Portfolio Decision Analysis for Evaluating Stakeholder Conflicts in Land Use Planning

The CAR method (Danielson et al. 2014; Danielson and Ekenberg 2016) extends an ordinal ranking of criteria with information regarding the strengths of the preferences between two criteria. The strength of a preference is denoted by \(\succ _i\) where i is the number of steps separating the pair of criteria. These steps can be expressed using semantic labels such asFor example, let \(G = \{G_1,\ldots ,G_4\}\) be a set of 4 criteria. A cardinal ranking of the criteria could be \(G_1 \succ _1 G_2 \succ _3 G_3 \succ _2 G_4\), i.e. criterion \(G_1\) is slightly more important than criterion \(G_2\), which is much more important than criterion \(G_3\), which in turn is more important than criterion \(G_4\). The preference statements are then transformed into weights by the following three-step procedure.

The method of CAR for conflict evaluations (Fasth et al. 2018) is an application of CAR for values (Danielson et al. 2014; Danielson and Ekenberg 2016), in which a do nothing action \(A_{\alpha }\) is added to the ranking. The purpose of the do nothing action is to enable making preference statements regarding an action’s negative or positive performance in comparison to this action. CAR-CE uses the same notation and semantic labels for values as the CAR described above.

For example, let \(A = \{A_1,\ldots ,A_4\}\) be a set of 4 actions evaluated in terms of criterion \(G_1\). Suppose that a stakeholder, using cardinal ranking, states that \(A_1 \succ _2 A_2 \succ _2 A_3 \succ _3 A_4\). Then the stakeholder inserts the do nothing action \(A_{\alpha }\) between \(A_2\) and \(A_{3}\), resulting in the CAR-CE ranking \(A_1 \succ _2 A_2 \succ _1 A_{\alpha } \succ _1 A_3 \succ _3 A_4\), i.e. \(A_1\) is better than \(A_2\), which is better than \(A_3\), which is much better than \(A_4\). \(A_1\) and \(A_2\) are positive relative to \(A_{\alpha }\), and \(A_3\) and \(A_4\) are negative relative to \(A_{\alpha }\). CAR-CE extends the CAR method of eliciting scores with a step where the do nothing action is included in the ranking. The preference statements are transformed into scores using the following procedure: Note that Eq. (3) transforms the values to an underlying interval scale where the worst action is assigned the value of 0 and the best action the value of 1. However, de Almeida et al. (2014) argue that this transformation can lead to scaling issues in multi-attribute portfolio problems. Instead, they suggest using Eq. (4), which results in a ratio scale type transformation of the values, where the worst action may have a value less than 0 and the best action a value greater than 1.We therefore replace Eq. (3) by Eq. (4) in the last step of the CAR-CE procedure described above.

Several methods for investigating conflict between stakeholder preferences have been suggested in the literature. For example, Ngwenyama et al. (1996) and Luè and Colorni (2015) suggest analysing the weights of the criteria, Cook et al. (1997) and Ray and Triantaphyllou (1998) suggest investigating the rankings, whereas Bana e Costa (2001) and Fasth et al. (2018) suggest analysing the action’s performance scores. The two latter approaches are similar in how the underlying conflicting stakeholder sets are defined. However, in Fasth et al. (2018) there are also defined two conflict indices: (i) a within-group conflict index, which measures the conflict within a single stakeholder group, and (ii) a between-group conflict index, which measures the conflict between two groups of stakeholders. Furthermore, the indices use a sum of squares approach similar to Ward’s clustering method (Rencher 2003, p. 466) and use the do nothing action \(A_{\alpha }\) to divide the stakeholders into two opposing stakeholder subsets.

