# Stochastic Multicriteria Acceptability Analysis for Evaluation of Combined Heat and Power Units

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

_{2}production and footprint. Uncertainties and imprecision are common both in criteria measurements and weights, therefore the stochastic multicriteria acceptability analysis (SMAA) model is used in aiding this decision making problem. These uncertainties are treated better using a probability distribution function and Monte Carlo simulation in the model. Moreover, the idea of “feasible weight space (FWS)” which represents the union of all preference information from decision makers (DMs) is proposed. A complementary judgment matrix (CJM) is introduced to determine the FWS. It can be found that the idea of FWS plus CJM is well compatible with SMAA and thus make the evaluation reliable.

## 1. Introduction

_{2}production and footprint. The data for these systems have been collected by a literature review [29,31]. SMAA is used for the evaluation of the 16 CHP units. The uncertainties of criteria PVs and weight vectors are treated using a probability distribution function and a Monte Carlo simulation which make FWS plus CJM well compatible with SMAA. The method presented in this paper is for the evaluation of CHP units, but it can be extended for the evaluation of other complex energy systems.

## 2. Combined Heat and Power Units and Evaluation Criteria

#### 2.1. Combined Heat and Power Units

- S1: compression engines diesel 200 kWe (industry).
- S2: compression engines diesel 20 MWe (industry).
- S3: gas engines—Otto cycle 1 kWe (household).
- S4: gas engines—Otto cycle 13 MWe (industry).
- S5: GT 500 kWe (industry).
- S6: GT 225 MWe (industry).
- S7: micro-turbines (CHP) 10 kWe (industry).
- S8: micro-turbines (CHP) 500 kWe (industry).
- S9: combined cycle gas turbines (CCGT) 8 MWe (industry).
- S10: CCGT 750 MWe (industry).
- S11: ST 500 kWe (coal) (hot water) (industry).
- S12: ST 500 kWe (fuel oil) (hot water) (industry).
- S13: ST 500 kWe (natural gas) (hot water) (industry).
- S14: ST 150 MWe (coal) (hot water) (industry).
- S15: ST 150 MWe (fuel oil) (hot water) (industry).
- S16: ST 150 MWe (natural gas) (hot water) (industry).

_{2}production and footprint are the environmental criteria.

CHP | Electrical output (kW_{e}) | Power to heat ratio | Efficiency (%) | Installation cost (€/kW_{e}) | Maintenance cost (c€/kW·h_{e}) | Electricity cost (c€/kW·h_{e}) | Heat cost (c€/kW·h_{th}) | CO_{2} production (kg/MW·h_{e}) | Footprint (m^{2}/kW_{e}) |
---|---|---|---|---|---|---|---|---|---|

