# A Risk-Aversion Approach for the Multiobjective Stochastic Programming Problem

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

#### 2.1. Multicriteria Decision-Making and Optimization under Uncertainty

**Definition**

**1**

- Weakly efficient if there is no $x\in X$, $x\ne \widehat{x}$, such that $f(x)<f(\widehat{x})$ i.e., ${f}_{k}(x)<{f}_{k}(\widehat{x})$ for all $k=1,\dots ,K$.
- Efficient or Pareto optimal if there is no $x\in X$ such that ${f}_{k}(x)\le {f}_{k}(\widehat{x})$ for all $k=1,\dots ,K$ and ${f}_{i}(x)<{f}_{i}(\widehat{x})$ for some $i\in \{1,\dots ,K\}$.
- Strictly efficient if there is no $x\in X$, $x\ne \widehat{x}$, such that $f(x)\le f(\widehat{x})$.

**Definition**

**2**

#### 2.2. Multiobjective Stochastic Programming

## 3. Methodology

**Definition**

**3**

**Remark**

**1.**

- If ${\lambda}_{i}=\frac{1}{n}$, the resulting OWA is the average of a.
- If ${\lambda}_{1}=1$, and ${\lambda}_{j}=0$ for $j>1$, the OWA is the maximum of a.
- If ${\lambda}_{n}=1$, and ${\lambda}_{j}=0$ for $j<n$, the OWA is the minimum of a.

**Definition**

**4**

- 1.
- Sort vector a such that ${a}_{(1)}\ge {a}_{(2)}\ge \dots \ge {a}_{(n)}$.
- 2.
- With $(\times )$ as the order induced by a, define ${T}_{j}={\sum}_{k=1}^{j}{u}_{(k)}$.
- 3.
- Let f be a function, such that $f:[0,1]\to [0,1]$ and $f(0)=0,f(1)=1$. This function is called weight generating function.
- 4.
- Obtain the weights as ${\lambda}_{j}=f({T}_{j})-f({T}_{j-1})$.

**Example**

**1**(of Definition 4).

- ${\lambda}_{1}=f({T}_{1})=f({u}_{(1)})=\frac{{u}_{(1)}}{r}$, assuming ${u}_{(1)}<r$
- ${\lambda}_{2}=f({T}_{2})-f({T}_{1})=f({u}_{(1)}+{u}_{(2)})-f({u}_{(1)})=\frac{{u}_{(1)}+{u}_{(2)}}{r}-\frac{{u}_{(1)}}{r}=\frac{{u}_{(2)}}{r}$, assuming ${u}_{(1)}+{u}_{(2)}<r$
- …
- ${\lambda}_{{j}^{*}}=f({T}_{{j}^{*}})-f({T}_{{j}^{*}-1})=1-\left(\frac{{u}_{(1)}+{u}_{(2)}+\dots +{u}_{({j}^{*}-1)}}{r}\right)$, since ${T}_{{j}^{*}}\ge r$
- ${\lambda}_{{j}^{*}+1}=f({T}_{{j}^{*}+1})-f({T}_{{j}^{*}})=1-1=0$
- …
- ${\lambda}_{n}=f({T}_{n})-f({T}_{n-1})=1-1=0$

**Definition**

**5**

**Remark**

**2**

**Example**

**2.**

- For $\beta =0.2$, the scenario $j=1$ is the only one needed to obtain the worst scenario with probability $0.2$, and hence ${g}_{k}^{\beta}(x)=\frac{0.2\times 10}{0.2}=10$.
- When β equals 0.3 it is necessary to include scenario 2, obtaining a β-average of $\frac{0.2\times 10+0.1\times 7}{0.3}=9$.
- Finally, if $\beta =0.5$ scenario 3 needs to be added as well, but only with the probability needed until reaching $0.5$: ${g}_{k}^{\beta}(x)=\frac{0.2\times 10+0.1\times 7+0.2\times 4}{0.5}=7$.

**Definition**

**6**(r-OWA, ${O}_{r}(\overrightarrow{x})$)

**Remark**

**3.**

**Example**

**3.**

- 1.
- As ${g}_{k}(x)$ are already ordered for largest to smallest, the values of ${T}_{k}$ are:$${T}_{1}=0.2,{T}_{2}=0.2+0.1=0.3,{T}_{3}=0.6,{T}_{4}=0.85,{T}_{5}=1$$
- 2.
- The values of ${T}_{k}$ under f:$$f({T}_{1})=\frac{0.2}{0.5},f({T}_{2})=\frac{0.3}{0.5},f({T}_{3})=f({T}_{4})=f({T}_{5})=1$$
- 3.
- The weights of the OWA:$${\lambda}_{1}=\frac{0.2}{0.5},{\lambda}_{2}=\frac{0.3-0.2}{0.5}=\frac{0.1}{0.5},{\lambda}_{3}=1-\frac{0.3}{0.5}=\frac{0.2}{0.5},{\lambda}_{4}={\lambda}_{5}=0$$
- 4.
- Consequently, the r-OWA is:$$r\text{-}OWA=\frac{0.2{g}_{(1)}(x)+0.1{g}_{(2)}(x)+0.2{g}_{(3)}(x)}{0.5}=\frac{0.2\times 10+0.1\times 7+0.2\times 4}{0.5}=7$$

**Remark**

**4.**

**Remark**

**5.**

**Definition**

**7**(Dominance).

**Reflexivity**- Given x, ${h}_{r}^{\beta}(x)\le {h}_{r}^{\beta}(x)$, and then $x\succsim x$, so ≿ is reflexive.
**Transitiveness**- Given $x\succsim y$, $y\succsim z$, we have ${h}_{r}^{\beta}(x)\le {h}_{r}^{\beta}(y)$ and ${h}_{r}^{\beta}(y)\le {h}_{r}^{\beta}(z)$, and then ${h}_{r}^{\beta}(x)\le {h}_{r}^{\beta}(z)$, which leads to $x\succsim z$, and we conclude that ≿ is transitive.
**Antisymmetry**- Given $x\succsim y$, $y\succsim x$, we have ${h}_{r}^{\beta}(x)\le {h}_{r}^{\beta}(y)$ and ${h}_{r}^{\beta}(y)\le {h}_{r}^{\beta}(x)$, but, from ${h}_{r}^{\beta}(x)={h}_{r}^{\beta}(y)$, it cannot be guaranteed that $x=y$, and, hence, ≿ is not antisymmetric.

#### 3.1. Idea of Solution and Dominance Properties

- For every $x\in X$ there is a function ${f}_{k}^{j}$ to be minimized which depends on the scenario j and the criterion k.
- The problem is transformed into a deterministic one with multiple objectives (MOP) while using the $\beta $-average concept.
- When computing the r-OWA, each $x\in X$ is assigned a scalar. The problem consists of finding the x, which minimizes this ${h}_{r}^{\beta}(x)$.

**Proposition**

**1.**

- 1.
- ${x}^{*}$ is not necessarily efficient of the MOP problem.
- 2.
- ${x}^{*}$ is weakly efficient of the MOP problem.
- 3.
- If ${x}^{*}$ is the only minimum of ${h}_{r}^{\beta}(x)$, then ${x}^{*}$ is efficient.
- 4.
- Given ${x}^{*}$ not efficient, an alternative ${y}^{*}$ can be found on a second phase, such that ${y}^{*}$ is efficient and ${h}_{r}^{\beta}({x}^{*})={h}_{r}^{\beta}({y}^{*})$.

