IJAHP Article: Patel, Mjema Godwin/ Linear programming models for estimating weights in Analytic Hierarchy Process and for optimization of human resource allocation International Journal of the Analytic Hierarchy Process 299 Vol. 8 Issue 2 2016 ISSN 1936-6744 http://dx.doi.org/10.13033/ijahp.v8i2.335 LINEAR PROGRAMMING MODELS FOR ESTIMATING WEIGHTS IN ANALYTIC HIERARCHY PROCESS AND FOR OPTIMIZATION OF HUMAN RESOURCE ALLOCATION Gokulananda Patel Professor of Decision Science, Birla Institute of Management Technology Greater Noida India gn.patel@bimtech.ac.in Godwin Mjema Institute of Finance Management Dar Es Salam, Tanzania gdmjema@gmail.com K M Godwin Institute of Finance Management Dar Es Salam, Tanzania casymj30@gmail.com ABSTRACT The Analytic Hierarchy Process (AHP) provides a way to rank the alternatives by deriving priorities. In this paper we used Linear Programming (LP) models to estimate the weights of a pairwise comparison matrix derived within the frame work of the Analytic Hierarchy Process. The priorities obtained for the alternatives served as the coefficients of the objective function of linear programming to optimize a human resource problem at Bakhresa Food Product Limited (BFPL). Keywords: AHP; linear programming; resource allocation; element dominance; sensitivity analysis 1. Introduction The success of any organization lies in its ability to make critical decisions on growth and sustainability. However, decision making is a complex process as it involves multiple stakeholders with different opinions and interests. To avoid making ad-hoc decisions, decision makers are required to evaluate every alternative to the problem. With the Analytical Hierarchy Process, the problem is modeled by the decision maker and is structurally decomposed into a hierarchy consisting of levels of criteria, sub-criteria and alternatives with homogeneous clusters of factors (Saaty, 1977, 2007). Subsequently, an assessment of the usefulness of elements at each hierarchical level is made. AHP is a very suitable multi-criteria decision making tool proving to be effective in different application areas such as planning, optimization, selection of the best alternative and allocation of resources. It is also a reliable tool in resolving conflicts (Saaty& Vargas, mailto:gn.patel@bimtech.ac.in mailto:gdmjema@gmail.com mailto:casymj30@gmail.com IJAHP Article: Patel, Mjema Godwin/ Linear programming models for estimating weights in Analytic Hierarchy Process and for optimization of human resource allocation International Journal of the Analytic Hierarchy Process 300 Vol. 8 Issue 2 2016 ISSN 1936-6744 http://dx.doi.org/10.13033/ijahp.v8i2.335 2006). AHP is critical in defining decision making processes taking into consideration decision maker’s input, judgments, views and feelings (Vaidya & Kumar, 2006). Specifying the hierarchy is of crucial importance; the hierarchical structure gives a clear overview of the complex relationships existing in the problem. This is important because it enables the decision maker to take into consideration every aspect in each level of the hierarchy. It also allows a decision maker to take into consideration a set of evaluation criteria, and alternative options from among which the best selection is to be made. The focus of the problem, usually the goal, is the highest level of the hierarchy. There are subsequent levels of criteria further down that include sub criteria, and finally the level of alternatives from which decisions are generated. Elements with a global composition may be included along the top levels of the hierarchy. In each hierarchical level division, a pairwise comparison matrix is developed with 𝑛 (𝑛 βˆ’ 1) 2⁄ number of comparisons, where β€˜n’ is the number of criteria or alternatives in each level (Garg, Rahman, Qureshi & Kumar, 2012). Using a fundamental scale developed by Saaty, decision makers are able to assign the corresponding importance of one criterion relative to the other (Saaty, 1977). Finally, the weights of the elements being compared are estimated. In the end all the pairwise comparison results are synthesized and the decision is made in accordance with the final overall ranking of the alternatives. (Saaty,1999). In summary, AHP applications in decision making involve four main procedures. These procedures include: 1) decomposition of the problem, 2) making judgments in the pairwise comparison matrices and checking their inconsistency, 3) improving it to derive the priority weights, and 4) the synthesis step where the final global weights for all the elements in the model are determined. With