# Possibility/Necessity-Based Probabilistic Expectation Models for Linear Programming Problems with Discrete Fuzzy Random Variables

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Fuzzy Set and Fuzzy Number

**Definition**

**1.**

**Definition**

**2.**

- 1.
- $L\left(t\right)$ and $R\left(t\right)$ are nonincreasing for any $t>0$.
- 2.
- $L\left(0\right)=R\left(0\right)=1$.
- 3.
- $L\left(t\right)=L(-t)$ and $R\left(t\right)=R(-t)$ for any $t\in \mathbb{R}$.
- 4.
- There exists a ${t}_{0}^{L}>0$ such that $L\left(t\right)=0$ holds for any t larger than ${t}_{0}^{L}$. Similarly, there exists a ${t}_{0}^{R}>0$ such that $R\left(t\right)=0$ holds for any t larger than ${t}_{0}^{R}$.

**Definition**

**3.**

- 1.
- $L\left(t\right)$ is nonincreasing for any $t>0$.
- 2.
- $L\left(0\right)=1$.
- 3.
- $L\left(t\right)=L(-t)$ for any $t\in \mathbb{R}$.
- 4.
- There exists a ${t}_{0}^{L}>0$ such that $L\left(t\right)=0$ holds for any t larger than ${t}_{0}^{L}$.

