed selection and evaluation systems are mainly focused on companies operating cost reduction, in order to optimize staffing requirements and layout planning [6], but is not enough in the safety field. Assessing the activities of private security, their complexity and the peculiarities of the hazard, the following complexity – hazard levels are distinguished: 1) protection of civil objects; 2) protection of critical and strategic objects, collection, and personal protection [7]. Evolution of the modern world generated the demand of private security services in military operations. However, armed security services in the "hot" spots of the world providing companies are often confronted with the staff inappropriate behavior in dangerous and emergency situations. Assessing the private security business diversity, personnel evaluation and selection procedure Copyright © 2006-2014 by CCC Publications 152 S. Dadelo, A. Krylovas, N. Kosareva, E.K. Zavadskas, R. Dadeliene for different tasks becomes necessity. There is a need to look for factors affecting the security staff competence and professional activities as a basis for generation of algorithms for employees evaluation and selection for different tasks procedures [8]. Evaluating of personnel engaged in security features in dangerous environment it is necessary to search for specific solutions [3]. Critical tasks in hazardous environments, which may result in loss of health and even life, must carry only the elite guards. Ryan et al. [9] note that only about 10 – 12 % of the total sample of candidates are able to meet the demands posed by members of special elite forces. Dessler [10] states that only 15% of the total sample of individuals can be evaluated in the maximum. Recent research reveals the elite diagnostic possibilities according to six sigma principle [11]. Private security personnel selection process should point competency assessment in two directions: external (evaluation of subordinate by his immediate superior) and internal (personal competence) [12]. Relevance of the problem occurs in the construction of selection algorithms for private security elite, capable of effectively carry out extremely important and dangerous tasks. This scientific problem is not paid enough attention. The aim of this study is construction approval of private security elite selection algorithm. A similar kind of research has not yet been done. An overview of possible MCDM methods for this problem solution is provided by Zavadskas and Turskis [13]. Recently fuzzy MCDM methods are becoming increasingly popular [14]. In this article heuristic methods, based on the function, minimizing non-compliance of ranks, calculated for the best security staff in accordance with objective and subjective criteria are proposed. 2 Internal and external evaluation criteria Participants. One hundred and eighteen security guards were randomly selected from the company G4S Lietuva; twenty two leader managers (experts) of G4S Lietuva with not less than 10 years of service at private security structures involving execution and organization of security ranked the competences chosen by authors of the article. Security guards internal evaluation (x) (objective testing, measuring, personal competence). Selected security guards were tested and evaluated according to 41 criteria. The data received were classified into six groups of competences (variables) regarding the features analysed [12] (Table 1): 