each comparison matrix, the decision maker commonly uses the eigenvector method (EM) or additive normalization (AN) or logarithmic least square method (LLS) to generate a priority vector. These methods give the estimated relative weights of the elements as a result of the judgments (Saaty, 1999). To produce the final weight for the alternatives, weights generated at different levels of the hierarchy are synthesized according to the principle of hierarchic structure (Saaty, 1980) In the next section, AN, EM, LLS, and LP methods are explained. The relative weights in a problem have been estimated using all these methods. In the subsequent section, the necessity of using AHP and LP models in human resource selection is covered. 2. Estimating weights Let 𝐴 = (π‘Žπ‘–π‘— )for all 𝑖, 𝑗 = 1,2, … , 𝑛 denote an 𝑛 π‘₯ 𝑛 pairwise comparison matrix, where π‘Žπ‘–π‘— is the importance of element 𝑖 over the j th element. All the entries in matrix 𝐴 are positive (π‘Žπ‘–π‘— > 0) and reciprocal (π‘Žπ‘–π‘— = 1/π‘Žπ‘—π‘– for all 𝑖, 𝑗 = 1,2, … . 𝑛) . The decision maker wants to compute a vector 𝑀 = (𝑀1,𝑀2, … . 𝑀𝑛 ) of weights associated to pairwise comparison matrix 𝐴 (Saaty, 1999). IJAHP Article: Patel, Mjema Godwin/ Linear programming models for estimating weights in Analytic Hierarchy Process and for optimization of human resource allocation International Journal of the Analytic Hierarchy Process 301 Vol. 8 Issue 2 2016 ISSN 1936-6744 http://dx.doi.org/10.13033/ijahp.v8i2.335 The matrix 𝐴 is considered to be consistent when π‘Žπ‘–π‘— = π‘Žπ‘–π‘˜ π‘Žπ‘˜π‘— π‘“π‘œπ‘Ÿπ‘Žπ‘™π‘™ 𝑖, 𝑗, π‘˜ = 1,2, … , 𝑛, which implies that the decision maker is coherent (no error) in his judgments to develop the comparison matrix (Saaty, 1999). Assuming A contains no error and 𝑀𝑖is the weight of the𝑖 π‘‘β„Ž element, we then have π‘Žπ‘–π‘— = 𝑀𝑖 𝑀𝑗 , 𝑖, 𝑗 = 1,2, … , 𝑛 (1) Summing over all j, we obtain βˆ‘ π‘Žπ‘–π‘— 𝑛 𝑗=1 𝑀𝑗 = 𝑛𝑀𝑖 , 𝑖 = 1,2, … , 𝑛 (2) Which, in matrix notation, is equivalent to 𝐴𝑀 = 𝑛𝑀. (3) The vector w is the principal eigenvector of the matrix 𝐴 corresponding to the eigenvalue n., alternatively, we can say that the matrix 𝐴 is consistent when 𝐴𝑀 = 𝑛𝑀 (Garg, Rahman, Qureshi & Kumar, 2012). 2.1 Additive normalization (AN) In obtaining the priority vector 𝑀 using the AN method, columns are first normalized such that elements of each column of the matrix 𝐴 is divided by the sum of that column; then in each resulting row, normalized elements are summed up and divided by the number of elements in each row which is arithmetic average of the row (Srdjevic, 2005). The following equations (4) to (6) describe the above process; π‘Žπ‘–π‘— β€² = π‘Žπ‘–π‘— βˆ‘ π‘Žπ‘–π‘— 𝑛 𝑖=1 ⁄ 𝑖, 𝑗 = 1,2, … 𝑛 (4) 𝑀𝑖 = ( 1 𝑛 ) βˆ‘ π‘Žπ‘–π‘— β€² 𝑛 𝑗=1 , 𝑖 = 1,2, … . , 𝑛 (5) It can be observed that βˆ‘ 𝑀𝑖 = 1 𝑛 𝑖=1 (6) If A is consistent, then the columns of the normalized matrix 𝑁 = (π‘Žπ‘–π‘— β€² ) of A are identical. If A is not consistent, then we can write 𝐴𝑀 = πœ†π‘šπ‘Žπ‘₯ 𝑀, whereπœ†π‘šπ‘Žπ‘₯ is the principal eigenvalue and given by IJAHP Article: Patel, Mjema Godwin/ Linear programming models for estimating weights in Analytic Hierarchy Process and for optimization of human resource allocation International Journal of the Analytic Hierarchy Process 302 Vol. 8 Issue 2 2016 ISSN 1936-6744 http://dx.doi.org/10.13033/ijahp.v8i2.335 πœ†π‘šπ‘Žπ‘₯ = βˆ‘ βˆ‘ π‘Žπ‘–π‘— 𝑀𝑗 𝑛 𝑗=1 𝑛 𝑖=1 (7) According to Saaty (1980) the consistency of the method can be checked as follows: The consistency index (CI) is given by 𝐢𝐼 = (πœ†π‘šπ‘Žπ‘₯ βˆ’ 𝑛) (𝑛 βˆ’ 1) ⁄ (8) while the consistency ratio (CR) is given by 𝐢𝑅 = 𝐢𝐼 𝑅𝐼⁄ (9) The random index (RI), which depends on the order of the matrix, is the average CI of a large number of randomly generated matrices. A CR of 0.10 or less is considered acceptable. 2.2 Eigenvector method (EM) The principal eigenvector πœ†π‘šπ‘Žπ‘₯ of 𝐴 is determined by solving the determinant, det(𝐴 βˆ’ πœ†π‘šπ‘Žπ‘₯ 𝐼) = 0 (10) Then using the value of πœ†π‘šπ‘Žπ‘₯, the eigenvector 𝑀 = (𝑀1,𝑀2, … . 𝑀𝑛 ) is find out from (𝐴 βˆ’ πœ†π‘šπ‘Žπ‘₯ 𝐼)𝑀 = 0 (11) The consistency of the matrix is checked using Equations (8) and (9), in Section 2.1. 2.3 Logarithmic least square method (LLS) This method has also been developed to estimate the vector of weights (Srdjevic, 2005). With LLS, the weights 𝑀𝑖,π‘“π‘œπ‘Ÿ 𝑖 = 1, … , 𝑛, are chosen to minimize the objective βˆ‘ βˆ‘[ln π‘Žπ‘–π‘— βˆ’ (ln 𝑀𝑖 + ln 𝑀𝑗 )] 2 𝑛 𝑗=1 𝑛 𝑖=1 (12) Given that π‘Žπ‘–π‘— = 1 π‘Žπ‘—π‘–β„ π‘“π‘œπ‘Ÿπ‘Žπ‘™π‘™π‘–, 𝑗 = 1,2, … , 𝑛, (13) the LLS is quite simple: 𝑀𝑖,π‘“π‘œπ‘Ÿ 𝑖 = 1, … , 𝑛 is given by the geometric mean of the row I (Saaty, 1980). 