**Example**

**4**(${x}^{*}$ is not necessarily efficient).

#### 3.2. An Illustrative Example

**Step 0**- Normalize all objective functions ${f}_{k}^{j}(x)$.
**Step 1**- Set values for $\beta ,r\in (0,1]$.
**Step 2**- For every $x\in X$ and every criterion define ${g}_{k}^{\beta}(x)$ as:$${g}_{k}^{\beta}(x)=\begin{array}{l}\mathrm{average}\mathrm{of}\mathrm{worst}\mathrm{scenarios}\mathrm{for}\mathrm{criterion}k\\ \mathrm{with}\mathrm{probabilities}\mathrm{adding}\mathrm{up}\mathrm{to}\beta \end{array}$$
**Step 3**- Define ${h}_{r}^{\beta}(x)$ as:$${h}_{r}^{\beta}(x)=\begin{array}{l}averageofworst{g}_{k}^{\beta}(x)values\\ withimportancesaddinguptor\end{array}$$
**Step 4**- Search for $x\in X$ minimizing ${h}_{r}^{\beta}(x)$.

- For the first criterion the worst scenario is ${j}_{5}$, which has probability $0.1$. The second worst is ${j}_{4}$, with a probability of $0.25$. As the sum of those probabilities exceeds the $\beta $ fixed, for computing the $\beta $-average just a probability of $0.2$ is considered:$${g}_{1}^{\beta}({x}_{1})=\frac{0.1\times 0.86+0.2\times 0.76}{0.3}=0.793$$
- ${g}_{2}^{\beta}({x}_{1})=\left(0.2\times 0.65+0.1\times 0.44\right)/0.3=0.580$
- ${g}_{3}^{\beta}({x}_{1})=\left(0.3\times 0.90\right)/0.3=0.900$
- ${g}_{4}^{\beta}({x}_{1})=0.833$, ${g}_{5}^{\beta}({x}_{1})=0.930$, ${g}_{6}^{\beta}({x}_{1})=0.728$

## 4. Computing the Minimum: Continuous Case

#### Mathematical Programming Model

**Remark**

**6.**

**Proposition**

**2.**

**Proof.**

**Proposition**

**3.**

**Proof.**

- For each feasible solution $(z,{v}_{k})$ of model (6), there is at least one feasible solution of model (7) with the same values $(z,{v}_{k})$, being so the same objective function.Let $({z}^{1},{v}_{k}^{1})$ be a feasible solution of model (6), and $({z}_{k}^{*},{y}_{kj}^{*})$ the optimal solution for each k minimizing ${g}_{k}^{\beta}(x)$ (right-hand-side of equation (6b)). Because constraints (7), (7c), (7d), and (7e) are satisfied in model (6), $({z}^{1},{v}_{k}^{1},{z}_{k}^{*},{y}_{kj}^{*})$ is a feasible solution or model (7).
- For each feasible solution $(z,{v}_{k},{z}_{k},{y}_{kj})$ of model (7), $(z,{v}_{k})$ is a feasible solution of model (6), hence being the same objective function. Let $({z}^{2},{v}_{k}^{2},{z}_{k}^{2},{y}_{kj}^{2})$ a feasible solution of model (7). Since constraints (7b), (7c) and (7d) are included in model (7), $({z}_{k}^{2},{y}_{kj}^{2})$ is feasible for the model that is included in the RHS of constraint (6b) and therefore greater than or equal to the minimum of that model, verifying:$${z}^{2}+{v}_{k}^{2}\ge {z}_{k}^{2}+\sum _{j=1}^{J}\frac{{\pi}_{j}}{\beta}{y}_{kj}^{2}\ge min\left\{{z}_{k}+\sum _{j=1}^{J}\frac{{\pi}_{j}}{\beta}{y}_{kj}\right\}$$

## 5. Application to the Knapsack Problem

**Definition**

**8**(Multiobjective stochastic knapsack problem).

#### 5.1. Computational Experiments

Algorithm 1 Generating random data, with $\mathcal{U}(a,b)$ the uniform distribution in $[a,b]$ | |

1: functionrandomInstance($\left|I\right|,\left|J\right|,\left|K\right|$) | |

2: $p\leftarrow \mathcal{U}(0.25,0.75)$ | ▹ proportion of objects that can fit on average |

3: $W\leftarrow \frac{1}{p\left|I\right|}$ | ▹ average weight of each object |

4: for $i\in I$ do | |

5: ${w}_{i}\leftarrow \mathcal{U}(0.5W,1.5W)$ | ▹ weight of each object |

6: for $j,k\in J\times K$ do | |

7: ${b}_{kj}^{i}\leftarrow \mathcal{U}(0,1)$ | ▹ value of each object |

8: end for | |

9: end for | |

10: end function |

- ${t}_{\mathrm{MSP}},{t}_{\mathrm{MIP}}$: solution time in seconds of models (MSP) and (MIP). With them, the following value is calculated:$${\Delta}_{\mathrm{time}}:=\frac{{t}_{\mathrm{MSP}}}{{t}_{\mathrm{MIP}}}\phantom{\rule{2.em}{0ex}}(timepenaltyfactor)$$${\Delta}_{\mathrm{time}}$, the time penalty factor, indicates the increase of computing time when solving model (MSP) rather than model (MIP).
- ${z}_{\mathrm{MSP}}^{*},{z}_{\mathrm{MIP}}^{*}$: optimal values of the models.
- ${f}_{\mathrm{MSP}}\left({x}_{\mathrm{MIP}}^{*}\right),{f}_{\mathrm{MIP}}\left({x}_{\mathrm{MSP}}^{*}\right)$: objective value of ${x}_{\mathrm{MIP}}^{*}$ in model (MSP) and vice versa.
- To grasp the difference between the MSP and the naive approach, the following will be calculated:$$\begin{array}{cc}\hfill {\Delta}_{\mathrm{avg}}& :=100\frac{{f}_{\mathrm{MIP}}\left({x}_{\mathrm{MSP}}^{*}\right)-{z}_{\mathrm{MIP}}^{*}}{{z}_{\mathrm{MIP}}^{*}}\phantom{\rule{2.em}{0ex}}(deterioratingrate)\hfill \\ \hfill {\Delta}_{\mathrm{tail}}& :=100\frac{{f}_{\mathrm{MSP}}\left({x}_{\mathrm{MIP}}^{*}\right)-{z}_{\mathrm{MSP}}^{*}}{{f}_{\mathrm{MSP}}\left({x}_{\mathrm{MIP}}^{*}\right)}\phantom{\rule{2.em}{0ex}}(improvementrate)\hfill \end{array}$$These quantities reflect what is the effect of making decision ${x}_{\mathrm{MSP}}^{*}$ instead of ${x}_{\mathrm{MIP}}^{*}$. Large values of ${\Delta}_{\mathrm{avg}}$ indicate high penalties for making decision ${x}_{\mathrm{MSP}}^{*}$ instead of ${x}_{\mathrm{MIP}}^{*}$ in average scenarios-criteria. Similarly, the larger ${\Delta}_{\mathrm{tail}}$, the higher benefit obtained from making decision ${x}_{\mathrm{MSP}}^{*}$ in tail events. They will be, respectively, called deteriorating rate and improvement rate.