**Definition**

**4.**

#### 2.2. Fuzzy Random Variable

**Definition**

**5.**

#### 2.3. Special Types of Fuzzy Random Variables Used in Decision Making

**Definition**

**6.**

**Example**

**1.**

**Example**

**2.**

**Definition**

**7.**

**Example**

**3.**

**Definition**

**8.**

**Example**

**4.**

## 3. Discrete Fuzzy Random Variable

#### Definitions of Discrete Fuzzy Random Variables

**Definition**

**9.**

**Definition**

**10.**

**Example**

**5.**

**Definition**

**11.**

**Example**

**6.**

## 4. Problem Formulation

#### 4.1. Model Using Discrete L-R Fuzzy Random Variables

#### 4.2. Model Using Discrete Triangular Fuzzy Random Variables

## 5. Possibility/Necessity-Based Probabilistic Expectation

#### 5.1. Preliminary: Possibility and Necessity Measures

#### 5.1.1. Possibility Measure

**Definition**

**12.**

#### 5.1.2. Necessity Measure

**Definition**

**13.**

#### 5.2. Optimization Criteria in Fuzzy Random Environments

**Definition**

**14.**

**Definition**

**15.**

## 6. Discrete Fuzzy Random Linear Programming Models Using Possibility/Necessity-Based Probabilistic Expectation

#### 6.1. Possibility-Based Probabilistic Expectation (PPE) Model

**Possibility-based probabilistic expectation model (PPE model)**]

**Definition**

**16.**

**Definition**

**17.**

**Maximin problem for PPE model**]

**Proposition**

**1.**

**Proof.**

**Augmented maximin problem for PPE model**]

**Proposition**

**2.**

**Proof.**

#### 6.2. Necessity-Based Probabilistic Expectation Model (NPE Model)

**Necessity-based probabilistic expectation model (NPE model)**]

**Definition**

**18.**

**Definition**

**19.**

#### Scalarization-Based Problems for Obtaining a Pareto Optimal Solution

**Maximin problem for NPE model**]

**Proposition**

**3.**

**Augmented maximin problem for NPE model**]

**Proposition**

**4.**

## 7. Solution Algorithms

#### 7.1. Solution Algorithm for the PPE Model

**Theorem**

**1.**

**Proof.**

- Case 1: If ${\mathit{d}}_{lk}\mathit{x}<{f}_{l}^{1}$ the value of $\mathrm{\Pi}\left({\tilde{\mathit{C}}}_{lk}\mathit{x}\stackrel{<}{\sim}{f}_{l}\right)$ is equal to 1, as shown in Figure 13.
- Case 2: If ${f}_{l}^{1}\le {\mathit{d}}_{lk}\mathit{x}\le {f}_{l}^{0}+{\mathit{\gamma}}_{lk}\mathit{x}$, the value of $\mathrm{\Pi}\left({\tilde{\mathit{C}}}_{lk}\mathit{x}\stackrel{<}{\sim}{f}_{l}\right)$ is calculated as the ordinate of the crossing point between the membership function of fuzzy goal ${\tilde{G}}_{l}$ and the objective function ${\mathit{C}}_{lk}\mathit{x}$, as shown in Figure 14. The abscissa of the crossing point of two functions (${\mu}_{{\tilde{\mathit{C}}}_{lk}\mathit{x}}$ and ${\mu}_{{\tilde{G}}_{l}}$) is obtained by solving the equation$$1-\frac{{\mathit{d}}_{lk}\mathit{x}-y}{{\mathit{\beta}}_{lk}\mathit{x}}=\frac{y-{f}_{l}^{0}}{{f}_{l}^{1}-{f}_{l}^{0}},\phantom{\rule{4pt}{0ex}}l=1,2,\dots ,q,\phantom{\rule{4pt}{0ex}}k=1,2,\dots ,{r}_{l}.$$Then, the solution ${y}_{lk}^{\mathrm{\Pi}*}$ of (37) is$${y}_{lk}^{\mathrm{\Pi}*}=\frac{{\displaystyle {f}_{l}^{1}\sum _{j=1}^{n}{\beta}_{ljk}{x}_{j}+({f}_{l}^{0}-{f}_{l}^{1})\sum _{j=1}^{n}{d}_{ljk}{x}_{j}}}{{\displaystyle \sum _{j=1}^{n}{\beta}_{ljk}{x}_{j}-{f}_{l}^{1}+{f}_{l}^{0}}},\phantom{\rule{4pt}{0ex}}l=1,2,\dots ,q,\phantom{\rule{4pt}{0ex}}k=1,2,\dots ,{r}_{l}.$$Consequently, the ordinate of the crossing point is calculated as$${\mu}_{{\tilde{G}}_{l}}\left({y}_{lk}^{\mathrm{\Pi}*}\right)={\mu}_{{\tilde{\mathit{C}}}_{lk}\mathit{x}}\left({y}_{lk}^{\mathrm{\Pi}*}\right)=\frac{{\displaystyle \sum _{j=1}^{n}({\beta}_{ljk}-{d}_{ljk}){x}_{j}+{f}_{l}^{0}}}{{\displaystyle \sum _{j=1}^{n}{\beta}_{ljk}{x}_{j}-{f}_{l}^{1}+{f}_{l}^{0}}}\left(\stackrel{\u25b5}{=}{g}_{lk}^{\mathrm{\Pi}}\left(\mathit{x}\right)\right)$$
- Case 3: If ${\mathit{d}}_{lk}\mathit{x}>{f}_{l}^{0}+{\mathit{\gamma}}_{lk}$, the value of $\mathrm{\Pi}\left({\tilde{\mathit{C}}}_{lk}\mathit{x}\stackrel{<}{\sim}{f}_{l}\right)$ is equal to 0, as shown in Figure 15.

**Augmented maximin problem for PPE model (linear membership function case)**]

**[An algorithm for obtaining a (strong) Pareto optimal solution of PPE model (linear membership function case)]**

**Step 1:**- (Calculation of possible objective function values)Using a linear programming technique, solve individual minimization problems (34) for $l=1,2,\dots ,q$, namely$$\left.\begin{array}{cc}\hfill \mathrm{minimize}& {\displaystyle \sum _{j=1}^{n}\sum _{k=1}^{{r}_{l}}{p}_{lk}{d}_{ljk}{x}_{j}}\hfill \\ \hfill \mathrm{subject}\phantom{\rule{4pt}{0ex}}\mathrm{to}& \mathit{x}\in X\hfill \end{array}\right\}\mathrm{for}\phantom{\rule{4pt}{0ex}}l=1,2,\dots ,q,$$
**Step 2:**- (Setting of membership functions of fuzzy goals)Ask the DM to specify the values of ${f}_{l}^{0}$ and ${f}_{l}^{1}$, $l=1,2,\dots ,q$. If the DM has no idea of how ${f}_{l}^{0}$ and ${f}_{l}^{1}$, $l=1,2,\dots ,q$ are determined, then the DM can set the following values calculated by (33) as$$\left.\begin{array}{ccc}{f}_{l}^{1}\hfill & =\hfill & {\displaystyle \sum _{j=1}^{n}\sum _{k=1}^{{r}_{l}}{p}_{lk}{d}_{ljk}{\widehat{x}}_{j}^{l*}}\hfill \\ {f}_{l}^{0}\hfill & =\hfill & {\displaystyle \underset{r\in \{1,2,\dots ,q\}}{\mathrm{max}}\sum _{j=1}^{n}\sum _{k=1}^{{r}_{l}}{p}_{lk}{d}_{ljk}{\widehat{x}}_{j}^{r*}}\hfill \end{array}\right\}\mathrm{for}\phantom{\rule{4pt}{0ex}}l=1,2,\dots ,q,$$
**Step 3:**- (Derivation of a strong Pareto optimal solution of PPE model)Using a nonlinear programming technique, solve the following augmented maximin problem (38):$$\begin{array}{cc}\hfill \mathrm{maximize}& {\displaystyle \underset{l\in \{1,2,\dots ,q\}}{\mathrm{min}}\sum _{k=1}^{{r}_{l}}{p}_{lk}\xb7\mathrm{min}\left[1,\phantom{\rule{4pt}{0ex}}\mathrm{max}\left\{0,\phantom{\rule{4pt}{0ex}}{g}_{lk}^{\mathrm{\Pi}}\left(\mathit{x}\right)\right\}\right]}\hfill \\ & {\displaystyle +\rho \sum _{l=1}^{q}\sum _{k=1}^{{r}_{l}}{p}_{lk}\xb7\mathrm{min}\left[1,\phantom{\rule{4pt}{0ex}}\mathrm{max}\left\{0,\phantom{\rule{4pt}{0ex}}{g}_{lk}^{\mathrm{\Pi}}\left(\mathit{x}\right)\right\}\right]}\hfill \\ \hfill \mathrm{subject}\phantom{\rule{4pt}{0ex}}\mathrm{to}& \mathit{x}\in X,\hfill \end{array}$$$${g}_{lk}^{\mathrm{\Pi}}\left(\mathit{x}\right)\stackrel{\u25b5}{=}\frac{{\displaystyle \sum _{j=1}^{n}({\beta}_{ljk}-{d}_{ljk}){x}_{j}+{f}_{l}^{0}}}{{\displaystyle \sum _{j=1}^{n}{\beta}_{ljk}{x}_{j}-{f}_{l}^{1}+{f}_{l}^{0}}},\phantom{\rule{4pt}{0ex}}l=1,2,\dots ,q,\phantom{\rule{4pt}{0ex}}k=1,2,\dots ,{r}_{l},$$

