This is a cross sectional descriptive-analytic study, conducted in 2017. The study area included 31 provinces of the country (Fig. 1). In this study, the data related to 4 years of the provinces of the country (2012 and 2013 before the development of the HTP), 2015 and 2016 (after the development of the Health Transformation Plan) were studied; and the required statistics and information from the data bases of Ministry of Health and Medical Education as well as statistics yearbook of the country were prepared. Considering that 2014 is the year of implementation of the plan, so the data of 2014 have not been included in the study analysis. On the other hand, based on the four studied years (2 years before implementing HTP and 2 years after that), in addition to perform the ranking, it was possible to examine the process of changes for each indicator in each province. In this regard, multi-criteria decision-making method was applied. This method is one of the decision-making methods in which the problem has multiple criteria and the purpose of the decision is based on these multiple criteria [21]. SPSS18 and Excel2013 software were used to analyze the data.
In the ranking of the provinces of the country, 11 effective indicators in the field of health and treatment (including paramedics, staffs, general practitioner, dentist, pharmacist, specialists, medical institutions, hospital beds, diagnostic laboratories, rehabilitation centers and pharmacies affiliated with the Ministry of Health, and Medical Education) whose data are available, were used. The selection of such indicators has been based on the objectives of the Health Transformation Plan and having access to available data. In addition, Shannon’s entropy method was used to calculate weights of each of the examined indicators in this study.
The “entropy” is a very important concept in the social sciences, physics and information theory, and when the data of a decision matrix is fully characterized, this method can be used to evaluate weights [22]. In this way, if \(m\) is the number of options and \(n\) is the number of indicators, the weight of the indicators is obtained briefly by taking the following steps [23]:
Step 1: Calculating the probability distribution through the following relationship:
$$P_{\,i\,j} = \frac{{a_{\,i\,j} }}{{\sum\limits_{i = 1}^{m} {a_{\,i\,j} } }}$$
(1)
Step 2: Calculating the amount of entropy in which \(k = \frac{1}{Lnm}\)
$$E_{\,j} = - k\sum\limits_{i = 1}^{m} {P_{\,i\,j} Ln(P_{\,i\,j} )}$$
(2)
Step 3: The amount of uncertainty is obtained.
$$d_{j} = 1 - E_{j}$$
(3)
Step 4: Calculating the weights of the indicators will be calculated by the following equation.
$$w_{j} = \frac{{d_{j} }}{{\sum\limits_{i = 1}^{n} {d_{j} } }}$$
(4)
The VIKOR method was used to rank the provinces of the country. The VIKOR method is one of the multi-criteria decision-making methods introduced by Opriukovich and Tsugon in 1998.
This method evaluates issues with inappropriate and incompatible criteria. It has been developed to multi criteria optimization of the developed complex systems; and is as a determiner of a list of compromise ratings, compromise and intervals solutions for fixing the weight of the criteria. This method focuses on ranking and selection of alternatives despite existing conflicting criteria.
VIKOR introduces a multi-criteria ranking index based on a specific criteria of proximity to the ideal solution. In this method, it is assumed that each alternative is evaluated according to the criteria function, and compromise ranking can be made by comparing the criteria of proximity with the ideal alternative.
An aggregate LP metric function is used for a compromise rating [24].
Suppose that the \(J\) alternative is specified with \(a_{1} \,,\,a_{2} \,,\, \ldots ,\,a_{j}\). For \(a_{j}\) alternative, the degree and the amount of degree of \(j\) is determined by \(f_{ij}\). That is, \(f_{ij}\) is the value of the criterion function of \(i\) for the \(a_{j}\) alternative, in a way that, \(n\) is the number of criteria. The development of the VIKOR’s method begins with the following form called the LP metric:
$$\begin{aligned} L_{P,j} = \left\{ {\left. {\sum\limits_{i = 1}^{n} {\left[ {w_{i} \,(f_{i}^{*} - f_{ij} )/(f_{i}^{*} - f_{i}^{ - } )} \right]^{P} } } \right\}} \right.^{{\frac{1}{P}}} \, \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,1 \le P \le \infty \,\,\,\,\,\,\,\,\,\,j = 1\,,\,2\,,\, \ldots \,,\,J\,\,\,\, \hfill \\ \, \hfill \\ \end{aligned}$$
(5)
Based on the VIKOR method, the value of \(L_{1,j}\) (as \(S_{j}\) in equation (1) and \(L_{\infty ,j}\) [as \(R_{j}\) in equation (2) is used to formulate the ranking criterion.