Within-group conflict index This is a measure of the conflict within a stakeholder group. Let \(A = \{A_{1},A_{2}\ldots ,A_{n}\}\) be a set of actions, \(G = \{G_{1},G_{2},\ldots ,G_{m}\}\) a set of criteria, \(S = \{S_{1},S_{2},\ldots ,S_{o}\}\) a set of stakeholders, and let the performance score of action \(A_j\) in terms of criterion \(G_i\) for stakeholder \(S_k\) be denoted by \(v^{k}_{ij}\). For each criterion \(G_i\) and action \(A_j\), the set of stakeholders S is divided into two sub-sets: the stakeholders who state that \(v^k_{ij} < v^k_{i\alpha }\) are assigned to the con-group \(S^-_{ij}\), and stakeholders who state that \(v^k_{ij} \ge v^k_{i\alpha }\) are assigned to the pro-group \(S^+_{ij}\), see Eqs. (5) and (6).The within-group conflict index is based on the value difference \(d^k_{ij}\) between the part-worth value \(q^k_{ij}\) of an action \(A_j\) and the part-worth value \(q^k_{i\alpha }\) of action \(A_{\alpha }\). The part-worth value \(q^k_{ij}\) of criterion \(G_i\) for action \(A_j\) for stakeholder \(S_k\) is given by \(q^k_{ij} = w^k_i v^k_{ij}\), where \(w^k_i\) is the weight (scaling constant) of criterion \(G_i\) under the condition that \(0 \le w^k_i \le 1\) and \(\sum _i^m w^k_i = 1\). The value difference in part-worth values is thenThe within-group conflict index uses a sum of squares approach calculated for the con-group (9), the pro-group (10), and a third group which includes the members of both groups (11). In the equations, we use a stakeholder scaling constant \(\lambda _k\) (where \(\lambda _k \ge 0\) and \(\sum _k^o \lambda _k= 1\)) to quantify the power of social influence, and for each stakeholder we calculate the sum of the squared differences between the value difference \(d^k_{ij}\) and the group’s mean distance. The conflict index of a set of stakeholders \({\varvec{S}}\) for criterion \(G_i\) and action \(A_j\) is then given by Equation (8), where \(\beta\) is a factor used to normalize the result to a [0,1] scale.whereThe conflict index \(d^S_{ij}\) in Eq. (8) is defined within the range [0, 1]. In Eq. (7), a maximum value of 1 is assigned when all members of the con-group state that \(q_{i\alpha } = 1\) and \(q_{ij} = 0\), all members of the pro-group state that \(q_{i\alpha } = 0\) and \(q_{ij} = 1\), and when the power balance of the two groups are equal, i.e. when the sum of the con-group’s and pro-group’s stakeholder scaling constants are equal, such that \(\sum _{S_{k} \in S^{-}_{ij}} \lambda _{k} - \sum _{S_{l} \in S^{+}_{ij}} \lambda _{l} = 0\). Furthermore, in Eq. (7), a minimum of 0 is assigned when either the con-group or the pro-group is empty. The power balance between the two groups affects the result since an evenly distributed power balance produces a greater conflict (as seen above), and since stakeholders with less power and social influence are more likely to accept (what they consider) counterproductive actions (Torrance 1957). Expressed differently, if all stakeholders have the same scaling constant, then the number of stakeholders in each group affects the magnitude of the conflict.

Between-group conflict index This is a measure of the conflict between two stakeholder groups. Let D and E be two subsets of S, e.g. two stakeholder groups from two separate residential areas. As in the within-group conflict index, for each criterion \(G_i\) and action \(A_j\), we partition each stakeholder group, D and E, into two subsets, such that stakeholders who stated that \(v^k_{ij} < v^k_{i\alpha }\) are assigned to the con-groups \({\varvec{S}}_{ij}^{D-}\) and \({\varvec{S}}_{ij}^{E-}\), and stakeholders who stated that \(v^k_{ij} \ge v^k_{i\alpha }\) are assigned to the pro-groups \({\varvec{S}}_{ij}^{D+}\) and \({\varvec{S}}_{ij}^{E+}\), see (12).As in the within-group conflict index, the between-group conflict index uses a sum of squares approach, here calculated for i) the con-groups \(C^D_{ij}, C^E_{ij}\), and the combined con-group \(C^{D,E}_{ij}\) (9), ii) the pro-groups \(P^D_{ij},P^E_{ij}\) and the combined pro-group \(P^{D,E}_{ij}\) (10), and iii) \(T^D_{ij}\) for stakeholder group D, \(T^E_{ij}\) for stakeholder group E, and \(T^{D,E}_{ij}\) for the combined D and E group (11). The between-group conflict index \(d^{D,E}_{ij}\) of stakeholder sets D and E regarding criterion \(G_i\) and alternative \(A_j\) is given bywhere \(\beta = \frac{1}{\sum _{S_{k} \in S} {\lambda _k}^{2}}.\)

The main idea of conflict constrained portfolios is to investigate how a change in overall conflict affects the composition of a portfolio. It is centred around solving a 0–1 knapsack optimization problem (Martello and Toth 1990) where the objective function maximizes the overall value of the portfolio using conflict as a resource constraint. The name is derived from the problem faced by a person who is interested in filling a knapsack (rucksack) of fixed size with items, each with a set of two attributes: a value that quantifies the level of desirability or importance of the item, and a volume. Since the knapsack is of fixed size, the problem is to figure out how to fill it with the optimal combination of items that yields the highest total value. The knapsack problem is formally described in Eq. (14), where \(x_{j}\) denotes a binary decision variable set to 1 if the item is included and 0 if it is not.The stakeholders’ aggregated value \(W(A_{j})\) of an action \(A_j\) is given byAn action’s overall conflict index \(d_j\), i.e. the sum of all criterion-specific conflict indices, is given byThe resource constraint B is the sum of all actions’ overall conflict indices, and is defined byNote that \(W(A_{j})\) becomes negative when the stakeholder group as a whole holds the opinion that action \(A_j\) is worse than the do nothing action. This means in practice that the action is deemed counterproductive.