1. Theoretical and practical preparation (x1): knowledge, skills, abilities, practical experience – acquired throughout life; 2. Professional activity (x2): carrying out required tasks; 3. Mental qualities (x3): individual psychical qualities vital for performance of professional activities; 4. Physical development (x4): morphological indications of a body; 5. Motor abilities (x5): personal physical conditions allowing carrying out physical tasks at work or home, during leisure, and reflecting the level of physical qualities; 6. Fighting efficiency (x6): a set of physical and mental qualities influencing the ability to effectively carry out actions fighting an adversary in direct contact. Table1: Security guards internal evaluation criteria (objective tests and measurements) Algorithm of Maximizing the Set of Common Solutions for Several MCDM Problems and it’s Application for Security Personnel Scheduling 153 Security guards Criteria (competences) x1 x2 x3 x4 x5 x6 max (highest values) 2.582 2.186 2.651 3.388 2.768 1.696 min (lowest values) −1.606 −2.567 −2.518 −1.948 −2.596 −1.709 a1 1.358 −1.503 0.607 0.520 1.381 1.696 a2 0.259 −0.549 −0.424 −1.487 −0.645 0.068 a3 −0.984 0.465 0.101 −0.479 0.690 −0.100 ... ... ... ... ... ... ... a116 0.225 −0.500 −0.837 −0.599 0.749 0.795 a117 0.561 −1.366 0.438 −0.017 0.013 −0.903 a118 −1.512 −1.391 −2.180 −1.666 −1.520 −0.845 Security guards external evaluation (y) (evaluation of subordinate by his immediate supe- rior). Team leaders evaluated their subordinates based on A. Sakalas [15] modified methodology (Table 2). They have been assessed on nine criteria that could affect security guards professional competences: 1. Specialty knowledge, professionalism (y1): versatility, knowledge about their and related occupations; 2. Diligence and positive attitude to work (y2): activeness, responsibility, discipline, zeal, vocation to work; 3. Behaviour with colleagues and supervisors (y3): the ability to cooperate and work in a team; 4. Reliability at work (y4): ability and willingness to perform tasks independently; 5. Quality of work (y5): the ability to avoid mistakes; 6. Workload performance (y6): the ability to carry out the maximum amount of work; 7. Image (y7): the self-representational skills (exterior, posture, language culture); 8. Development rate (y8): the ability to quickly adapt to new requirements and new working conditions; 9. Being promissing (y9): potential for career. Table 2: Security guards external evaluation criteria (evaluation of subordinate by his imme- diate superior) Security guards Criteria (competences) y1 y2 y3 y4 y5 y6 y7 y8 y9 max (highest values) 5 5 5 5 5 5 5 5 5 min (lowest values) 0 0 0 0 0 0 0 0 0 a1 3 4 4 4 4 4 4 4 4 a2 4 4 4 4 5 3 3 4 4 a3 3 4 4 4 4 4 4 4 4 ... ... ... ... ... ... ... ... ... ... a116 2 3 3 4 3 3 3 3 2 a117 1 2 2 2 2 3 2 1 1 a118 3 3 4 3 3 3 3 3 2 Security staff competency assessment is depicted in Figure 1. 3 Mathematical model Definition. Suppose that vector of real components X = (x1,x2, . . . ,xn) ,(∀j)xj > 0, has the meaning of some measurable values and is the set of n criteria values. Any function 154 S. Dadelo, A. Krylovas, N. Kosareva, E.K. Zavadskas, R. Dadeliene Figure 1: Research progress φ : Rn → R will be called criteria convolution, if it nondecreasing for each variable, i. e. (∀j = 1,2, . . .n,∀δ > 0) φ(. . . ,xj + δ, . . .) > φ(. . . ,xj, . . .) . (1) Note, that if for the two alternatives X(1) and X(2) the