2.4 Linear programming approach (LP) There are two desirable properties of a pairwise comparison matrix – element dominance (ED) and row dominance (RD). IJAHP Article: Patel, Mjema Godwin/ Linear programming models for estimating weights in Analytic Hierarchy Process and for optimization of human resource allocation International Journal of the Analytic Hierarchy Process 303 Vol. 8 Issue 2 2016 ISSN 1936-6744 http://dx.doi.org/10.13033/ijahp.v8i2.335 ED is said to be preserved if π‘Žπ‘–π‘— > 1 π‘–π‘šπ‘π‘™π‘–π‘’π‘  𝑀𝑖 β‰₯ 𝑀𝑗. If π‘Žπ‘–π‘— is exactly equal to 1, then an argument can be made for either 𝑀𝑖 β‰₯ 𝑀𝑗 π‘œπ‘Ÿπ‘€π‘— β‰₯ 𝑀𝑖 . RD is said to be preserved if π‘Žπ‘–π‘˜ β‰₯ π‘Žπ‘—π‘˜ π‘“π‘œπ‘Ÿ π‘Žπ‘™π‘™ π‘˜ π‘Žπ‘›π‘‘π‘Žπ‘–π‘˜ > π‘Žπ‘—π‘˜ π‘“π‘œπ‘Ÿ π‘ π‘œπ‘šπ‘’ π‘˜ π‘–π‘šπ‘π‘™π‘–π‘’π‘  𝑀𝑖 β‰₯ 𝑀𝑗 . If the comparison matrix has cardinal inconsistency, that is, π‘Žπ‘–π‘— β‰₯ 1, π‘Žπ‘—π‘˜ β‰₯ 1, π‘Žπ‘›π‘‘π‘Žπ‘˜π‘– β‰₯ 1, then the only feasible solution is 𝑀𝑖 = 𝑀𝑗 = π‘€π‘˜ . Such a comparison matrix would be highly inconsistent. EM and LLS both warrant RD (but not the ED). In the LP approach we can incorporate ED and RD as constraints, which have additional benefit of detecting cardinal inconsistency by ED constraints. The two stage LP approach (Chandran, B., et.al, 2005) is described in sub-sections 2.4.1 to 2.4.3. 2.4.1 First stage: LP to establish the consistency bound In general, any estimate of relative preference π‘Žπ‘–π‘— can be written as 𝑀𝑖 𝑀𝑗 = π‘Žπ‘–π‘— πœ€π‘–π‘— , 𝑖, 𝑗 = 1,2, … , 𝑛 (14) If the decision maker is consistent then πœ€π‘–π‘— is equal to 1. Defining three transformed decision variables for the model: π‘₯𝑖 = ln(𝑀𝑖 ) , 𝑦𝑖𝑗 = ln(πœ€π‘–π‘— ) , and𝑧𝑖𝑗 = |𝑦𝑖𝑗 | The first stage LP can be written as: π‘€π‘–π‘›π‘–π‘šπ‘–π‘§π‘’ βˆ‘ βˆ‘ 𝑧𝑖𝑗 𝑛 𝑗=𝑖+1 (15) π‘›βˆ’1 𝑖=1 Subject to π‘₯𝑖 βˆ’ π‘₯𝑗 βˆ’ 𝑦𝑖𝑗 = ln π‘Žπ‘–π‘— , 𝑖, 𝑗 = 1,2, … , 𝑛; 𝑖 β‰  𝑗, (16) 𝑧𝑖𝑗 β‰₯ 𝑦𝑖𝑗 , 𝑖, 𝑗 = 1,2, … , 𝑛; 𝑖 < 𝑗, (17) 𝑧𝑖𝑗 β‰₯ 𝑦𝑗𝑖 , 𝑖, 𝑗 = 1,2, … , 𝑛; 𝑖 < 𝑗, (18) π‘₯1 = 0 (19) π‘₯𝑖 βˆ’ π‘₯𝑗 β‰₯ 0 𝑖, 𝑗 = 1,2, … , 𝑛; π‘Žπ‘–π‘— > 1 (20) π‘₯𝑖 βˆ’ π‘₯𝑗 β‰₯ 0 𝑖, 𝑗 = 1,2, … , 𝑛; π‘Žπ‘–π‘— β‰₯ π‘Žπ‘—π‘˜ , π‘“π‘œπ‘Ÿπ‘Žπ‘™π‘™π‘˜; π‘Žπ‘–π‘˜ > π‘Žπ‘—π‘˜ π‘“π‘œπ‘Ÿπ‘ π‘œπ‘šπ‘’π‘˜ (21) 𝑧𝑖𝑗 β‰₯ 0 𝑖, 𝑗 = 1,2, … , 𝑛 (22) π‘₯𝑖 , 𝑦𝑖𝑗 , π‘’π‘›π‘Ÿπ‘’π‘ π‘‘π‘Ÿπ‘–π‘π‘‘π‘’π‘‘π‘–, 𝑗 = 1,2, … , 𝑛 (23) The objective function (15) which is βˆ‘ βˆ‘ 𝑧𝑖𝑗 𝑛 𝑗=𝑖+1 π‘›βˆ’1 𝑖=1 , minimizes the sum of logarithms of positive errors in natural log space, whereas the constraint (16) is defining the errors. Equations (17) and (18) are the degree of over estimation, (19) sets one of the weight 𝑀1 to zero, (20) preserves element dominance and (21) for row dominance. For a perfectly consistent comparison matrix, π‘§βˆ— is equal to zero. IJAHP Article: Patel, Mjema Godwin/ Linear programming models for estimating weights in Analytic Hierarchy Process and for optimization of human resource allocation International Journal of the Analytic Hierarchy Process 304 Vol. 8 Issue 2 2016 ISSN 1936-6744 http://dx.doi.org/10.13033/ijahp.v8i2.335 The objective function provides consistency index 𝐢𝐼(𝐿𝑃) = 2π‘§βˆ— 𝑛(𝑛 βˆ’ 1) (24) 2.4.2 Second stage: LP to generate a priority vector The first stage LP minimizes the product of all errors πœ€π‘–π‘—, but multiple optimal solutions may exist. In the second stage LP, the solution that minimizes the maximum errors πœ€π‘–π‘— is selected. The second stage LP can be presented as: π‘€π‘–π‘›π‘–π‘šπ‘§π‘’π‘§π‘šπ‘Žπ‘₯ (25) Subject to βˆ‘ βˆ‘ 𝑧𝑖𝑗 𝑛 𝑗=𝑖+1 = π‘§βˆ— (26) π‘›βˆ’1 𝑖=1 π‘§π‘šπ‘Žπ‘₯ β‰₯ 𝑧𝑖𝑗 , 𝑖, 𝑗 = 1,2, … , 𝑛; 𝑖 < 𝑗, (27) and all first stage LP constraints. π‘§βˆ—is the optimal first stage solution value, π‘§π‘šπ‘Žπ‘₯ is the maximum value of error 𝑧𝑖𝑗 . Constraint (26) ensures that only those solutions that are optimal in the first stage LP are feasible in the second stage model. 2.4.3 Advantages of the LP approach In AN or EM or LLS, the error from Equation (14) is πœ€π‘–π‘— = 𝑀𝑖 𝑀𝑗 π‘Žπ‘—π‘– , 𝑖, 𝑗 = 1,2, … , 𝑛 (28) and then can be decided which π‘Žπ‘–π‘— can reduce the inconsistency. In the first stage of the LP model, the decision maker may go for a sensitivity analysis on inputs to know which entry in the pair wise comparison matrix should be changed to reduce inconsistency. The question is, how much should the entry should be changed? The values of the dual variables (shadow prices) at optimality provide an indication of the incorrect entries in the pair wise comparison matrix. It is a well-known fact in LP that a shadow price with value k at optimality has the following interpretation: β€œif we increase the right hand side of the dual variable’s corresponding