- Experiment 1

- $\left|I\right|\in \{50,100,200\}$
- $\left|J\right|\in \{5,25,100\}$
- $\left|K\right|\in \{3,6,9\}$
- $r\in \{0.33,0.5,0.67\}$
- $\beta \in \{0.05,0.1,0.5\}$

- Experiment 2

#### 5.2. Results

- Experiment 1

- Experiment 2

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

MCDM | Multicriteria decision making |

VaR | Value-at-risk |

CVaR | Conditional value-at-risk |

MSP | Multiobjective stochastic programming |

OWA | Ordered weighted averaging |

MOP | Multiple objective problem |

## Appendix A

Criteria | ||||||||
---|---|---|---|---|---|---|---|---|

${\mathit{w}}_{\mathbf{1}}=\mathbf{0.20}$ | ${\mathit{w}}_{\mathbf{2}}=\mathbf{0.10}$ | ${\mathit{w}}_{\mathbf{3}}=\mathbf{0.20}$ | ${\mathit{w}}_{\mathbf{4}}=\mathbf{0.25}$ | ${\mathit{w}}_{\mathbf{5}}=\mathbf{0.15}$ | ${\mathit{w}}_{\mathbf{6}}=\mathbf{0.10}$ | |||

${\mathit{k}}_{\mathbf{1}}$ | ${\mathit{k}}_{\mathbf{2}}$ | ${\mathit{k}}_{\mathbf{3}}$ | ${\mathit{k}}_{\mathbf{4}}$ | ${\mathit{k}}_{\mathbf{5}}$ | ${\mathit{k}}_{\mathbf{6}}$ | |||

scenarios | ${\pi}_{1}=0.15$ | ${j}_{1}$ | 0.40 | 0.58 | 0.39 | 0.45 | 0.54 | 0.18 |

${\pi}_{2}=0.20$ | ${j}_{2}$ | 0.68 | 0.74 | 0.70 | 0.15 | 0.54 | 0.72 | |

${\pi}_{3}=0.30$ | ${j}_{3}$ | 0.93 | 0.52 | 0.23 | 0.82 | 0.21 | 0.03 | |

${\pi}_{4}=0.25$ | ${j}_{4}$ | 0.37 | 0.85 | 0.07 | 0.42 | 0.52 | 0.22 | |

${\pi}_{5}=0.10$ | ${j}_{5}$ | 0.92 | 0.13 | 0.71 | 0.39 | 0.90 | 0.87 | |

$\beta $-average, $\beta =0.30$ | 0.930 | 0.832 | 0.703 | 0.820 | 0.660 | 0.770 | ||

r-OWA, $r=0.17$ | 0.930 |

Criteria | ||||||||
---|---|---|---|---|---|---|---|---|

${\mathit{w}}_{\mathbf{1}}=\mathbf{0.20}$ | ${\mathit{w}}_{\mathbf{2}}=\mathbf{0.10}$ | ${\mathit{w}}_{\mathbf{3}}=\mathbf{0.20}$ | ${\mathit{w}}_{\mathbf{4}}=\mathbf{0.25}$ | ${\mathit{w}}_{\mathbf{5}}=\mathbf{0.15}$ | ${\mathit{w}}_{\mathbf{6}}=\mathbf{0.10}$ | |||

${\mathit{k}}_{\mathbf{1}}$ | ${\mathit{k}}_{\mathbf{2}}$ | ${\mathit{k}}_{\mathbf{3}}$ | ${\mathit{k}}_{\mathbf{4}}$ | ${\mathit{k}}_{\mathbf{5}}$ | ${\mathit{k}}_{\mathbf{6}}$ | |||

scenarios | ${\pi}_{1}=0.15$ | ${j}_{1}$ | 0.80 | 0.90 | 0.61 | 0.28 | 0.94 | 0.09 |

${\pi}_{2}=0.20$ | ${j}_{2}$ | 0.29 | 0.48 | 0.26 | 0.23 | 0.21 | 0.07 | |

${\pi}_{3}=0.30$ | ${j}_{3}$ | 0.73 | 0.65 | 0.32 | 0.56 | 0.95 | 0.65 | |

${\pi}_{4}=0.25$ | ${j}_{4}$ | 0.58 | 0.39 | 0.21 | 0.66 | 0.70 | 0.93 | |

${\pi}_{5}=0.10$ | ${j}_{5}$ | 0.73 | 0.22 | 0.33 | 0.31 | 0.32 | 0.38 | |

$\beta $-average, $\beta =0.30$ | 0.765 | 0.775 | 0.468 | 0.643 | 0.950 | 0.883 | ||

r-OWA, $r=0.17$ | 0.943 |

Criteria | ||||||||
---|---|---|---|---|---|---|---|---|

${\mathit{w}}_{\mathbf{1}}=\mathbf{0.20}$ | ${\mathit{w}}_{\mathbf{2}}=\mathbf{0.10}$ | ${\mathit{w}}_{\mathbf{3}}=\mathbf{0.20}$ | ${\mathit{w}}_{\mathbf{4}}=\mathbf{0.25}$ | ${\mathit{w}}_{\mathbf{5}}=\mathbf{0.15}$ | ${\mathit{w}}_{\mathbf{6}}=\mathbf{0.10}$ | |||

${\mathit{k}}_{\mathbf{1}}$ | ${\mathit{k}}_{\mathbf{2}}$ | ${\mathit{k}}_{\mathbf{3}}$ | ${\mathit{k}}_{\mathbf{4}}$ | ${\mathit{k}}_{\mathbf{5}}$ | ${\mathit{k}}_{\mathbf{6}}$ | |||

scenarios | ${\pi}_{1}=0.15$ | ${j}_{1}$ | 0.30 | 0.52 | 0.12 | 0.68 | 0.46 | 0.73 |

${\pi}_{2}=0.20$ | ${j}_{2}$ | 1.00 | 0.57 | 0.46 | 0.82 | 0.90 | 0.72 | |

${\pi}_{3}=0.30$ | ${j}_{3}$ | 0.18 | 0.76 | 0.30 | 0.34 | 0.54 | 0.99 | |

${\pi}_{4}=0.25$ | ${j}_{4}$ | 0.53 | 0.21 | 0.13 | 0.12 | 0.66 | 0.86 | |

${\pi}_{5}=0.10$ | ${j}_{5}$ | 0.98 | 0.46 | 0.50 | 0.29 | 0.27 | 0.40 | |

$\beta $-average, $\beta =0.30$ | 0.993 | 0.760 | 0.473 | 0.773 | 0.820 | 0.990 | ||

r-OWA, $r=0.17$ | 0.993 |

**Table A4.**All instances of first experiment. The three instances with 200 objects, 100 scenarios, 6 criteria and $\beta =0.05$ did not reach the optimal solution in 2 h. The integrality gaps of the solution shown are $0.31\%,0.24\%$ and $0.19\%$ for $r=0.33,0.5$ and $0.67$ respectively.