Pareto dominance is valid X(1) ≻ X(2) (∀j = 1,2, . . .n : x(1)j > x (2) j and ∃i : x (1) i > x (2) i ), then inequality (1) will be true for the criteria convolution: φ ( X(1) ) > φ ( X(2) ) . Suppose, that permutation (j1,j2, . . . ,jn) of natural numbers {1,2, . . . ,n} determine the criterion X components xj preferences xj1 ≻ xj2 ≻ · · · ≻ xjn . For example, permutation (2,3,1) means that the most important influence has the second component of criterion X = (x1,x2,x3), and the least influence has the first component. Definition. Let’s say that criteria convolution φ has preferences feature (j1,j2, . . . ,jn), if all the inequalities are valid: (∀i < k & δ > 0) φ(. . . ,xi + δ, . . . ,xk, . . .) > φ(. . . ,xi, . . . ,xk + δ, . . .) , (2) when ji > jk. Note, that criteria convolutions determined on the basis of weighted averages have preferences feature defined by (2). For example, criteria convolution φ(X) = w1x1 + w2x2 + w3x3 + w4x4 w1 + w2 + w3 + w4 , when w2 > w3 > w1 > w4 > 0 has preferences feature (2,3,1,4). Convolution φ(X) = x p1 1 x p2 2 x p3 3 , when p2 > p1 > p3 > 0 has preferences feature (2,1,3). 3.1 Formulation of optimization problems Suppose that for the certain alternatives criteria-referenced evaluations are known X(k) =( x (k) 1 ,x (k) 2 , . . . ,x (k) n ) , Y (k) = ( y (k) 1 ,y (k) 2 , . . . ,y (k) m ) , k = 1,2, . . . ,K and criteria X, Y preferences are determined: (j1,j2, . . . ,jn), (i1, i2, . . . , im). Suppose that φ and ψ are convolutions of criteria X and Y respectively, having correspond- ing preferences features. For each alternative ( X(k),Y (k) ) values of both criteria convolutions Algorithm of Maximizing the Set of Common Solutions for Several MCDM Problems and it’s Application for Security Personnel Scheduling 155 φ ( X(k) ) and ψ ( Y (k) ) are calculated and numbered in ascending order to obtain ranks of alter- natives: r(k)x , r (k) y . Note that the lower is the rank, the better is alternative. Let’s denote AKx and BKy sets of the best alternatives – those subsets of the set {1,2, . . . ,K} whose items ranks r(k)x < Kx and r (k) y < Ky respectively. Here Kx, Ky mean the proportion of the best alternatives to be selected (generally 10–15%, i. e. Kx,Ky ∈ [0.10K,0.15K]). Denote S(j1,j2,...,jn) class of convolutions having preferences feature (j1,j2, . . . ,jn). In this class of functions we’ll search for the function φ (analogously ψ). Denote ∣∣∣AKx ∩ BKy∣∣∣ number of elements in intersection of sets AKx and BKy and formulate the optimization problem: max φ ∈ S(j1,j2,...,jn) ψ ∈ S(i1,i2,...,im) ∣∣∣AKx ∩ BKy∣∣∣ . (3) Notice that if Kx = Ky = K the problem (3) has trivial solution AKx ∩ BKy = {1,2, . . . ,K} for any convolutions φ, ψ. Decreasing values of parameters Kx, Ky imply the decreasing number of common elements (intersection) of the sets AKx , BKy . If this number is less than necessary, we mus increase values of parameters Kx, Ky. This means that criteria X and Y don’t match. Choose the numbers K′x 6 Kx, K′y 6 Ky. The incompatibility of criteria X and Y is quan- titatively described by number of all elements of the union of sets AK′x and BK′y which don’t belong to the intersection in the right side of formula (3), i. e. the alternatives with high rank according to the one and low rank according to another criterion. So the problem (3) could be supplemented with additional one min φ ∈ S(j1,j2,...,jn) ψ ∈ S(i1,i2,...,im) ∣∣∣(AK′x \ BKy) ∪ (BK′y \ AKx)∣∣∣ . (4) Notice that the problem (4) also has trivial solution AK′x ∪ BK′y = ∅, if K ′ x = K ′ y = 1. In general, this means a certain amount of "fine" for the ignorance of dubious alternatives and numbers K′x 6 Kx, K′y 6 Ky must to be the larger. 