constraint by one unit, then the value of the objective function increases by k units” IJAHP Article: Patel, Mjema Godwin/ Linear programming models for estimating weights in Analytic Hierarchy Process and for optimization of human resource allocation International Journal of the Analytic Hierarchy Process 305 Vol. 8 Issue 2 2016 ISSN 1936-6744 http://dx.doi.org/10.13033/ijahp.v8i2.335 The dual variable that corresponds to each constraint in (16) changes the value of the objective function for unit change on the right hand side (that is, ln π‘Žπ‘–π‘— ). Since it is in natural logarithm space, a dual variable with value k at optimality will increase the objective function value by k units when the corresponding π‘Žπ‘–π‘— increase by a factor of e, the base of natural logarithm, which is a useful information for a decision maker (Chandran, B., et.al, 2005). It is now possible to identify which π‘Žπ‘–π‘— to change in order to decrease the inconsistency value by the greatest amount. A dual variable with negative value indicates that π‘Žπ‘–π‘— should increase, while a positive value indicates that π‘Žπ‘–π‘— should be decreased. The following section describes how the AHP was used to analyze the selection process in BFPL in Tanzania. 2.5 Numerical example In this section, relative weights (principal eigenvector) of a 4x4 comparison matrix (Table 1) have been estimated using the AN, EM, LLS and LP approaches. The entries in Table 1 are the geometric means of a set of responses from 1/9 to 9. The principal eigenvalue πœ†π‘šπ‘Žπ‘₯, consistency index CI, and consistency ratio CR are calculated using Equations (7), (8) and (9) respectively and shown in Table 2. Relative weights obtained using LP shows that it preserves ED. Table 1 4 x 4 pairwise comparison matrix 1 1 0.34 0.53 1 1 1.58 0.60 1/0.34 1/1.58 1 1.24 1/0.53 1/0.60 1/1.24 1 Table 2 Priority vectors, eigenvalues, consistency index and consistency ratio AN EM LLS LP 𝑀1 0.1567 0.1560 0.1577 0.1583 𝑀2 0.2450 0.2490 0.2388 0.2716 𝑀3 0.3035 0.3017 0.2982 0.2716 𝑀4 0.2948 0.2933 0.3053 0.2985 πœ†π‘šπ‘Žπ‘₯ 4.25 4.2486 4.1328 4.2667 𝐢𝐼 0.0833 0.0829 0.0443 0.0889 𝐢𝑅 0.0842 0.0837 0.0447 0.0898 The consistency index within the LP framework Equation (24) is 0.377. It can be observed that π‘Ž23 = 1.58 has been violated in the AN, EM and LLS approaches whereas it has been preserved in the LP approach. Any highly overrated cell can be detected using Equation (28) for all the four methods. It does not directly give how much the CR has improved. But in the LP approach, the highly overrated cell can be detected with the help IJAHP Article: Patel, Mjema Godwin/ Linear programming models for estimating weights in Analytic Hierarchy Process and for optimization of human resource allocation International Journal of the Analytic Hierarchy Process 306 Vol. 8 Issue 2 2016 ISSN 1936-6744 http://dx.doi.org/10.13033/ijahp.v8i2.335 of a dual variable, and how much the improvement in first stage can be known determined. In this numerical example, cell π‘Ž23 = 1.58 has been detected as the most overrated cell and the dual value corresponding equation from (16) has reduced the value of π‘§βˆ— from 2.262 to 1.307. 3. Human resource allocation at BFPL Bakhresa Food Product Limited (BFPL) is a carbonated soft drinks manufacturing company owned and founded by Mr. Said Salim Awadh Bakhresa in the 1970s. The company, among several others, together make up the Azam Group of Companies, a conglomerate that deals with the production and provision of different products and services. It is among the largest family owned businesses in East and Central Africa. BFPL generates revenue of a minimum of three million dollars annually. With its headquarters located in the coastal region of Dar Es Salaam, Tanzania, this multi-national corporation faces different decision making challenges in its day to day operations and it is in the company’s best interest to manage risky situations that may come from wrong decision making choices. Proper decision making choices will ensure the company’s sustainability and prosperity. Since a company’s activities help to ensure its longevity, managers and decision makers at BFPL understand the need to evaluate weights and priorities of each activity in relation to its consequent outcome. Their aim is to equip the companies with strategic decision making techniques for sustainable growth, and consequently to achieve a competitive upper hand in the market. Decision makers must therefore adopt and apply different business practices, methods, and various tools that prove effective in decision making. In any