$\mathit{\beta}\to $ | 0.05 | 0.1 | 0.5 | |||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{r}\to $ | 0.33 | 0.5 | 0.67 | 0.33 | 0.5 | 0.67 | 0.33 | 0.5 | 0.67 | |||||||||||||||||||||||||||||

|I| | |J| | |K| | ${\mathit{t}}_{\mathbf{MSP}}$ | ${\mathit{t}}_{\mathbf{MIP}}$ | ${\Delta}_{\mathbf{avg}}$ | ${\Delta}_{\mathbf{tail}}$ | ${\mathit{t}}_{\mathbf{MSP}}$ | ${\mathit{t}}_{\mathbf{MIP}}$ | ${\Delta}_{\mathbf{avg}}$ | ${\Delta}_{\mathbf{tail}}$ | ${\mathit{t}}_{\mathbf{MSP}}$ | ${\mathit{t}}_{\mathbf{MIP}}$ | ${\Delta}_{\mathbf{avg}}$ | ${\Delta}_{\mathbf{tail}}$ | ${\mathit{t}}_{\mathbf{MSP}}$ | ${\mathit{t}}_{\mathbf{MIP}}$ | ${\Delta}_{\mathbf{avg}}$ | ${\Delta}_{\mathbf{tail}}$ | ${\mathit{t}}_{\mathbf{MSP}}$ | ${\mathit{t}}_{\mathbf{MIP}}$ | ${\Delta}_{\mathbf{avg}}$ | ${\Delta}_{\mathbf{tail}}$ | ${\mathit{t}}_{\mathbf{MSP}}$ | ${\mathit{t}}_{\mathbf{MIP}}$ | ${\Delta}_{\mathbf{avg}}$ | ${\Delta}_{\mathbf{tail}}$ | ${\mathit{t}}_{\mathbf{MSP}}$ | ${\mathit{t}}_{\mathbf{MIP}}$ | ${\Delta}_{\mathbf{avg}}$ | ${\Delta}_{\mathbf{tail}}$ | ${\mathit{t}}_{\mathbf{MSP}}$ | ${\mathit{t}}_{\mathbf{MIP}}$ | ${\Delta}_{\mathbf{avg}}$ | ${\Delta}_{\mathbf{tail}}$ | ${\mathit{t}}_{\mathbf{MSP}}$ | ${\mathit{t}}_{\mathbf{MIP}}$ | ${\Delta}_{\mathbf{avg}}$ | ${\Delta}_{\mathbf{tail}}$ |

50 | 5 | 3 | 0.12 | 0.13 | 3.75 | 8.01 | 0.15 | 0.12 | 3.75 | 8.01 | 0.18 | 0.14 | 3.51 | 4.54 | 0.14 | 0.14 | 3.75 | 6.59 | 0.12 | 0.12 | 3.75 | 6.59 | 0.20 | 0.17 | 3.51 | 3.79 | 0.12 | 0.12 | 3.75 | 5.84 | 0.12 | 0.12 | 3.75 | 5.84 | 0.13 | 0.12 | 3.16 | 3.64 |

6 | 0.22 | 0.11 | 3.79 | 5.07 | 0.25 | 0.11 | 3.79 | 5.07 | 0.18 | 0.11 | 1.03 | 3.01 | 0.30 | 0.13 | 3.79 | 4.01 | 0.25 | 0.13 | 3.79 | 4.01 | 0.23 | 0.15 | 1.48 | 2.10 | 0.23 | 0.16 | 3.77 | 3.58 | 0.23 | 0.14 | 3.77 | 3.58 | 0.18 | 0.17 | 1.48 | 1.47 | ||

9 | 0.28 | 0.14 | 1.76 | 6.67 | 0.36 | 0.14 | 1.76 | 6.67 | 0.20 | 0.15 | 1.58 | 3.26 | 0.22 | 0.14 | 2.02 | 5.66 | 0.21 | 0.14 | 2.02 | 5.66 | 0.22 | 0.14 | 1.24 | 2.10 | 0.20 | 0.15 | 2.02 | 4.39 | 0.22 | 0.13 | 2.02 | 4.39 | 0.25 | 0.15 | 1.20 | 1.37 | ||

25 | 3 | 0.76 | 0.18 | 2.47 | 5.37 | 0.66 | 0.17 | 2.47 | 2.85 | 0.26 | 0.17 | 2.00 | 1.55 | 0.64 | 0.18 | 2.47 | 5.08 | 0.62 | 0.16 | 2.47 | 2.51 | 0.25 | 0.15 | 1.00 | 0.97 | 0.60 | 0.17 | 2.47 | 4.93 | 0.51 | 0.16 | 1.79 | 2.52 | 0.26 | 0.14 | 1.00 | 0.69 | |

6 | 1.20 | 0.16 | 1.87 | 5.77 | 0.91 | 0.29 | 1.96 | 3.70 | 0.49 | 0.18 | 0.79 | 1.81 | 1.15 | 0.18 | 1.87 | 4.93 | 0.74 | 0.18 | 1.14 | 3.51 | 0.41 | 0.18 | 0.70 | 1.55 | 0.93 | 0.16 | 1.17 | 4.33 | 0.78 | 0.18 | 1.17 | 3.45 | 0.38 | 0.16 | 0.71 | 1.22 | ||

9 | 0.69 | 0.16 | 0.61 | 4.21 | 0.78 | 0.16 | 0.43 | 2.78 | 0.57 | 0.16 | 0.67 | 0.92 | 0.52 | 0.15 | 0.61 | 3.90 | 0.96 | 0.16 | 0.44 | 2.14 | 0.61 | 0.16 | 0.67 | 0.65 | 0.69 | 0.15 | 0.61 | 3.25 | 1.02 | 0.18 | 0.39 | 1.75 | 0.47 | 0.18 | 0.51 | 0.68 | ||

100 | 3 | 1.15 | 0.15 | 0.07 | 2.43 | 0.78 | 0.15 | 0.07 | 2.15 | 0.44 | 0.14 | 0.07 | 0.16 | 1.07 | 0.14 | 0.07 | 2.02 | 0.83 | 0.19 | 0.07 | 1.74 | 0.34 | 0.14 | 0.07 | 0.16 | 1.14 | 0.15 | 0.07 | 1.81 | 0.85 | 0.16 | 0.07 | 1.53 | 0.36 | 0.14 | 0.07 | 0.14 | |

6 | 2.51 | 0.20 | 0.77 | 2.24 | 4.45 | 0.22 | 0.78 | 1.19 | 3.52 | 0.20 | 0.23 | 0.47 | 2.62 | 0.25 | 0.77 | 1.99 | 5.09 | 0.18 | 0.31 | 1.10 | 5.45 | 0.20 | 0.19 | 0.29 | 2.70 | 0.22 | 0.77 | 1.38 | 3.31 | 0.19 | 0.47 | 0.63 | 5.59 | 0.19 | 0.23 | 0.27 | ||

9 | 4.06 | 0.18 | 0.03 | 0.41 | 1.47 | 0.17 | 0.03 | 0.30 | 1.07 | 0.16 | 0.00 | 0.00 | 2.75 | 0.17 | 0.03 | 0.29 | 1.12 | 0.16 | 0.03 | 0.30 | 1.11 | 0.16 | 0.00 | 0.00 | 2.06 | 0.19 | 0.03 | 0.32 | 1.10 | 0.16 | 0.03 | 0.18 | 1.16 | 0.17 | 0.00 | 0.00 | ||

100 | 5 | 3 | 1.24 | 0.26 | 3.81 | 7.63 | 1.29 | 0.20 | 3.81 | 7.63 | 0.37 | 0.22 | 2.08 | 4.47 | 0.72 | 0.18 | 3.81 | 5.22 | 0.66 | 0.18 | 3.81 | 5.22 | 0.32 | 0.27 | 1.56 | 2.33 | 0.74 | 0.19 | 3.38 | 4.02 | 1.13 | 0.23 | 3.38 | 4.02 | 0.28 | 0.20 | 0.63 | 1.36 |