S1 | 200 | 1.00 | 85 | 500 | 1 | 5.64 | 7.74 | 623.53 | 0.02 |

S2 | 20,000 | 1.23 | 88 | 1,500 | 0.5 | 2.30 | 4.84 | 545.96 | 0.011 |

S3 | 1 | 0.45 | 85 | 500 | 2 | 32.82 | 17.85 | 758.17 | 0.3 |

S4 | 13,000 | 0.90 | 88 | 2,500 | 0.7 | 3.81 | 4.84 | 479.80 | 0.014 |

S5 | 500 | 0.45 | 80 | 500 | 0.8 | 15.93 | 7.74 | 805.56 | 0.015 |

S6 | 225,000 | 0.70 | 90 | 1,200 | 0.2 | 4.37 | 4.72 | 539.68 | 0.0045 |

S7 | 10 | 0.29 | 75 | 1,500 | 1 | 33.84 | 9.61 | 1,186.21 | 0.05 |

S8 | 500 | 0.60 | 85 | 1,100 | 0.5 | 10.12 | 7.74 | 627.45 | 0.02 |

S9 | 8,000 | 0.96 | 73 | 1,000 | 0.8 | 5.78 | 5.18 | 559.36 | 0.03 |

S10 | 750,000 | 1.25 | 90 | 500 | 0.2 | 1.73 | 4.72 | 400 | 0.025 |

S11 | 500 | 0.25 | 82 | 2,000 | 0.5 | 2.23 | 0.51 | 2,042.68 | 0.06 |

S12 | 500 | 0.25 | 82 | 2,000 | 0.45 | 8.23 | 2.18 | 1,615.85 | 0.05 |

S13 | 500 | 0.25 | 82 | 2,000 | 0.4 | 28.35 | 7.74 | 1,219.51 | 0.027 |

S14 | 150,000 | 0.60 | 85 | 1,100 | 0.25 | 0.77 | 0.51 | 1,050.98 | 0.06 |

S15 | 150,000 | 0.60 | 85 | 1,100 | 0.2 | 2.82 | 2.18 | 831.37 | 0.05 |

S16 | 150,000 | 0.60 | 85 | 1,100 | 0.15 | 5.98 | 4.72 | 627.45 | 0.027 |

#### 2.2. Properties and Measurements of Criteria

_{2}production and footprint, which are listed in Table 2.

Criteria | Efficiency | Installation cost | Maintenance cost | Electricity cost | Heat cost | CO_{2} production | Footprint |
---|---|---|---|---|---|---|---|

Criteria No. | C_{1} | C_{2} | C_{3} | C_{4} | C_{5} | C_{6} | C_{7} |

Property ^{1} | ▲ | ▼ | ▼ | ▼ | ▼ | ▼ | ▼ |

Uncertainty | ±5% | ±10% | ±10% | ±10% | ±10% | ±10% | ±10% |

^{1}Property “▲” means the larger the better, i.e., positive or benefit criteria; property “▼” means the smaller the better, i.e., negative or cost criteria.

_{ij}stands for the criteria PVs and then the positive criteria can be normalized as:

_{i}in relation to criterion j, and x

_{ij}

^{+}and x

_{ij}

^{−}are the maximum and minimum values of alternative x

_{i}corresponding to criterion j.

## 3. Weighting Method Based on a Complementary Judgment Matrix

#### 3.1. Complementary Judgment Matrix

**A**, should be constructed via consultation and/or a questionnaire using the binary grading values shown in Table 3 [34].

Description | a_{ij} | a_{ji} |
---|---|---|

ith criterion is equally important compared with jth | 0.5 | 0.5 |

ith criterion is a little more important compared with jth | 0.6 | 0.4 |

ith criterion is important compared with jth | 0.7 | 0.3 |

ith criterion is very important compared with jth | 0.8 | 0.2 |

ith criterion is extremely important compared with jth | 0.9 | 0.1 |

_{ij}is the preference proportion of the ith criterion compared with the jth criterion. Assume that the weights of the ith and jth criteria are w

_{i}and w

_{j}, respectively. Then a

_{ij}would take the form:

_{ij}has the following two properties: a

_{ii}= 0.5 and a

_{ij}= 1 − a

_{ji}, $\forall $i, j = 1, 2, …, n. In addition, the following definitions are quite important for the use of CJM.

**A**= (a

_{ij})

_{n}

_{×n}, has ordinal consistency if any one of the following relationships hold true:

**A**= (a

_{ij})

_{n}

_{×n}, has complementary consistency if the following relationship holds true [34]:

_{ik}a

_{kj}a

_{ji}= a

_{ki}a

_{jk}a

_{ij}

_{ij}.

**A**consistent, because Equation (6) or even Equation (5) is not easy to satisfy. Therefore, an inconsistency check [35] is necessary prior to eliciting weight vectors. However, if the inconsistency only varies slightly and can be deemed “satisfactorily consistent”, then the CJM is still acceptable and can be used to calculate the weight vector by means of the weighted least square method [35]. The advantage of this check is that we can determine where (between each pair of comparison) the inconsistency is and the extent of it so that we can ask the DM to rethink about his preferences in a CJM if necessary. In addition, the CJM method is also very helpful in case of too many criteria making the DMs struggle when giving the preference information directly. DMs just need to compare every two of the criteria, which is very straightforward.