organization, selection of employees to fill different departmental posts is a crucial and sensitive matter (Kashi & Friedrich, 2013). Decision makers are well aware of the costs involved in the improper selection of employees on one hand, and the associated benefits of equipping the company with the right employees on the other. It is therefore vital for companies to not only select and hire employees whose skills and capabilities align well with the goals and objectives of the company, but also to achieve optimality in the return of their investment in human resources (Saaty, Peniwati & Shang, 2007). Having limited resources, the need for selection of potential employees must also incorporate the need to provide the company with employees whose values, mission and vision match those of the companies. An effective tool is required, one that will be able to take into consideration both tangible and intangible aspects of criteria to take into consideration when selecting employees. The key is to find proper measurement of the weights and priorities of different criteria such that error and bias in decision making is minimized. With this tool, the right kind of employees will be chosen to man the right kind of tasks in the organization and ultimately help the organization to achieve the optimum value of its investment in human resources (Saaty, Peniwati, & Shang, 2007). AHP proves to be such a useful tool for selection of employees, and the application of Linear Programming will ensure the company achieves the optimality it requires (Saaty, Peniwati & Shang, 2007; Dye & Forman, 1992). Our purpose here is to illustrate AHP by deriving priorities and applying LP models and to formulate LP models to optimize returns. The company has identified IJAHP Article: Patel, Mjema Godwin/ Linear programming models for estimating weights in Analytic Hierarchy Process and for optimization of human resource allocation International Journal of the Analytic Hierarchy Process 307 Vol. 8 Issue 2 2016 ISSN 1936-6744 http://dx.doi.org/10.13033/ijahp.v8i2.335 the General Management and Quality Assurance Departments to fill different positions for BFPL as listed in Table 3. With the example given, we seek to obtain the optimal number of posts that need to be filled, and to select only the most qualified potential employees for every post whose contribution will result in the achievement of quality products, increase efficiency, expand capacity, and product development for BFPL (Saaty, Peniwati & Shang, 2007).The selection hierarchy for any human resource allocation problem of BFPL can be represented through Figure 1 with a 5-level of hierarchy. The 5 th level of hierarchy of alternatives is not shown, which can be on the bottom. 3.1Illustration of human resource requirement and selection problem The problem has been elaborated with hierarchical disintegration, creation of the pair- wise comparison matrix, priority weights development and the global weights development. Finally, optimal recruitment satisfying constraints imposed by the company is determined (Saaty, Peniwati & Shang, 2007).The company has many departments including Human Resources, Production, General Management, Quality Assurance, and Finance and Sales. For one of the plants situated near the port of Dar Es Salam, the company wants to employ a technical manager, a driver and assistant(s) within the General Management Department, as well as lab-technicians and micro-biologists within Quality Assurance Department. The company is looking to invest not more than 11 million TSH to be used as monthly salary packages for all the posts. The particulars of the posts including qualifications, salary packages and other details are given in Table 3 (Saaty, Peniwati & Shang, 2007). To begin with, a hierarchical structure with 5 levels is created after analyzing the problem. The highest level (goal) is the selection of potential candidates. The following level is the department level where in this case, there are two departments with vacanies, the General Management and Quality Assurance departments. The third level is the posts to be filled; level four contains criteria for consideration in each of the posts within the departments. For example, the Technical Manager for the General Management department is evaluated based on experience, educational qualification, technical skill and communication skills. This level establishes different intensities for each selection criteria. The lowest level, the alternatives, consists of potential candidates who have shown interest in the post (see Figure 1). IJAHP Article: Patel, Mjema Godwin/ Linear programming models for estimating weights in Analytic Hierarchy Process and for optimization of human resource allocation International Journal of the Analytic Hierarchy Process 308 Vol. 8 Issue 2 2016 ISSN 1936-6744 http://dx.doi.org/10.13033/ijahp.v8i2.335 Table 3 Personnel requirements Variable Post Department Number of positions Criteria of selection Monthly salary in Tzs (m) 𝑦1 Technical Manager General Management 1 Experience Education Technical Skill Communication Skill 6.0 𝑦2 Driver General Management 1 Experience Education Mechanical Skill 0.8 𝑦3 Assistant General Management 1 - 3 Experience Education Office Management Communication Skill 1.2 𝑦4 Lab Technician Quality Assurance 0 - 1 Experience Education Technical Skill Communication Skill 1.0 𝑦5 Micro Biologist Quality Assurance 0 - 2 Experience Education Technical Skill Communication Skill 0.9 A pair wise comparison matrix is developed for each hierarchical level using the fundamental scale. Five senior level personnel from BFPL were selected and a questionnaire was given to them. Instead of responding individually, persons responsible for selection helped the authors in providing the ratings in consensus. The priority weights for any level are obtained using the LP approach described in Section 2.4. These weights are mentioned in Figure1. Level 4 demonstrates the criteria considered under each post. These are usually set by the organization to ensure the right candidates are selected for the right kind of job positions. Each post has its own specific set of selection criteria. For instance, the technical manager’s post in the General Management department has four selection criteria namely; experience, education, technical skills and communication skills. To obtain the weight of each of the criterion a pair wise comparison matrix is developed and the LP approach is further applied for local weights. The global weights of criteria are obtained by synthesizing with the global weight of the root level. For example, the global weight of criteria for technical manager is obtained as (0.473, 0.0118, 0.059, 0.067). 3.2 Intensities and scores of criteria To implement the absolute measurement mode in AHP, each selection criterion for every post is further broken down into levels of intensity. These intensities should be located at level 6 (Saaty, Peniwati & Shang, 2007). For example, the evaluation of the Technical IJAHP Article: Patel, Mjema Godwin/ Linear programming models for estimating weights in Analytic Hierarchy Process and for optimization of human resource allocation International Journal of the Analytic Hierarchy Process 309 Vol. 8 Issue 2 2016 ISSN 1936-6744 http://dx.doi.org/10.13033/ijahp.v8i2.335 Manager consists of the following intensities: (i) experience is divided into three intensities of high (corresponds to 3+ years of experience), medium (1-3 years), and low (less than one year); (ii) education is divided into master, degree and diploma; (iii) technical skills are divided into excellent, good and fair; (iv) communication skills into high, medium and low (Saaty, Peniwati & Shang, 2007). The rating for each position is given in Table 4.Using the LP approach, priorities of the intensities are obtained from pair wise comparisons and idealized by dividing each by the highest value so that it becomes 1 and the rest follow proportionally (Table 5). The evaluation of each candidate is done by a group of experts who conduct multiple interviews relating to required job skills, personality and character, communication skills and ability to work in a group. This evaluation is done according to the criteria required for the successful selection of a potential employee. The experts (not less than 3 in number) generate a pairwise comparison matrix after obtaining the geometric mean of their judgments. The candidate’s synthesized score for each post they applied for is calculated according to the corresponding local weights of the selection criteria shown in Table 4.Their results are organized into a list in Table 6. The total scores for each employee in Table 6 have been adjusted multiplying by 3/5 for the General Management department and 2/5 for the Quality Assurance department. IJAHP Article: Patel, Mjema Godwin/ Linear programming models for estimating weights in Analytic Hierarchy Process and for optimization of human resource allocation International Journal of the Analytic Hierarchy Process 310 Vol. 8 Issue 2 2016 ISSN 1936-6744 http://dx.doi.org/10.13033/ijahp.v8i2.335 Figure 1.Selection hierarchy for BFPL with local (global) weights In