rel="nofollow" href="#ulink_7cc59da7-09bc-5e07-9cf6-dcc6bc1e53e4">(2.4)
For negative criteria:(2.5)
in which r(i,j) = the normalized performance value for the ith alternatives with respect to the jth criterion.
Form II: In this normalizing procedure, which is slightly more advanced than the former technique, both ideal and inferior alternatives are used to normalize the performance values, as follows (Ma et al. 1999):
For positive criteria:(2.6)
For negative criteria:(2.7)
The cited normalization procedure yields dimensionless performance values of the decision‐matrix in which the [r(i,j)] range between 0 and 1.
Through the normalization procedure, the decision‐maker transforms the elements of a decision‐matrix into commensurable values. The next step is for the decision‐maker to combine these values in a way that the alternatives’ overall preferences can be evaluated. Herein, assume that the decision‐maker evaluated the importance of each criterion and derive the set of weights that best reflect the stakeholder’s priorities. Let wj denote the assigned weight to the jth criterion; the following holds for the assigned weight set:
(2.8)
Assigning the proper weight to each criterion is a challenging procedure that is discussed later in the appendix section. The challenge remains, however, on how these values can be combined to form an overall preference for each given alternative. Sections 2.2 and 2.3 describe, in detail, two basic methods to aggregate the alternatives’ performances and obtain the alternatives’ overall preference.
2.2 The Weighted Sum Method
The weighted sum method (WSM), also referred to as the simple additive weighting (SAW) method, is the best known and simplest MADM method for evaluating a number of alternatives in terms of a number of decision criteria. The basic logic of WSM, which was perhaps the first logical solution that enabled the decision‐makers to cope with the MADM problems, is to obtain a weighted sum of the performance values of each alternative's overall attributes. Churchman and Ackoff (1954) were among the first to employ the WSM method to cope with a portfolio selection problem (Tzeng and Huang 2011). Ever since, due to the simplistic nature of the method, it quickly became a popular tool to cope with a MADM problem (Zanakis et al. 1995, 1998). Notable examples of applying WSM in different fields would be in agroecosystem management (Andrews and Carroll 2001), airlines' strategic planning and management (Chang and Yeh 2001), energy planning and management (San Cristóbal 2011), construction management (Jato‐Espino et al. 2014), environmental assessments (Kang 2002; Zhou et al. 2006), forestry management (Howard 1991), industrial management (Ma et al. 1999), industrial robot selection (Athawale and Chakraborty 2011), landfill site selection (Şener et al. 2006), mobile network selection (Savitha and Chandrasekar 2011), software evaluation (Olson et al. 1995), wastewater management (Zarghami 2011), and water supply planning (Goicoechea et al. 1992; Hobbs et al. 1992), to name a few.
The following is a detailed stepwise instruction to implement WSM as an MADM solving method:
2.2.1 Step 1: Defining the Decision‐making Problem
The initial step of the WSM would be for the decision‐maker to determine the elements of the decision‐matrix. It goes without saying that the integrity of the final result would rely heavily on this step. A well‐defined decision‐matrix is the basic requirement of any MADM method.
2.2.2 Step 2: Normalizing the Elements of the Decision‐matrix
At this junction, the decision‐maker creates a commensurable decision‐matrix by normalizing the elements of the matrix, using (Eqs. 2.4–2.7). This step is vital to compose a logical and viable decision‐making process, for otherwise, combining the nonnormalized values to obtain the overall scores of the alternatives would yield meaningless elements.
2.2.3 Step 3: Aggregating the Preference of Alternatives
The following equation is employed to aggregate the normalized preference values of alternatives (Churchman and Ackoff 1954):
in which Vi = the overall preference of the ith alternative. Equation (2.9) is found on most compensatory methods and implies that poor performance of an alternative with respect to some criteria can be compensated for by high performance by other criteria.
Despite its simplicity, the WSM remains a top choice of decision‐makers for evaluating an MADM problem. A conceptually similar method to WSM is the weighted product method (WPM), which is discussed in Section 2.3.
2.3 The Weighted Product Method
The WPM employs multiplication for synthesizing the attributes' performance values, each of which is raised to the power of the corresponding attributes' weights. In essence, compared to the WSM, the WPM penalizes alternatives with poor attribute preference values more heavily (Triantaphyllou and Mann 1989; Chang and Yeh 2001; San Cristóbal 2012). WPM has been successfully implemented in bidding strategies (Wang et al. 2010), business strategic planning (Chang and Yeh 2001), energy resources management (Pohekar and Ramachandran 2004), environmental evaluations (Zhou et al. 2006), technological instrument selection (Savitha and Chandrasekar 2011), supply chain management (Chou et al. 2008), and waste management (Cheng et al. 2003).
The WPM follows the same basic steps applied in the WSM. The following is a stepwise procedure to implement the WPM in an MADM problem:
Step 1: Defining the decision‐making problem
Step 2: Normalizing the elements of the decision‐matrix
Step 3: Aggregating the preference of alternatives
Equation (2.10) synthesizes the alternatives' performance values (Chang and Yeh 2001):
This method is also considered as a compensatory MADM method where the assumption of criteria independency is made (Saaty and Ergu 2015).