3.2 Restrictions and general scheme of the research Formulated optimization problems (3) and (4) aren’t easy. Construction of the effective algorithms for their solving is interesting, but poorly investigated issue. Efficiency of these algorithms should strongly depend on the class of criteria convolutions S(j1,j2,...,jn). In this article we restrict our investigation with the functions from the class of the weighted averages: φ(j1,j2,...,jn) (x1,x2, . . . ,xn) = n∑ i=1 wjixji n∑ i=1 wi . (5) here wj1 > wj2 > · · · > wjn > 0. Preferences feature (j1,j2, . . . ,jn) is determined by experts estimates of the form xj1 ≻ xj2 ≻ · · · ≻ xjn . In this article we limit ourselves to calculation of the average estimates for criteria ranks and numbering of xj accordingly. There are many various ranking methods and their comparison will be our further investigation. 156 S. Dadelo, A. Krylovas, N. Kosareva, E.K. Zavadskas, R. Dadeliene 3.3 Solving the personnel selection problem (case study) The particular managerial task is solved in the paper. It requires finding an acceptable alter- native sets having a clear and easily verifiable interpretation of the subject matter. Therefore, the comparison of solutions received by various methods could be their interesting practical suitability criteria and have a long-term value. Task parameters are as follows: the number of alternatives (security personnel compared with each other) – K = 118; the number of vector criterion X components (physical measurements of emploees) n = 6; the number of vector criterion Y components (managers assessment) m = 9. Criteria X and Y components priority was determined from expert estimates. Each of 22 experts had to specify priority permutations xj1 ≻ xj2 ≻ · · · ≻ xjn and yi1 ≻ yi2 ≻ · · · ≻ yim . These evaluations are shown in the Table 3: Table 3: Criteria X and Y components ranks Expert x1 x2 x3 x4 x5 x6 y1 y2 y3 y4 y5 y6 y7 y8 y9 1 6 1 5 3 2 4 7 9 4 8 5 3 6 2 1 2 5 2 3 1 6 4 7 9 5 6 8 2 3 4 1 3 5 1 3 2 6 4 5 9 4 6 8 1 7 3 2 4 6 1 5 3 4 2 5 9 7 8 6 3 4 2 1 5 6 1 5 2 3 4 7 9 6 8 5 3 4 2 1 6 5 1 4 2 6 3 5 9 6 8 7 3 4 2 1 7 4 2 3 1 6 5 4 8 5 9 7 2 6 3 1 8 6 2 5 3 4 1 6 8 5 9 7 3 4 2 1 9 5 4 6 3 2 1 7 9 4 8 6 3 5 1 2 10 5 4 6 1 2 3 6 7 4 9 8 3 5 2 1 11 4 1 3 2 6 5 5 8 6 9 7 2 4 3 1 12 6 2 4 1 5 3 6 9 5 7 8 4 3 2 1 13 6 4 5 1 3 2 5 9 6 8 7 3 4 2 1 14 6 3 5 1 4 2 7 8 6 9 5 4 3 2 1 15 4 1 3 2 6 5 5 9 7 8 6 3 2 4 1 16 4 1 3 2 6 5 7 9 8 6 5 4 2 3 1 17 6 1 4 2 3 5 6 8 7 9 4 5 2 3 1 18 6 3 4 1 5 2 6 9 8 7 4 5 2 3 1 19 4 1 5 2 6 3 4 9 8 6 7 1 5 3 2 20 6 2 4 1 5 3 5 9 6 7 8 3 4 2 1 21 5 2 6 3 4 1 6 8 7 9 5 4 3 2 1 22 4 1 6 3 2 5 6 9 7 8 5 4 2 3 1 Σ 114 41 97 42 96 72 127 190 131 172 138 68 84 55 25 ri 5.18 1.86 4.41 1.91 4.36 3.27 5.77 8.64 5.96 7.82 6.27 3.09 3.82 2.50 1.14 rank 1 6 2 5 3 4 5 1 4 2 3 7 6 8 9 prefs X : (1,3,5,6,4,2) Y : (2,4,5,3,1,7,6,8,9) Component’s X and Y ranks preferences were calculated by method of sum (average) of ranks (i. e. the average place of each component was calculated and preferences drawn up. 3.4 Optimization algorithm The heuristic is used for solving problem (3). Functions φ(X) and ψ(Y ) will be selected from the class of functions (5) as follows: φ(X) = α1x1 + α3x3 + α5x5 + α6x6 + α4x4 + α2x2, α1 > α3 > α5 > α6 > α4 > α2 > 0, Algorithm of Maximizing the Set of Common Solutions for Several MCDM Problems and it’s Application for Security Personnel Scheduling 157 by analogy accordingly to the preference feature given in the last row of Table 3: ψ(Y ) = β2y2 + β4y4 + · · · + β9y9, β2 > β4 > β5 > · · · > β8 > β9 > 0. Weighted coefficients αj and βi are determined so as to satisfy the normalizing condition α1 + · · · + α6 = β1 + · · · + β9 = 1. (6) Designate R(k)(α) and R(k)(β),k = 1,2, . . . ,K ranks of numbers { φ ( X(1) ) ,φ ( X(2) ) , . . . ,φ ( X(K) )} and { ψ ( Y (1) ) ,ψ ( Y (2) ) , . . . ,ψ ( Y (K) )} (K = 118). Determine function of ranks discrepancies, i. e. sum of squares of highest ranks differences according to criteria X and Y : CRKx,Ky (α,β) = ∑ k ∈ {1, 2, . . . , 118} : R(k) (α) 6 Kx & R(k) (β) 6 Ky ( R(k) (α) − R(k) (β) )2 (7) and minimize this function min φ ∈ S(j1,j2,...,jn) ψ ∈ S(i1,i2,...,im) CRKx,Ky (α,β) . (8) Here Kx and Ky are chosen so as ∣∣∣AKx ∩ BKy∣∣∣ = 12, since the goal is to select the top 12 security guards. Minimization problem (8) is solved by re-selecting of values αj, βi. For the search of values α1, α3, α5 and β2, β4, β5 evenly divide intervals: α1 ∈ [ 1 6 ,1 ) , α3 ∈ [ 1 − α1 5 ,min{α1,1 − α1} ) , α5 ∈ [ 1 − α1 − α3 4 ,min{α3,1 − α1 − α3} ) , β2 ∈ [ 1 9 ,1 ) , β4 ∈ [ 1 − β2 8 ,min{β2,1 − β2} ) , β5 ∈ [ 1 − β2 − β4 7 ,min{β4,1 − β2 − β4} ) . Values α6, α4, β3, β1, β7, β6, β8 are chosen randomly in the intervals constructed analogously. For example, α4 is uniformly distributed in the interval α4 ∈ [ 1 − α1 − α3 − α5 − α6 2 ,min{α6,1 − α1 − α3 − α5 − α6} ) . The remaining values α2 and β9 are determined from the normalizing condition (6). General scheme of the research is represented in Figure 2. 4 Results and Discussion This heuristic enabled to achieve the criterion (8) value CR12,12 (α,β) = 429 with the follow- ing values of weights: α1 α3 α5 α6 α4 α2 0.31 0.19 0.17 0.17 0.12 0.04 β2 β4 β5 β3 β1 β7 β6 β8 β9 0.41 0.24 0.08 0.08 0.06 0.05 0.03 0.03 0.02 158 S. Dadelo, A. Krylovas, N. Kosareva, E.K. Zavadskas, R. Dadeliene Figure 2: Research progress Top 12 security guards are those belonging to the intersection AKx ∩ BKy : a21, a34, a36, a47, a56, a76, a77, a81, a91, a102, a106, a112. According to another heuristics we are looking for such convolutions among all φ ∈ S(j1,j2,...,jn) and ψ ∈ S(i1,i2,...,im) which maximize (3) and minimize (4). Since there are many solutions of problem (3)–(4), among the entire sample of solutions we are looking for one that minimize the sum (7). Let’s choose K′x = K ′ y = 4. Then we reached even lower criterion (8) value CR12,12 (α,β) = 406 and the following X and Y weights were obtained: α1 α3 α5 α6 α4 α2 0.26 0.18 0.18 0.17 0.13 0.08 β2 β4 β5 β3 β1 β7 β6 β8 β9 0.21 0.12 0.12 0.12 0.10 0.10 0.09 0.07 0.07 Top 12 security guards were selected: a21, a34, a36, a47, a56, a76, a77, a81, a91, a102, a106, a111. This solution differs from the first solution by only one employee – in place of a112 is an employee a111. Another 4 security guards were distinguished having high rank (> 4) by one and low rank by another criterion: a6, a26, a46, a54. Attention is drawn to the fact that future studies are interesting for the aspect of development of general design of the study for the similar problems, since a very different assessments have similar structure. Briefly this scheme can be described as follows. 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