The front of efficient portfolios is generated by following the procedure described in Fasth et al. (2016) and Lourenço et al. (2012), in which the problem is solved multiple times with different values set for the resource constraint B. The problem is first solved by setting B to the sum of all actions’ conflict indices, then in the subsequent executions, B is reduced by a small number \(\tau\), e.g. \(\tau =B/|S|^2\). A difference between our procedure and the aforementioned is that in addition to maximizing the objective function, we also minimize it to find possible negative or counterproductive portfolios. The concept of counterproductive portfolios gives an additional insight into the decision problem, as it highlights actions that are especially bad and should be discarded or further investigated. The procedure is illustrated in Algorithm 1.

The aim of a sensitivity analysis is to investigate how a change in some variable affects the stability of a solution. Kleinmuntz (2007) describes two methods of sensitivity analysis. In the first, the actions under consideration are forced to be either excluded or included in the portfolio. In the second, the budget constraint is varied to analyse how the portfolio composition changes. As described in the previous section, we use the latter method by solving the optimization problem with different levels of the budget constraint. A simple method for measuring an action’s degree of inclusion in a set of optimal portfolios is to use the concept of the core index (Liesiö et al. 2007, 2008). A core index is defined bywhere the denominator is equal to the number of portfolios p which are included in the set of non-dominated portfolios P, given that \(A_j\) is included in p, and the numerator is equal to the cardinality of the set of non-dominated portfolios P. The core index \({\text {CI}}(A_j)\) of an action \(A_j\) lies within \(0 \le {\text {CI}}(A_j) \le 1\). Three semantic labels are used to distinguish between the core index of different actions. First, an action \(A_j\) included in all portfolios \({\text {CI}}(A_j) = 1\) is called a core action. Second, an action \(A_j\) included in no portfolios \({\text {CI}}(A_j) = 0\) is called an exterior action. Third, an action \(A_j\) included in more than one but not all portfolios \(0< {\text {CI}}(A_j) < 1\) is called a borderline action. In general, a decision-maker is advised to select core actions, discard exterior actions, and to focus on borderline actions. In the calculation of the core index we separate negative from positive portfolios, which results in negative and positive core indices.

To further distinguish between the borderline actions, we introduce the concept of borderline action subtypes. First, we denote by a the lower bound and by b the upper bound of the core index interval. Then, we define n points \(\{x_1,x_2,x_3,\ldots ,x_n\}\) along the interval to divide it into \(n + 1\) equal sized bins \([a,x_1],[x_2,x_3],\ldots ,[x_n,b]\). Next, to identify actions whose conflict index values between two stakeholder groups differ by more than a predefined threshold value, we introduce the concept of core index slopes. Let \(CI(A_j^D)\) and \(CI(A_j^E)\) be the core indices of action \(A_j\) for group D and E respectively. A core index slope exists if \(| CI(A_j^D) - CI(A_j^E)| > \gamma\), where \(\gamma\) is a predefined threshold value in the range (0,1). For example, in the next section, where we demonstrate the framework, we divide the core index into three equal sized bins: a lower bin with the range (0, 0.33], a middle bin with the range (0.33, 0.66], and an upper bin with the range (0.66, 1). We use a threshold value \(\gamma\) equal to one-half the borderline subtype range, i.e. \(\gamma = 1/2 \cdot 1/3 = 1/6\). The exact number of bins is obviously a judgment call that depends on the context and purpose of the sensitivity analysis. A slope graph (Tufte 2001) can effectively support the analysis and communicate the results to stakeholders, see Fig. 3 in the next section.

The slope graph in Fig. 3 displays the core index values of group D and group E for all 50 actions. Core index slopes, actions with a big core index difference between the two groups, are indicated with a red downward facing triangle. Actions with a small core index difference and a high core index value in both groups are indicated with a green upward facing triangle. The remaining actions are indicated with a light grey circle. Let us first focus on the red downward facing triangles, the core index slopes. As can be seen, action 8d, “more modern IT in education”, has the steepest slope, indicating the largest disagreement between the two groups. In group D, action 8d is found in the lower bin [0, 33] with a core index of 0.14, whereas in group E it is found in the upper bin [0.66, 1] with a core index of 0.77. This result suggests action 8d is a good candidate for further investigation, but not an action suitable for implementation. Other high conflict actions include 3e “build underground car parks in residential buildings” (core index difference 0.61), 10a “reduce energy consumption” (0.51), 4e “improve public transport to and from Stockholm” (0.48), and 6e “Improve the education in high schools” (0.34).

Turning to the actions that are indicated with a green upward triangle, we find good candidates for implementation:The sensitivity analysis suggests several important insights and areas for further investigation. First, the city council should focus on improving the public education system: strengthening teachers’ professional development, raising the quality of teaching, and improving primary school education. Second, the city council should improve the safety around the train station and in the city centre, for example by installing better street lighting since many people feel uneasy in inadequately lit outdoor urban environments. Action 10e, “Reduce toxins and hazardous chemicals in the environment” has, by far, the smallest core index difference among all considered actions. The municipal office can confidently implement this action in terms of stakeholder disagreement (as it is negligible), but it should probably prioritize other more pressing issues discussed above.