_{ij}is the errors of the elements in the CJM

**A**. They can be seen as statistically random variables with mean value expectations of zero. Basically, the more important a criterion is, the lower its error should be. Following this reasoning, we can define the objective function as the sum of the weighted square of ω

_{ij}and then minimize it subject to the weight constraints. Based on this, we can check the inconsistency extent and then calculate the weight vectors if the CJM is satisfactorily consistent. The problem can be expressed as:

#### 3.2. Feasible Weight Space

**w**represents weight vectors with nonnegative values and the summation of them is 1. This means that the general weight space is a union of all weight vectors. Follow this logic; a deterministic weight vector can be represented as a specific point in this space. Furthermore, in a real life MCDA problem like the one addressed in this paper, we have seven criteria, which means that the corresponding general weight space is a hyper-space or a simplex. In such a hyper-space, only one point is essentially not enough to represent the preferences of a group of DMs. This is the main motivation for the authors to propose FWS, which can be seen as a sub-space of the general weight space. FWS is not a totally new concept, but it narrows the weight space by assuming the weight vectors as random variables or variables with certain probability distributions that span only in the feasible sub-space. This means that the sample weight vectors are taken in random or with other probability distributions from the FWS in the Monte Carlo simulation. Therefore, FWS can concentrate on the weight vectors that are most probably used in real life.

**w**

_{A}is represented by one point

**A**on this plane. However, a possible FWS with interval constraints on each criterion can be demonstrated as a polygon shaded area on the same plane. This FWS can be expressed as in Equation (8) and shown in Figure 1b:

**Figure 1.**(

**a**) General weight space and a deterministic weight vector A of a three criteria case; and (

**b**) a feasible weight space (FWS) with interval constraints on each criterion.

**A**, shown in Figure 2a and the FWS in Figure 2b. Specifically, the upper and lower whiskers in Figure 2b represent the maximum and minimum constraints of weight values, but they cannot reach the maximum simultaneously because of the normalization property in Equation (9). In conclusion, a CJM is used to elicit the weight vector and then the FWS extends the weight vector from only one point in the weight space to a sub-space. For group decision making, it is necessary to get this sub-space to cover all DMs’ preference information. DMs give their CJMs and then the inconsistency check is implemented to get satisfactorily consistent CJMs or to determine whether a second round of judgment is needed for some of the DMs. Subsequently, all “consistent” CJMs are used to calculate the weight vectors and then we can obtain the FWS by merging all these weight vectors. However, if there are too few DMs in some situations, we can set an interval for each criterion based on the calculated weight vector as shown in Figure 2b.

**Figure 2.**Bar representation of: (

**a**) a deterministic weight vector A; and (

**b**) an FWS with interval constraints on each criterion.

## 4. Stochastic Multicriteria Acceptability Analysis

#### 4.1. The Stochastic Multicriteria Acceptability Analysis-2 Model

**A**= {x

_{1}, x

_{2}, x

_{3}, …, x

_{m}}, which needs to be evaluated in terms of n criteria. Assume that the DM’s preference structure can be represented by a utility function, which maps the different alternatives to the utility values for u(x

_{i},

**w**). The SMAA-2 method introduces a rank acceptability index to describe the overall acceptability of each alternative. A ranking function is presented to compute the rank of each alternative from the best rank in Equation (1) to the worst rank (m) as [37]:

**ξ**to denote criteria PVs with a stochastic distribution of f

_{X}(

**ξ**), and

**w**has a stochastic distribution of f

_{W}(

**w**). Then the SMAA-2 is based on analyzing sets of favorable rank weights, W

_{i}

^{r}(

**ξ**), which are defined as:

**w**$\in $ W

_{i}

^{r}(

**ξ**), assigns utilities for the alternatives so that alternative x

_{i}obtains rank r. The rank acceptability index, b

_{i}

^{r}, is then defined as the expected volume of the set of favorable rank weight space for each alternative. This is done as follows:

_{i}with a rank r. In reality, rank acceptability means the percentage of all Monte Carlo simulations among which a given alternative i obtains rank r. The SMAA-2 method extends the original SMAA model by considering all ranks in the analysis based on a holistic acceptability index in order to examine the overall acceptability of each alternative. A holistic acceptability index is defined to consider all rank acceptability indices as follows:

_{r}are meta-weights, which indicate the contribution of each rank acceptability index to the evaluation of an alternative. It is natural that the first ranks contribute most and the worst ranks contribute very little to the holistic acceptability index. Therefore the meta-weights

**α**can be obtained by a descending vector:

**w**

_{i}

^{c}, is defined as the expected center of gravity of the favorable weight space. The central weight vector is computed as an integral of the weight vector over the criteria and weight distributions by:

_{i}, given the assumed weight distribution, which can be found in Section 5.2.

**w**

_{i}

^{c}is actually the average of the finite used weight vectors favoring alternative i in the Monte Carlo simulations.