6 | 8.68 | 0.22 | 5.68 | 7.12 | 8.69 | 0.19 | 5.68 | 7.12 | 0.43 | 0.17 | 4.46 | 2.92 | 0.65 | 0.20 | 4.30 | 5.64 | 1.06 | 0.19 | 4.30 | 5.64 | 0.35 | 0.17 | 0.81 | 1.28 | 1.18 | 0.22 | 3.93 | 4.20 | 0.64 | 0.18 | 3.93 | 4.20 | 0.28 | 0.18 | 0.53 | 0.88 | ||

9 | 3.31 | 0.18 | 2.19 | 3.14 | 3.26 | 0.20 | 2.19 | 3.14 | 0.67 | 0.18 | 0.97 | 2.90 | 1.04 | 0.20 | 2.19 | 2.86 | 0.96 | 0.20 | 2.19 | 2.86 | 0.23 | 0.14 | 0.64 | 1.88 | 1.02 | 0.17 | 0.92 | 2.31 | 0.90 | 0.17 | 0.92 | 2.31 | 0.24 | 0.18 | 0.41 | 1.18 | ||

25 | 3 | 10.65 | 0.17 | 2.96 | 6.52 | 3.39 | 0.18 | 2.07 | 4.00 | 0.29 | 0.15 | 0.48 | 1.73 | 7.09 | 0.18 | 2.96 | 5.46 | 1.83 | 0.19 | 2.96 | 3.67 | 0.30 | 0.14 | 0.48 | 0.95 | 3.46 | 0.19 | 2.19 | 4.98 | 1.30 | 0.16 | 2.96 | 3.50 | 0.34 | 0.16 | 0.41 | 0.59 | |

6 | 32.12 | 0.20 | 2.78 | 4.47 | 9.18 | 0.19 | 0.78 | 3.53 | 0.44 | 0.18 | 0.52 | 1.31 | 26.53 | 0.18 | 2.59 | 3.06 | 3.64 | 0.22 | 0.61 | 2.47 | 0.32 | 0.15 | 0.26 | 0.79 | 12.77 | 0.17 | 0.60 | 2.62 | 0.90 | 0.17 | 0.50 | 2.00 | 0.41 | 0.17 | 0.26 | 0.65 | ||

9 | 8.58 | 0.18 | 0.72 | 5.52 | 1.90 | 0.17 | 0.97 | 3.49 | 0.42 | 0.18 | 0.24 | 0.82 | 6.32 | 0.19 | 0.72 | 4.75 | 1.24 | 0.16 | 0.97 | 3.06 | 0.51 | 0.20 | 0.24 | 0.44 | 1.60 | 0.19 | 1.12 | 3.83 | 0.88 | 0.19 | 1.12 | 2.53 | 0.59 | 0.17 | 0.50 | 0.23 | ||

100 | 3 | 51.23 | 0.22 | 2.21 | 1.12 | 1.67 | 0.21 | 0.27 | 1.36 | 0.82 | 0.18 | 0.09 | 0.25 | 22.70 | 0.21 | 2.21 | 1.16 | 1.22 | 0.19 | 0.34 | 1.05 | 0.81 | 0.18 | 0.05 | 0.17 | 18.75 | 0.16 | 2.21 | 1.17 | 0.84 | 0.22 | 0.34 | 0.92 | 0.75 | 0.18 | 0.05 | 0.13 | |

6 | 48.25 | 0.18 | 0.76 | 2.56 | 31.87 | 0.17 | 0.62 | 2.05 | 62.14 | 0.15 | 0.42 | 0.70 | 24.73 | 0.18 | 0.71 | 2.48 | 27.08 | 0.18 | 0.62 | 1.81 | 42.18 | 0.19 | 0.28 | 0.55 | 20.26 | 0.17 | 0.60 | 2.10 | 22.09 | 0.20 | 0.75 | 1.48 | 7.79 | 0.20 | 0.17 | 0.50 | ||

9 | 2.16 | 0.19 | 0.37 | 1.48 | 3.34 | 0.18 | 0.29 | 0.77 | 1.84 | 0.17 | 0.18 | 0.41 | 1.80 | 0.17 | 0.34 | 1.25 | 2.87 | 0.20 | 0.28 | 0.69 | 2.22 | 0.19 | 0.26 | 0.22 | 1.67 | 0.18 | 0.34 | 1.14 | 2.77 | 0.19 | 0.20 | 0.63 | 3.09 | 0.16 | 0.08 | 0.13 | ||

200 | 5 | 3 | 146.24 | 0.23 | 1.61 | 3.81 | 140.12 | 0.20 | 1.61 | 3.81 | 7.71 | 0.23 | 1.30 | 1.61 | 151.22 | 0.21 | 1.61 | 3.30 | 135.09 | 0.24 | 1.61 | 3.30 | 4.60 | 0.21 | 1.30 | 1.58 | 83.44 | 0.22 | 1.10 | 3.06 | 89.20 | 0.21 | 1.10 | 3.06 | 4.21 | 0.22 | 1.30 | 1.55 |

6 | 88.70 | 0.19 | 1.08 | 2.50 | 89.69 | 0.19 | 1.08 | 2.50 | 5.14 | 0.17 | 0.72 | 0.83 | 96.44 | 0.19 | 1.08 | 1.91 | 91.66 | 0.18 | 1.08 | 1.91 | 2.93 | 0.18 | 0.91 | 0.58 | 39.26 | 0.18 | 0.94 | 1.68 | 32.92 | 0.18 | 0.94 | 1.68 | 0.70 | 0.17 | 0.58 | 0.44 | ||

9 | 468.37 | 0.15 | 3.73 | 9.18 | 484.89 | 0.14 | 3.73 | 9.18 | 29.46 | 0.16 | 1.74 | 4.99 | 304.04 | 0.16 | 3.69 | 6.24 | 305.90 | 0.16 | 3.69 | 6.24 | 2.71 | 0.16 | 1.74 | 3.54 | 110.03 | 0.15 | 3.38 | 4.92 | 107.34 | 0.14 | 3.38 | 4.92 | 0.91 | 0.17 | 1.20 | 2.28 | ||

25 | 3 | 5629.58 | 0.33 | 2.75 | 7.84 | 4765.42 | 0.24 | 2.24 | 5.33 | 4.86 | 0.24 | 0.81 | 1.40 | 5430.90 | 0.25 | 2.75 | 6.73 | 3394.56 | 0.24 | 2.75 | 5.05 | 5.32 | 0.28 | 0.81 | 1.22 | 6896.05 | 0.25 | 2.75 | 6.15 | 2546.43 | 0.21 | 2.75 | 4.91 | 5.66 | 0.34 | 0.81 | 1.13 | |

6 | 2886.13 | 0.19 | 1.67 | 4.36 | 146.48 | 0.17 | 1.93 | 2.77 | 0.57 | 0.17 | 0.19 | 0.79 | 1651.91 | 0.21 | 1.67 | 4.20 | 15.06 | 0.22 | 1.93 | 2.29 | 0.71 | 0.21 | 0.19 | 0.81 | 93.66 | 0.19 | 1.46 | 3.64 | 19.36 | 0.19 | 1.93 | 2.02 | 0.55 | 0.18 | 0.12 | 0.40 | ||

9 | 1235.12 | 0.32 | 2.05 | 2.59 | 342.32 | 0.22 | 0.96 | 1.22 | 1.99 | 0.21 | 0.22 | 0.26 | 404.70 | 0.29 | 1.90 | 2.12 | 28.09 | 0.21 | 0.88 | 0.76 | 0.82 | 0.20 | 0.13 | 0.17 | 99.73 | 0.21 | 1.99 | 1.58 | 2.23 | 0.22 | 0.39 | 0.58 | 0.87 | 0.20 | 0.06 | 0.08 | ||