The obtained result from \(\hbox{min} \,S_{j}\) expresses the maximizing group desirability (the rule of the majority) and the obtained results of \(\hbox{min} \,R_{j}\), expresses minimizing the regret of individuals from the opponent alternatives.
An agreement solution of \(F^{c}\) is a justifiable answer to an ideal solution of \(F^{*}\), and compromise means an agreement reached through bilateral negotiations as depicted in Fig. 2:
$$\Delta f_{1} = f_{1}^{*} - f_{1}^{c} \,\,\,\,\,\,\,\,\Delta f_{2} = f_{2}^{*} - f_{2}^{c} \,$$
(6)
The VIKOR’s compromise ranking algorithm includes the following steps: [24]
Step 1: Determining the best and worst value of \(f_{i}^{ - }\) for all criteria functions \((i = 1\,,\,2\,,\, \ldots ,\,n\,)\) if \(i\) th function represents an advantage (positive aspect) in this case \(f_{i}^{*} = \mathop {\hbox{max} }\limits_{j} \,f_{ij}\) and \(f_{i}^{ - } = \mathop {\hbox{min} }\limits_{j} \,f_{ij}\).
Step 2: Calculating the values \(S_{j}\) and \(R_{j}\) for \(j = 1\,,\,2\,,\, \ldots ,\,J\) using the following relationships:
$$P = 1\,\,\, \Rightarrow \,\,S_{j} = \sum\limits_{i = 1}^{n} {w_{i} } \,\frac{{f_{i}^{*} - f_{ij} }}{{f_{i}^{*} - f_{i}^{ - } }}$$
(7)
$$P \to \infty \,\,\, \Rightarrow \,\,R_{j} = \hbox{max} \,\left[ {w_{i} \,\frac{{f_{i}^{*} - f_{ij} }}{{f_{i}^{*} - f_{i}^{ - } }}\,} \right]$$
(8)
In a way that \(w_{i}\) expresses the relative weight of each criteria and indicates the relative importance of each one.
Step 3: Calculating the amount of \(Q_{j}\) for \(j = 1\,,\,2\,,\, \ldots \,,\,J\) using the following equation:
$$Q_{j} = \,v\,\left( {\frac{{S_{j} - S^{*} }}{{S^{ - } - S^{*} }}} \right) + (1 - v)\,\left( {\frac{{R_{j} - R^{*} }}{{R^{ - } - R^{*} }}} \right)$$
(9)
In a way that:
$$S^{*} = \hbox{min} \,S_{j}, \quad S^{ - } = \hbox{max} \,S_{j}, \quad R^{*} = \hbox{min} R_{j}, \quad R^{*} = \hbox{max} R_{j} \,$$
In this regard, the weight of the strategy is introduced “the majority of the criteria” (or the maximum utility of the group), which is here \(v = 0.5\).
Step 4: Ranking alternatives by sorting descending values of \(Q\,,\,R\,,\,S\). So the results of the three lists are ranked.
Step 5: It is suggested that a compromise solution to be considered for the alternative \((a^{\prime})\) ranking by the minimum criteria of \(Q\) if two conditions are met.
Condition 1. Acceptable advantage: it should be \(Q(a^{\prime\prime}) - Q(a^{\prime}) \ge \,DQ\). In a way that \((a^{\prime\prime})\) is an option having the second position in the classified list of \(Q\) and \(DQ = \frac{1}{J - 1}\) so that \(J\) is the number of alternatives.
Condition 2. Acceptable stability in decision making: Alternatives \((a^{\prime})\) should have the best rank in the list \(\,R\,,\,S\). That is, this reconciliation must be stable in a decision making process. So that under any circumstances (majority voting \(v > 0.5\), agreement with \(v = 0.5\), veto with \(v < 0.5\)) to be set.
If one of the conditions is not met, a set of compromise solutions is suggested:
- A.
Options \((a^{\prime})\) and \((a^{\prime\prime})\) if the only second condition is not meet.
- B.
Options \((a^{\prime})\), \((a^{\prime\prime}), \ldots ,(a^{M} )\) if the first condition does not occur and \((a^{M} )\) is determined from the following equation: \(Q(a^{M} ) - Q(a^{\prime}) < \,DQ\) M for maximum.
The best alternative ranked by the \(Q\) index is the amount with the minimum \(Q\) quality. The method of VIKOR is a useful tool in multi-criteria decision making, especially when the decision maker is not able to express its preferences at the beginning of the system design. In order to compare the data before and after the Health Transformation Plan, after examining the normality of the data, the paired t-test was used.