_{i}

^{c}, is defined as the probability that a particular alternative is the most preferred alternative when a particular central weight vector is chosen. Namely, only the first rank acceptability b

_{i}

^{1}correspond to the confidence factor, other ranks don’t have confidence factors at all. It is computed as an integral over the criteria distributions by:

#### 4.2. Handling the Uncertainties

#### 4.2.1. Uncertainties in Criteria Measurements

#### 4.2.2. Uncertainties in Weighting

**Figure 3.**(

**a**) FWS with no weight information of a 3-criterion case using uniform distribution; and (

**b**) projection onto w

_{1}-w

_{2}plane.

_{j}

^{min}, w

_{j}

^{max}]. They may result from direct preference statements of the DMs or from the CJMs. The intervals can be represented as a distribution by restricting the uniform weight distribution with linear inequality constraints based on the intervals. The restricted distribution weights can easily be generated by modifying the above procedure to reject weights that do not satisfy the interval constraints. Figure 4 illustrates the resulting weight distribution in a 3-criterion case.

**Figure 4.**(

**a**) FWS with interval weight information of a 3-criterion case using uniform distribution; and (

**b**) projection onto w

_{1}-w

_{2}plane.

_{2}production with a weight percentage of 16.7%. In addition, the rest of the criteria have weight percentages lower than 1/7; of these criteria, installation cost and maintenance cost are important, heat cost is less important, while the weight of footprint is small. It can be concluded that the ordinal sequence for these criteria weights is: C

_{1}> C

_{4}= C

_{6}> C

_{2}= C

_{3}> C

_{5}> C

_{7}. However, an arbitrary weight vector in this FWS may have a different ordinal sequence because of the uncertainties. This FWS is used in the SMAA-2 model for the evaluation of CHP units.

**Figure 5.**FWS with ±50% interval constraints using uniform distribution on each criterion for evaluation of CHP units.

## 5. Results and Discussion

Weight type | No. | Description |
---|---|---|

No weight information | (a) | The FWS is the general weight space with seven criteria, i.e., a six-dimensional simplex |

Interval constraints of weights | (b) | The FWS shown in Figure 5 |

#### 5.1. Results

^{h}> 50% and/or p

^{c}> 20% appear in boldface.

**Table 5.**Confidence factors (p

^{c}) and holistic (a

^{h}) and rank acceptability indices (b

^{r}) in percentages using type (a) weight bound.

CHP | p^{c} | a^{h} | b^{1} | b^{2} | b^{3} | b^{4} | b^{5} | b^{6} | b^{7} |

S1 | 6.2 | 30.7 | 0.4 | 8.8 | 7.8 | 6.9 | 7.5 | 9.2 | 12.6 |

S2 | 7.1 | 39.9 | 1.1 | 5.6 | 21.8 | 15.9 | 14.5 | 16.7 | 9.5 |

S3 | 0 | 2.5 | 0 | 0 | 0 | 0.1 | 0.1 | 0.1 | 0.1 |

S4 | 4.2 | 22.7 | 0.3 | 1.1 | 3.8 | 8.0 | 6.2 | 7.4 | 11.7 |

S5 | 6.3 | 16.3 | 0.1 | 1.0 | 2.5 | 2.0 | 2.2 | 2.8 | 3.9 |

S6 | 23.9 | 62.4 | 10.3 | 47.6 | 17.2 | 11.0 | 6.9 | 3.7 | 1.9 |

S7 | 0 | 1.9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

S8 | 0.7 | 21.9 | 0 | 0.1 | 0.7 | 2.1 | 4.6 | 8.4 | 15.6 |

S9 | 0.5 | 14.7 | 0 | 0.2 | 0.6 | 1.3 | 2.0 | 3.2 | 5.2 |

S10 | 99.4 | 92.5 | 80.1 | 12.8 | 3.9 | 1.8 | 0.8 | 0.3 | 0.2 |

S11 | 11.0 | 10.9 | 0.1 | 0.3 | 0.4 | 0.7 | 1.0 | 1.8 | 3.4 |

S12 | 0 | 10.1 | 0 | 0 | 0.1 | 0.1 | 0.2 | 0.6 | 1.4 |

S13 | 0 | 5.6 | 0 | 0 | 0 | 0 | 0 | 0 | 0.1 |

S14 | 53.1 | 40.8 | 5.4 | 10.0 | 12.2 | 13.4 | 13.9 | 12.2 | 10.0 |

S15 | 11.5 | 40.3 | 1.6 | 7.4 | 14.7 | 18.6 | 18.8 | 15.5 | 10.8 |

S16 | 4.8 | 39.0 | 0.6 | 5.1 | 14.2 | 18.1 | 21.2 | 18.1 | 13.5 |

CHP | b^{8} | b^{9} | b^{10} | b^{11} | b^{12} | b^{13} | b^{14} | b^{15} | b^{16} |