100 | 3 | 703.05 | 0.23 | 2.11 | 4.15 | 373.65 | 0.22 | 2.03 | 2.70 | 1.42 | 0.22 | 0.47 | 0.63 | 731.09 | 0.22 | 2.11 | 2.92 | 157.29 | 0.20 | 2.03 | 1.96 | 1.11 | 0.22 | 0.54 | 0.47 | 596.88 | 0.27 | 2.06 | 2.29 | 349.78 | 0.30 | 2.13 | 1.58 | 3.22 | 0.29 | 0.53 | 0.44 | |

6 | 7222.95 | 0.22 | 0.60 | 3.43 | 1814.25 | 0.18 | 0.48 | 2.08 | 22.11 | 0.21 | 0.13 | 0.44 | 7217.64 | 0.14 | 0.47 | 2.57 | 916.42 | 0.21 | 0.48 | 1.75 | 7.04 | 0.24 | 0.28 | 0.27 | 7216.94 | 0.15 | 0.47 | 2.11 | 656.48 | 0.20 | 0.37 | 1.41 | 7.89 | 0.22 | 0.24 | 0.20 | ||

9 | 3321.23 | 0.34 | 0.07 | 0.28 | 16.40 | 0.20 | 0.02 | 0.18 | 2.33 | 0.17 | 0.08 | 0.08 | 198.14 | 0.19 | 0.07 | 0.32 | 14.84 | 0.21 | 0.02 | 0.13 | 2.31 | 0.21 | 0.08 | 0.05 | 47.16 | 0.23 | 0.08 | 0.33 | 9.77 | 0.21 | 0.01 | 0.12 | 2.63 | 0.20 | 0.06 | 0.04 |

**Table A5.**All instances of second experiment. $\left|I\right|=100,\left|J\right|=25,\left|K\right|=6,r=0.5,\beta =0.1$.

${\mathit{t}}_{\mathbf{MSP}}$ | ${\mathit{t}}_{\mathbf{MIP}}$ | ${\mathbf{\Delta}}_{\mathbf{avg}}$ | ${\mathbf{\Delta}}_{\mathbf{tail}}$ | ${\mathit{t}}_{\mathbf{MSP}}$ | ${\mathit{t}}_{\mathbf{MIP}}$ | ${\mathbf{\Delta}}_{\mathbf{avg}}$ | ${\mathbf{\Delta}}_{\mathbf{tail}}$ |
---|---|---|---|---|---|---|---|

31.15 | 0.23 | 1.53 | 3.20 | 2.15 | 0.16 | 2.24 | 2.09 |

1.92 | 0.21 | 1.66 | 6.17 | 20.09 | 0.16 | 1.80 | 6.14 |

8.75 | 0.24 | 0.52 | 3.07 | 7.18 | 0.16 | 2.13 | 1.93 |

28.06 | 0.23 | 5.08 | 2.86 | 1.02 | 0.16 | 3.03 | 3.61 |

1.36 | 0.30 | 1.00 | 1.80 | 3.58 | 0.24 | 1.81 | 6.12 |

3.67 | 0.20 | 2.27 | 2.50 | 3.64 | 0.19 | 1.19 | 3.07 |

2.00 | 0.20 | 2.51 | 2.03 | 128.69 | 0.23 | 3.27 | 2.98 |

192.11 | 0.16 | 2.61 | 8.23 | 0.89 | 0.18 | 1.45 | 0.93 |

0.94 | 0.20 | 0.43 | 2.23 | 1.62 | 0.23 | 1.85 | 3.56 |

0.80 | 0.18 | 1.64 | 2.55 | 4.19 | 0.22 | 2.10 | 1.97 |

16.40 | 0.19 | 2.23 | 2.45 | 2.16 | 0.19 | 0.16 | 1.46 |

1.21 | 0.18 | 2.82 | 1.50 | 1.46 | 0.24 | 2.48 | 2.00 |

1.79 | 0.20 | 0.72 | 2.77 | 0.69 | 0.20 | 1.79 | 2.54 |

21.78 | 0.21 | 4.50 | 4.61 | 20.73 | 0.20 | 2.26 | 3.50 |

1.35 | 0.19 | 0.69 | 0.86 | 1.86 | 0.24 | 1.77 | 2.63 |

31.11 | 0.19 | 0.98 | 3.21 | 14.92 | 0.17 | 1.99 | 8.57 |

8.44 | 0.19 | 1.82 | 3.81 | 0.78 | 0.20 | 0.85 | 1.92 |

1.75 | 0.21 | 0.88 | 0.92 | 10.48 | 0.23 | 2.50 | 2.29 |

1.94 | 0.21 | 2.18 | 2.65 | 1.63 | 0.24 | 2.08 | 2.29 |

0.98 | 0.20 | 0.87 | 3.27 | 10.78 | 0.18 | 0.34 | 1.80 |

27.72 | 0.22 | 2.03 | 5.20 | 38.80 | 0.20 | 1.96 | 4.69 |

14.72 | 0.15 | 3.34 | 0.99 | 19.74 | 0.24 | 0.65 | 2.20 |

0.67 | 0.24 | 0.81 | 2.69 | 1.37 | 0.30 | 2.82 | 2.92 |

3.54 | 0.20 | 2.64 | 2.75 | 6.28 | 0.19 | 2.02 | 2.08 |

6.37 | 0.21 | 2.79 | 6.35 | 22.27 | 0.34 | 1.91 | 3.13 |

1.86 | 0.23 | 0.93 | 2.09 | 1.69 | 0.20 | 2.21 | 2.42 |

1.54 | 0.20 | 2.00 | 3.45 | 27.77 | 0.19 | 0.76 | 3.28 |

40.16 | 0.17 | 2.06 | 3.44 | 2.00 | 0.21 | 2.57 | 1.93 |

7.23 | 0.21 | 3.17 | 3.17 | 2.61 | 0.18 | 2.14 | 3.33 |

5.77 | 0.17 | 2.84 | 1.98 | 40.93 | 0.18 | 1.53 | 4.61 |

2.10 | 0.19 | 1.39 | 3.00 | 18.84 | 0.16 | 0.89 | 4.37 |

404.70 | 0.19 | 1.50 | 2.82 | 11.26 | 0.16 | 3.98 | 4.76 |

24.26 | 0.18 | 4.81 | 3.42 | 14.41 | 0.18 | 1.82 | 5.87 |

0.76 | 0.20 | 1.28 | 3.88 | 12.14 | 0.16 | 2.75 | 2.75 |

0.64 | 0.20 | 0.87 | 1.39 | 12.58 | 0.17 | 1.42 | 3.46 |

0.97 | 0.23 | 1.77 | 2.19 | 0.84 | 0.18 | 0.41 | 2.15 |

0.53 | 0.18 | 1.95 | 2.04 | 5.20 | 0.19 | 3.80 | 2.02 |

7.24 | 0.22 | 2.21 | 1.68 | 28.16 | 0.15 | 4.68 | 3.56 |

0.87 | 0.25 | 0.71 | 1.42 | 39.10 | 0.16 | 3.47 | 3.59 |

8.51 | 0.20 | 2.48 | 4.06 | 19.22 | 0.16 | 3.78 | 3.13 |

13.06 | 0.20 | 4.44 | 2.80 | 0.56 | 0.20 | 0.63 | 3.22 |

59.78 | 0.20 | 5.67 | 4.91 | 0.68 | 0.17 | 1.92 | 2.44 |

67.50 | 0.19 | 2.96 | 3.02 | 0.70 | 0.17 | 1.86 | 1.40 |

3.80 | 0.17 | 0.79 | 1.20 | 15.20 | 0.17 | 0.78 | 2.66 |

3.25 | 0.20 | 2.24 | 1.57 | 0.88 | 0.17 | 2.31 | 2.13 |

5.23 | 0.16 | 1.14 | 4.91 | 0.59 | 0.15 | 1.78 | 1.68 |

0.71 | 0.17 | 0.89 | 3.01 | 1.08 | 0.20 | 2.21 | 3.14 |

4.19 | 0.17 | 3.09 | 2.32 | 1.14 | 0.17 | 0.76 | 2.48 |

3.53 | 0.18 | 1.37 | 6.33 | 1.58 | 0.19 | 1.45 | 3.14 |

19.99 | 0.14 | 3.48 | 5.05 | 13.48 | 0.18 | 1.71 | 5.41 |

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**Figure 1.**The results from illustrative example. (