Table 4, the applicants are rated for quality for each criterion of the position for which they are applying. The standards for quality from which their rating is chosen are given in Table 5. TS 0.082 (0.059) EX 0.683(0.081) EX 0.696(0.027) EX 0.696(0.058) TS 0.139 (0.006) OM 0.087 (0.003) EX 0.557 (0.023) Selection of candidate QA 0.125 GM 0.875 EX 0.659 (0.473) DV 0.135(0.118) ED 0.165 (0.118) LT 0.667(0.083) MB 0.333(0.042) TM0.82 (0.718) AS 0.045 (0.039) CMS 0.094 (0.067) ED 0.20 (0.024) MS 0.117 (0.014) ED 0.174(0.015) TS 0.087(0.007) CMS 0.043 (0.004) CMS 0.076 (0.003) ED 0.174(0.007) CMS 0.043 (0.002) ED 0.228 (0.009) IJAHP Article: Patel, Mjema Godwin/ Linear programming models for estimating weights in Analytic Hierarchy Process and for optimization of human resource allocation International Journal of the Analytic Hierarchy Process 311 Vol. 8 Issue 2 2016 ISSN 1936-6744 http://dx.doi.org/10.13033/ijahp.v8i2.335 Table 4 Criteria for positions to be filled with an example of applicant x1’s ratings for the Tech. Manager position Applicants Tech. Manager Experience Education Tech. skill Comm. skill 0.473 0.118 0.059 0.067 π‘₯1 High Master Good High π‘₯2 π‘₯3 π‘₯4 Driver Experience Education Mech.skill 0.081 0.024 0.014 π‘₯5 π‘₯6 π‘₯7 π‘₯8 Assistant Experience Education Office mgmt Comm. skill 0.027 0.007 0.003 0.002 π‘₯9 π‘₯10 π‘₯11 π‘₯12 Lab-Technician Experience Education Tech. skill Mech. skill 0.058 0.015 0.007 0.004 π‘₯13 π‘₯14 π‘₯15 π‘₯16 Micro-Biologist Experience Education Tech. skill Mech. skill 0.023 0.009 0.006 0.003 π‘₯17 π‘₯18 π‘₯19 π‘₯20 IJAHP Article: Patel, Mjema Godwin/ Linear programming models for estimating weights in Analytic Hierarchy Process and for optimization of human resource allocation International Journal of the Analytic Hierarchy Process 312 Vol. 8 Issue 2 2016 ISSN 1936-6744 http://dx.doi.org/10.13033/ijahp.v8i2.335 Table 5 Ideal priorities of the intensities used for rating alternatives in Table 4 Criteria by job (global priorities) Intensities and idealized priorities Technical Mangers Experience(0.473) Education(0.165) Technical Skills(0.082) Communication Skills(0.094) High (1) Medium (0.55) Low (0.302) Masters (1) Degree (0.363) Diploma (0.132) Excellent (1) Good (0.25) Fair (0.125) High (1) Medium (0.573) Low (0.219) Driver Experience (0.081) Education (0.024) Mechanical Skills (0.014) High (1) Medium (0.55) Low (0.302) Diploma (0.142) Certificate (0.705) Secondary (1) High (1) Medium (0.210) Low (0.088) Assistant Experience (0.027) Education (0.007) Office Management Skills (0.003) Communication Skills (0.002) High (1) Medium (0.55) Low (0.302) Degree (0.142) Diploma (0.705) Certificate (1) Best (1) Good (0.363) Fair (0.132) High (1) Medium (0.573) Low (0.219) Lab Technician Experience (0.058) Education (0.015) Technical Skills (0.007) Communication Skills (0.004) High (1) Medium (0.55) Low (0.302) Masters (1) Degree (0.363) Diploma (0.132) Excellent (1) Good (0.25) Fair (0.125) High (1) Medium (0.573) Low (0.219) Micro Biologist Experience (0.023) Education (0.009) Technical Skills (0.006) Communication Skills (0.003) High (1) Medium (0.55) Low (0.302) Masters (1) Degree (0.363) Diploma (0.132) Excellent (1) Good (0.25) Fair (0.125) High (1) Medium (0.573) Low (0.219) IJAHP Article: Patel, Mjema Godwin/ Linear programming models for estimating weights in Analytic Hierarchy Process and for optimization of human resource allocation International Journal of the Analytic Hierarchy Process 313 Vol. 8 Issue 2 2016 ISSN 1936-6744 http://dx.doi.org/10.13033/ijahp.v8i2.335 Table 6 Applicant’s priorities obtained from their rating totals Number Applicants Post applied for Score 1 π‘₯1 Technical Manager 0.4040 2 π‘₯2 Technical Manager 0.3680 3 π‘₯3 Technical Manager 0.2403 4 π‘₯4 Technical Manager 0.3506 5 π‘₯5 Driver 0.0295 6 π‘₯6 Driver 0.0648 7 π‘₯7 Driver 0.0332 8 π‘₯8 Driver 0.0453 9 π‘₯9 Assistant 0.0225 10 π‘₯10 Assistant 0.0069 11 π‘₯11 Assistant 0.0145 12 π‘₯12 Assistant 0.0084 13 π‘₯13 Lab. Technician 0.0169 14 π‘₯14 Lab. Technician 0.0150 15 π‘₯15 Lab. Technician 0.0271 16 π‘₯16 Lab. Technician 0.0085 17 π‘₯17 Micro. Biologist 0.0123 18 π‘₯18 Micro. Biologist 0.0056 19 π‘₯19 Micro. Biologist 0.068 20 π‘₯20 Micro. Biologist 0.0046 3.3 Manpower allocation for BFPL Two comparable linear models are presented in this section for the best human resource allocation for BFPL. Model 1: Optimization for individual applicants The objective function coefficients are the scores given in Table 6. The decision variables π‘₯1, π‘₯2, … , π‘₯20are binary, subject to salary constraint, upper and lower bound