S1 | 14.4 | 14.0 | 7.1 | 5.4 | 3.3 | 1.6 | 1.0 | 0 | 0 |

S2 | 5.7 | 4.6 | 3.7 | 0.8 | 0 | 0 | 0 | 0 | 0 |

S3 | 0.2 | 0.5 | 2.0 | 4.5 | 5.2 | 3.3 | 7.6 | 15.1 | 61.0 |

S4 | 10.7 | 10.9 | 9.2 | 13.1 | 6.4 | 6.0 | 2.5 | 1.2 | 1.4 |

S5 | 6.1 | 11.1 | 18.7 | 17.6 | 12.7 | 17.4 | 1.9 | 0 | 0 |

S6 | 1.0 | 0.3 | 0.1 | 0 | 0 | 0 | 0 | 0 | 0 |

S7 | 0 | 0 | 0 | 0.3 | 1.8 | 3.5 | 7.1 | 60.2 | 27.1 |

S8 | 24.0 | 22.2 | 12.7 | 5.6 | 3.1 | 0.8 | 0.1 | 0 | 0 |

S9 | 8.6 | 11.3 | 19.5 | 15.1 | 10.1 | 9.5 | 9.9 | 3.4 | 0.2 |

S10 | 0.1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

S11 | 4.8 | 5.5 | 10.1 | 13.2 | 15.1 | 16.6 | 12.5 | 7.1 | 7.3 |

S12 | 2.9 | 4.9 | 8.9 | 16.9 | 28.5 | 23.7 | 7.6 | 3.8 | 0.3 |

S13 | 0.3 | 0.7 | 1.7 | 4.4 | 13.4 | 17.6 | 49.8 | 9.2 | 2.6 |

S14 | 7.8 | 7.8 | 4.3 | 2.6 | 0.3 | 0.1 | 0 | 0 | 0 |

S15 | 7.0 | 3.8 | 1.5 | 0.3 | 0 | 0 | 0 | 0 | 0 |

S16 | 6.3 | 2.3 | 0.6 | 0.1 | 0 | 0 | 0 | 0 | 0 |

**Table 6.**Confidence factors (p

^{c}) and holistic (a

^{h}) and rank acceptability indices (b

^{r}) in percentages using type (b) weight bound.

CHP | p^{c} | a^{h} | b^{1} | b^{2} | b^{3} | b^{4} | b^{5} | b^{6} | b^{7} |

S1 | 0 | 27.1 | 0 | 0 | 1.3 | 6.2 | 9.5 | 14.5 | 25.4 |

S2 | 0 | 51.6 | 0 | 2.0 | 73.3 | 14.8 | 6.4 | 2.9 | 0.6 |

S3 | 0 | 3.8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

S4 | 0 | 27.7 | 0 | 0 | 1.2 | 11.9 | 10.6 | 12.2 | 18.5 |

S5 | 0 | 13.4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

S6 | 2.5 | 70.0 | 1.8 | 95.8 | 2.2 | 0.1 | 0 | 0 | 0 |

S7 | 0 | 0.0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

S8 | 0 | 20.7 | 0 | 0 | 0 | 0.2 | 1.0 | 3.0 | 9.4 |

S9 | 0 | 7.8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

S10 | 99.1 | 99.5 | 98.2 | 1.8 | 0 | 0 | 0 | 0 | 0 |

S11 | 0 | 8.9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

S12 | 0 | 9.9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

S13 | 0 | 4.2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

S14 | 0 | 33.8 | 0 | 0.1 | 5.8 | 17.4 | 21.2 | 22.8 | 18.9 |

S15 | 0 | 37.1 | 0 | 0.1 | 8.5 | 25.4 | 26.4 | 21.3 | 12.6 |

S16 | 0 | 36.5 | 0 | 0.1 | 7.6 | 23.9 | 24.9 | 23.3 | 14.6 |

CHP | b^{8} | b^{9} | b^{10} | b^{11} | b^{12} | b^{13} | b^{14} | b^{15} | b^{16} |