**a**) Optimal alternative for some values of r and $\beta $, where each of the four alternatives is colour-coded. (

**b**) Optimal values of function ${h}_{r}^{\beta}(x)$ for some values of r and $\beta $.

**Figure 2.**Values ${\Delta}_{\mathrm{avg}}$ and ${\Delta}_{\mathrm{tail}}$ for each of the 243 instances, grouped by values of $(r,\beta )$.

**Figure 3.**Single instance with 100 scenarios and three criteria. For each k, sorted values of ${f}_{k}^{j}(x)$, where $x={x}_{\mathrm{MIP}}^{*}$ in blue squares and $x={x}_{\mathrm{MSP}}^{*}$ in orange circles.

Scenario | $\mathit{\beta}$ | |||||||
---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 0.2 | 0.3 | 0.5 | |

${\pi}_{j}$ | 0.2 | 0.1 | 0.3 | 0.25 | 0.15 | 10 | 9 | 7 |

${f}_{k}^{j}(x)$ | 10 | 7 | 4 | 3 | 2 |

Criterion | r | |||||||
---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 0.2 | 0.3 | 0.5 | |

${w}_{k}$ | 0.2 | 0.1 | 0.3 | 0.25 | 0.15 | 10 | 9 | 7 |

${g}_{k}(x)$ | 10 | 7 | 4 | 3 | 2 |

**Table 3.**Values of two alternatives for each scenario j and criterion k, together with their $\beta $-averages ($\beta ={\textstyle \frac{1}{2}}$) and r-OWAs ($r={\textstyle \frac{2}{3}}$).

(a) Alternative 1 | (b) Alternative 2 | ||||||
---|---|---|---|---|---|---|---|

${k}_{1}$ | ${k}_{2}$ | ${k}_{3}$ | ${k}_{1}$ | ${k}_{2}$ | ${k}_{3}$ | ||

${j}_{1}$ | 0.80 | 0.40 | 0.30 | ${j}_{1}$ | 0.70 | 0.45 | 0.65 |

${j}_{2}$ | 0.60 | 0.20 | 0.65 | ${j}_{2}$ | 0.80 | 0.30 | 0.50 |

$\beta $-average | 0.80 | 0.40 | 0.65 | $\beta $-average | 0.80 | 0.45 | 0.65 |

r-OWA | 0.725 | r-OWA | 0.725 |

Criteria | ||||||||
---|---|---|---|---|---|---|---|---|

${\mathit{w}}_{\mathbf{1}}=\mathbf{0.20}$ | ${\mathit{w}}_{\mathbf{2}}=\mathbf{0.10}$ | ${\mathit{w}}_{\mathbf{3}}=\mathbf{0.20}$ | ${\mathit{w}}_{\mathbf{4}}=\mathbf{0.25}$ | ${\mathit{w}}_{\mathbf{5}}=\mathbf{0.15}$ | ${\mathit{w}}_{\mathbf{6}}=\mathbf{0.10}$ | |||

${\mathit{k}}_{\mathbf{1}}$ | ${\mathit{k}}_{\mathbf{2}}$ | ${\mathit{k}}_{\mathbf{3}}$ | ${\mathit{k}}_{\mathbf{4}}$ | ${\mathit{k}}_{\mathbf{5}}$ | ${\mathit{k}}_{\mathbf{6}}$ | |||

scenarios | ${\pi}_{1}=0.15$ | ${j}_{1}$ | 0.51 | 0.27 | 0.39 | 0.45 | 0.75 | 0.76 |

${\pi}_{2}=0.20$ | ${j}_{2}$ | 0.58 | 0.65 | 0.47 | 0.26 | 0.90 | 0.24 | |

${\pi}_{3}=0.30$ | ${j}_{3}$ | 0.48 | 0.44 | 0.90 | 0.50 | 0.93 | 0.65 | |

${\pi}_{4}=0.25$ | ${j}_{4}$ | 0.76 | 0.18 | 0.01 | 0.90 | 0.56 | 0.02 | |

${\pi}_{5}=0.10$ | ${j}_{5}$ | 0.86 | 0.36 | 0.21 | 0.28 | 0.63 | 0.72 |

Criteria | ||||||||
---|---|---|---|---|---|---|---|---|

${\mathit{w}}_{\mathbf{1}}=\mathbf{0.20}$ | ${\mathit{w}}_{\mathbf{2}}=\mathbf{0.10}$ | ${\mathit{w}}_{\mathbf{3}}=\mathbf{0.20}$ | ${\mathit{w}}_{\mathbf{4}}=\mathbf{0.25}$ | ${\mathit{w}}_{\mathbf{5}}=\mathbf{0.15}$ | ${\mathit{w}}_{\mathbf{6}}=\mathbf{0.10}$ | |||

${\mathit{k}}_{\mathbf{1}}$ | ${\mathit{k}}_{\mathbf{2}}$ | ${\mathit{k}}_{\mathbf{3}}$ | ${\mathit{k}}_{\mathbf{4}}$ | ${\mathit{k}}_{\mathbf{5}}$ | ${\mathit{k}}_{\mathbf{6}}$ | |||

scenarios | ${\pi}_{1}=0.15$ | ${j}_{1}$ | 0.51 | 0.27 | 0.39 | 0.45 | 0.75 | 0.76 |

${\pi}_{2}=0.20$ | ${j}_{2}$ | 0.58 | 0.65 | 0.47 | 0.26 | 0.90 | 0.24 | |

${\pi}_{3}=0.30$ | ${j}_{3}$ | 0.48 | 0.44 | 0.90 | 0.50 | 0.93 | 0.65 | |

${\pi}_{4}=0.25$ | ${j}_{4}$ | 0.76 | 0.18 | 0.01 | 0.90 | 0.56 | 0.02 | |

${\pi}_{5}=0.10$ | ${j}_{5}$ | 0.86 | 0.36 | 0.21 | 0.28 | 0.63 | 0.72 | |

$\beta $-average, $\beta =0.30$ | 0.793 | 0.580 | 0.900 | 0.833 | 0.930 | 0.728 | ||

r-OWA, $r=0.17$ | 0.927 |

$\mathit{\beta}$-Averages | r-OWA | ||||||
---|---|---|---|---|---|---|---|

${\mathit{g}}_{\mathbf{1}}^{\mathit{\beta}}(\mathit{x})$ | ${\mathit{g}}_{\mathbf{2}}^{\mathit{\beta}}(\mathit{x})$ | ${\mathit{g}}_{\mathbf{3}}^{\mathit{\beta}}(\mathit{x})$ | ${\mathit{g}}_{\mathbf{4}}^{\mathit{\beta}}(\mathit{x})$ | ${\mathit{g}}_{\mathbf{5}}^{\mathit{\beta}}(\mathit{x})$ | ${\mathit{g}}_{\mathbf{6}}^{\mathit{\beta}}(\mathit{x})$ | ${\mathit{h}}_{\mathit{r}}^{\beta}(\mathit{x})$ | |