constraints on the number of people given in Table 3. π‘€π‘Žπ‘₯ 0.4040π‘₯1 + 0.3680π‘₯2 + 0.2403π‘₯3 + 0.3506π‘₯4 + 0.0295π‘₯5 + 0.0648π‘₯6 + 0.0332π‘₯7 + 0.0453π‘₯8 + 0.0225π‘₯9 + 0.0069π‘₯10 + 0.0145π‘₯11 + 0.0084π‘₯12 + 0.0169π‘₯13 + 0.0150π‘₯14 + 0.0271π‘₯15 + 0.0085π‘₯16 + 0.0123π‘₯17 + 0.0056π‘₯18 + 0.0068π‘₯19 + 0.0046π‘₯20 ≀ 11 Subject to 6π‘₯1 + 6π‘₯2 + 6π‘₯3 + 6π‘₯4 + 0.8π‘₯5 + 0.8π‘₯6 + 0.8π‘₯7 + 0.8π‘₯8 + 1.2π‘₯9 + 1.2π‘₯10 + 1.2π‘₯11 + 1.2π‘₯12 + 1.0π‘₯13 + 1.0π‘₯14 + 1.0π‘₯15 + 1.0π‘₯16 + 0.9π‘₯17 + 0.9π‘₯18 + 0.9π‘₯19 + 0.9π‘₯20 ≀ 11 (Salary constraint) IJAHP Article: Patel, Mjema Godwin/ Linear programming models for estimating weights in Analytic Hierarchy Process and for optimization of human resource allocation International Journal of the Analytic Hierarchy Process 314 Vol. 8 Issue 2 2016 ISSN 1936-6744 http://dx.doi.org/10.13033/ijahp.v8i2.335 π‘₯1 + π‘₯2 + π‘₯3 + π‘₯4 = 1 (π‘‡π‘’π‘β„Žπ‘›π‘–π‘π‘Žπ‘™π‘€π‘Žπ‘›π‘Žπ‘”π‘’π‘Ÿ) 0 ≀ π‘₯5 + π‘₯6 + π‘₯7 + π‘₯8 ≀ 1 (π·π‘Ÿπ‘–π‘£π‘’π‘Ÿ) 1 ≀ π‘₯9 + π‘₯10 + π‘₯11 + π‘₯12 ≀ 3 (π΄π‘ π‘ π‘–π‘ π‘‘π‘Žπ‘›π‘‘) 0 ≀ π‘₯13 + π‘₯14 + π‘₯15 + π‘₯16 ≀ 1 (πΏπ‘Žπ‘π‘‡π‘’π‘β„Žπ‘›π‘–π‘π‘–π‘Žπ‘›) 0 ≀ π‘₯17 + π‘₯18 + π‘₯19 + π‘₯20 ≀ 3 (π‘€π‘–π‘π‘Ÿπ‘œ π΅π‘–π‘œπ‘™π‘œπ‘”π‘–π‘ π‘‘) π‘₯𝑗 , j = 1,2, … ,20 are binary Model 1 was solved using Excel Solver to maximize the goal of BFPL and solution is given in Table 7. Table 7 The optimal solution of LP Model 1 Selected Applicant Position Salary in million (Tsh) π‘₯1 Technical Manager 6 π‘₯6 Driver 0.8 π‘₯9 Assistant 1.2 π‘₯15 Lab Technician 1 π‘₯17 Micro Biologist 0.9 π‘₯19 Micro Biologist 0.9 Total Salaries 10.8 Model 2: Optimizing different positions This approach consequently provides the priorities of the five positions found in level four of the hierarchy of Figure 1 as coefficients of the objective function. The posts are represented by 𝑦1 π‘‘π‘œ 𝑦5, such that their values integers denote the vacancy of jobs posts. The previous model made the selection of applicants based on their rating taking the relative importance of the post. With these models we are able to obtain the optimal number of jobs and the selection of the best applicants for those job posts. Coefficients of objective function from the fourth level of Figure1 are adjusted by multiplying 3/5 with the General Management department and 2/5 with the Quality Assurance department. The model is π‘€π‘Žπ‘₯ 0.492𝑦1 + 0.081𝑦2 + 0.027𝑦3 + 0.2668𝑦4 + 0.1332𝑦5 Subject to 𝑦1 + 0.8𝑦2 + 1.2𝑦3 + 𝑦4 + 0.9𝑦5 ≀ 11 π‘ π‘Žπ‘™π‘Žπ‘Ÿπ‘¦ π‘π‘œπ‘›π‘ π‘‘π‘Ÿπ‘Žπ‘–π‘›π‘‘ 𝑦1 = 1 π‘‡π‘’π‘β„Žπ‘›π‘–π‘π‘Žπ‘™ π‘€π‘Žπ‘›π‘Žπ‘”π‘’π‘Ÿ 𝑦2 = 1 π·π‘Ÿπ‘–π‘£π‘’π‘Ÿ 𝑦3 β‰₯ 1 π΄π‘ π‘ π‘–π‘ π‘‘π‘Žπ‘›π‘‘, πΏπ‘œπ‘€π‘’π‘Ÿ π‘π‘œπ‘’π‘›π‘‘ 𝑦3 ≀ 3 π΄π‘ π‘ π‘–π‘ π‘‘π‘Žπ‘›π‘‘, π‘ˆπ‘π‘π‘’π‘Ÿ π‘π‘œπ‘’π‘›π‘‘ 𝑦4 β‰₯ 0 πΏπ‘Žπ‘ π‘‡π‘’π‘β„Žπ‘›π‘–π‘π‘–π‘Žπ‘›, πΏπ‘œπ‘€π‘’π‘Ÿ π‘π‘œπ‘’π‘›π‘‘ 𝑦4 ≀ 1 πΏπ‘Žπ‘ π‘‡π‘’π‘β„Žπ‘›π‘–π‘π‘–π‘Žπ‘›, π‘ˆπ‘π‘π‘’π‘Ÿ π‘π‘œπ‘’π‘›π‘‘ 𝑦5 β‰₯ 0 π‘€π‘–π‘π‘Ÿπ‘œ π΅π‘–π‘œπ‘™π‘œπ‘”π‘–π‘ π‘‘, πΏπ‘œπ‘€π‘’π‘Ÿ π‘π‘œπ‘’π‘›π‘‘ 𝑦5 ≀ 2 π‘€π‘–π‘π‘Ÿπ‘œ π΅π‘–π‘œπ‘™π‘œπ‘”π‘–π‘ π‘‘, π‘ˆπ‘π‘π‘’π‘Ÿ π‘π‘œπ‘’π‘›π‘‘ 𝑦𝑗 are integers IJAHP Article: Patel, Mjema Godwin/ Linear programming models for estimating weights in Analytic Hierarchy Process and for optimization of human resource allocation International Journal of the Analytic Hierarchy Process 315 Vol. 8 Issue 2 2016 ISSN 1936-6744 http://dx.doi.org/10.13033/ijahp.v8i2.335 Table 8 Optimal solution second model Position Selection Position salary Total salary Technical Manager 1 6000000 Tzs 6000000 Tzs Driver 1 800000 Tzs 800000 Tzs Assistant 1 1200000 Tzs 1200000 Tzs Lab Technician 1 1000000 Tzs 1000000 Tzs Micro Biologist 2 900000 Tzs 1800000 Tzs Total salaries 10800000 Tzs The optimal solution of model 2 is given in Table 8. The solution of model 2 is consistent with the model 1 solution. All the posts can be filled if the company can spend 13.2 million Tzs in place of 11 million Tzs per month. 4. Summary and conclusions AHP can measure intangibles and LP proves to be effective in optimizing the resource allocation problem by also considering tangible measurements. This paper has used both tangible and intangible measures. After converting intangibles by using the AHP technique, priority has also been calculated using LP. Element dominance and row dominance have been incorporated as constraints in LP. LP has several advantages over the additive normalization or eigenvectors or LSS methods for determining priorities. In Section 3, the BFPL case has been presented to fill the vacant post. Combined AHP and LP models seem to provide an effective tool. 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