S1 | 27.9 | 15.1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

S2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

S3 | 0 | 0 | 0.2 | 4.6 | 4.9 | 10.4 | 23.8 | 55.6 | 0.3 |

S4 | 22.4 | 23.0 | 0.2 | 0 | 0 | 0 | 0 | 0 | 0 |

S5 | 0 | 0.2 | 69.1 | 17.9 | 11.8 | 1.0 | 0 | 0 | 0 |

S6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

S7 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.3 | 99.7 |

S8 | 30.7 | 55.7 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

S9 | 0 | 0.1 | 8.0 | 25.9 | 11.7 | 27.3 | 12.2 | 15.0 | 0 |

S10 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

S11 | 0 | 0 | 12.4 | 19.6 | 29.0 | 27.1 | 7.7 | 4.2 | 0 |

S12 | 0 | 0 | 10.1 | 31.3 | 39.7 | 16.7 | 1.8 | 0.3 | 0 |

S13 | 0 | 0 | 0 | 0.7 | 2.9 | 17.4 | 54.4 | 24.6 | 0 |

S14 | 9.8 | 4.1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

S15 | 4.5 | 1.1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

S16 | 4.7 | 0.8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

**Figure 6.**Rank acceptability indices (b

^{r}): (

**a**) for evaluation of CHP units with no weight information; and (

**b**) with interval constraints of weights.

**Figure 7.**Central weights (w

^{c}): (

**a**) favoring different CHP units with no weight information; and (

**b**) with interval constraints of weights.

#### 5.2. Discussion

## 6. Conclusions

_{2}production and footprint. Data of these CHP units were collected by a literature review and this problem has been addressed by some previous studies. However, we notice that the evaluation can be improved to some extent. First, uncertainties and imprecision are common both in criteria PVs and weights, therefore the SMAA model is adopted in this paper. Moreover, the FWS which represents the union of preference information from DMs is proposed. A CJM is introduced to determine the FWS. Subsequently, two different types of FWSs are used for the evaluation of CHP units. The first one is the general weight space which reduces subjectivity to a minimum level and the second one is the FWS with interval constraints on criteria.

## Acknowledgments

## Author Contributions

## Abbreviations

AHP | Analytical hierarchy process |

CC | Combined cycle |

CCGT | Combined cycle gas turbine |

CHP | Combined heat and power |

CJM | Complementary judgment matrix |

DM | Decision maker |

FWS | Feasible weight space |

GT | Gas turbine |

MCDA | Multicriteria decision analysis |

SMAA | Stochastic multicriteria acceptability analysis |

ST | Steam turbine |

## Letter Symbols

A | Complementary judgment matrix |

a_{ij} | Element of a judgment matrix |

a_{i}^{h} | Holistic acceptability index of alternative i |

p_{i}^{c} | Confidence factor of alternative i, % |

W | Weight space |

W_{i}^{r} | Favorable ranking weights |

w | Weight |

w | Weight vector |

w_{i}^{c} | Central weight vector |

## Subscripts and Superscripts

e | Electricity |

th | Thermal |

c | Cent |

## Conflicts of Interest

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## Share and Cite

**MDPI and ACS Style**

Wang, H.; Jiao, W.; Lahdelma, R.; Zhu, C.; Zou, P.
Stochastic Multicriteria Acceptability Analysis for Evaluation of Combined Heat and Power Units. *Energies* **2015**, *8*, 59-78.
https://doi.org/10.3390/en8010059

**AMA Style**

Wang H, Jiao W, Lahdelma R, Zhu C, Zou P.
Stochastic Multicriteria Acceptability Analysis for Evaluation of Combined Heat and Power Units. *Energies*. 2015; 8(1):59-78.
https://doi.org/10.3390/en8010059

**Chicago/Turabian Style**

Wang, Haichao, Wenling Jiao, Risto Lahdelma, Chuanzhi Zhu, and Pinghua Zou.
2015. "Stochastic Multicriteria Acceptability Analysis for Evaluation of Combined Heat and Power Units" *Energies* 8, no. 1: 59-78.
https://doi.org/10.3390/en8010059