Alternative 1 | 0.793 | 0.580 | 0.900 | 0.833 | 0.930 | 0.728 | 0.927 |

Alternative 2 | 0.930 | 0.832 | 0.703 | 0.820 | 0.660 | 0.770 | 0.930 |

Alternative 3 | 0.765 | 0.775 | 0.468 | 0.643 | 0.950 | 0.883 | 0.943 |

Alternative 4 | 0.993 | 0.760 | 0.473 | 0.773 | 0.820 | 0.990 | 0.993 |

**Table 7.**Objective values by scenario-criterion of solutions obtained with the multiobjective stochastic programming problem (MSP) and MIP models, for the first instance of the first experiment.

(a) MSP Solution | (b) MIP Solution | ||||||
---|---|---|---|---|---|---|---|

${k}_{1}$ | ${k}_{2}$ | ${k}_{3}$ | ${k}_{1}$ | ${k}_{2}$ | ${k}_{3}$ | ||

${j}_{1}$ | 11.97 | 11.26 | 11.00 | ${j}_{1}$ | 9.65 | 10.75 | 9.71 |

${j}_{2}$ | 11.96 | 9.92 | 13.71 | ${j}_{2}$ | 11.19 | 10.17 | 14.90 |

${j}_{3}$ | 13.48 | 13.51 | 10.92 | ${j}_{3}$ | 14.05 | 13.55 | 9.40 |

${j}_{4}$ | 13.62 | 13.13 | 13.71 | ${j}_{4}$ | 14.12 | 13.00 | 13.35 |

${j}_{5}$ | 12.94 | 11.35 | 13.47 | ${j}_{5}$ | 13.64 | 10.33 | 11.42 |

|I| | |J| | |K| | r | $\mathit{\beta}$ | |
---|---|---|---|---|---|

${t}_{\mathrm{MSP}}$ | 0.34 | 0.09 | −0.11 | −0.05 | −0.19 |

${t}_{\mathrm{MIP}}$ | 0.51 | 0.18 | −0.14 | −0.03 | −0.07 |

${\Delta}_{\mathrm{time}}$ | 0.31 | 0.11 | −0.08 | −0.02 | −0.18 |

${\Delta}_{\mathrm{avg}}$ | −0.05 | −0.57 | −0.28 | −0.09 | −0.36 |

${\Delta}_{\mathrm{tail}}$ | −0.07 | −0.56 | −0.18 | −0.21 | −0.50 |

$\mathit{\beta}$ | ${\mathit{t}}_{\mathbf{MSP}}$ | ${\mathbf{\Delta}}_{\mathbf{time}}$ | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

Min | Mean | Median | Max | Std | Min | Mean | Median | Max | Std | |

0.05 | 0.12 | 659.49 | 6.32 | 7222.95 | 1787.07 | 0.94 | 3188.96 | 32.77 | 50473.04 | 9472.55 |

0.10 | 0.12 | 212.47 | 2.23 | 4765.42 | 728.49 | 0.98 | 1002.35 | 11.09 | 20192.48 | 3245.85 |

0.50 | 0.13 | 3.49 | 0.67 | 62.14 | 9.05 | 1.06 | 19.14 | 3.75 | 414.29 | 55.51 |

**Table 10.**Values of ${\Delta}_{\mathrm{avg}}$ and ${\Delta}_{\mathrm{tail}}$, grouped by r and $\beta $.

r | $\mathit{\beta}$ | ${\mathbf{\Delta}}_{\mathbf{avg}}$ | ${\mathbf{\Delta}}_{\mathbf{tail}}$ | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Min | Mean | Median | Max | Std | Min | Mean | Median | Max | Std | ||

0.33 | 0.05 | 0.03 | 1.94 | 1.87 | 5.68 | 1.42 | 0.28 | 4.37 | 4.21 | 9.18 | 2.43 |

0.10 | 0.02 | 1.70 | 1.61 | 5.68 | 1.44 | 0.18 | 3.54 | 2.85 | 9.18 | 2.42 | |

0.50 | 0.00 | 0.93 | 0.52 | 4.46 | 1.08 | 0.00 | 1.57 | 0.92 | 4.99 | 1.46 | |

0.50 | 0.05 | 0.03 | 1.87 | 1.90 | 4.30 | 1.30 | 0.29 | 3.58 | 3.30 | 6.73 | 1.89 |

0.10 | 0.02 | 1.65 | 1.14 | 4.30 | 1.37 | 0.13 | 2.87 | 2.47 | 6.59 | 1.86 | |

0.50 | 0.00 | 0.72 | 0.54 | 3.51 | 0.75 | 0.00 | 1.07 | 0.79 | 3.79 | 1.01 | |

0.67 | 0.05 | 0.03 | 1.64 | 1.17 | 3.93 | 1.24 | 0.32 | 3.04 | 3.06 | 6.15 | 1.62 |

0.10 | 0.01 | 1.50 | 1.10 | 3.93 | 1.31 | 0.12 | 2.43 | 2.02 | 5.84 | 1.58 | |

0.50 | 0.00 | 0.60 | 0.50 | 3.16 | 0.66 | 0.00 | 0.80 | 0.59 | 3.64 | 0.81 |

${\mathit{t}}_{\mathbf{MSP}}$ | ${\mathit{t}}_{\mathbf{MIP}}$ | ${\mathbf{\Delta}}_{\mathbf{time}}$ | ${\mathbf{\Delta}}_{\mathbf{avg}}$ | ${\mathbf{\Delta}}_{\mathbf{tail}}$ | |
---|---|---|---|---|---|

mean | 16.98 | 0.20 | 91.31 | 2.03 | 3.09 |

std | 46.57 | 0.03 | 254.68 | 1.12 | 1.49 |

min | 0.53 | 0.14 | 2.81 | 0.16 | 0.86 |

25% | 1.37 | 0.17 | 6.73 | 1.18 | 2.09 |

50% | 3.74 | 0.19 | 19.72 | 1.93 | 2.81 |

75% | 15.50 | 0.21 | 86.19 | 2.52 | 3.51 |

max | 404.70 | 0.34 | 2175.82 | 5.67 | 8.57 |

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**MDPI and ACS Style**

León, J.; Puerto, J.; Vitoriano, B. A Risk-Aversion Approach for the Multiobjective Stochastic Programming Problem. *Mathematics* **2020**, *8*, 2026.
https://doi.org/10.3390/math8112026

**AMA Style**

León J, Puerto J, Vitoriano B. A Risk-Aversion Approach for the Multiobjective Stochastic Programming Problem. *Mathematics*. 2020; 8(11):2026.
https://doi.org/10.3390/math8112026

**Chicago/Turabian Style**

León, Javier, Justo Puerto, and Begoña Vitoriano. 2020. "A Risk-Aversion Approach for the Multiobjective Stochastic Programming Problem" *Mathematics* 8, no. 11: 2026.
https://doi.org/10.3390/math8112026