#### 7.2. Solution Algorithm for the NPE Model

**Theorem**

**2.**

**Proof.**

- Case 1: If $({\mathit{d}}_{lk}+{\mathit{\gamma}}_{lk})\mathit{x}<{f}_{l}^{1}$, the value of $N\left({\tilde{\mathit{C}}}_{lk}\mathit{x}\stackrel{<}{\sim}{f}_{l}\right)$ is equal to 1, as shown in Figure 16.
- Case 2: If ${f}_{l}^{1}-{\mathit{\gamma}}_{lk}\mathit{x}\le {\mathit{d}}_{lk}\mathit{x}\le {f}_{l}^{0}+{\mathit{\gamma}}_{lk}\mathit{x}$, the value of $N\left({\tilde{\mathit{C}}}_{lk}\mathit{x}\stackrel{<}{\sim}{f}_{l}\right)$ is calculated as the ordinate of the crossing point between the membership functions of fuzzy goal ${\tilde{G}}_{l}$ and the objective function ${\mathit{C}}_{lk}\mathit{x}$, as shown in Figure 17. The abscissa of the crossing point of two functions (${\mu}_{{\tilde{\mathit{C}}}_{lk}\mathit{x}}$ and ${\mu}_{{\tilde{G}}_{l}}$) is obtained by solving the equation$$\frac{y-{\mathit{d}}_{lk}\mathit{x}}{{\mathit{\gamma}}_{lk}\mathit{x}}=\frac{y-{f}_{l}^{0}}{{f}_{l}^{1}-{f}_{l}^{0}},\phantom{\rule{4pt}{0ex}}l=1,2,\dots ,q,\phantom{\rule{4pt}{0ex}}k=1,2,\dots ,{r}_{l}.$$Then, the solution of (41) is$${y}_{lk}^{N*}=\frac{{\displaystyle {f}_{l}^{0}\sum _{j=1}^{n}{\gamma}_{ljk}{x}_{j}+({f}_{l}^{0}-{f}_{l}^{1})\sum _{j=1}^{n}{d}_{ljk}{x}_{j}}}{{\displaystyle \sum _{j=1}^{n}{\gamma}_{ljk}{x}_{j}-{f}_{l}^{1}+{f}_{l}^{0}}},\phantom{\rule{4pt}{0ex}}l=1,2,\dots ,q,\phantom{\rule{4pt}{0ex}}k=1,2,\dots ,{r}_{l}.$$Consequently, the ordinate of the crossing point is calculated as$${\mu}_{{\tilde{G}}_{l}}\left({y}_{lk}^{N*}\right)=1-{\mu}_{{\tilde{\mathit{C}}}_{lk}\mathit{x}}\left({y}_{lk}^{N*}\right)=\frac{{\displaystyle -\sum _{j=1}^{n}{d}_{ljk}{x}_{j}+{f}_{l}^{0}}}{{\displaystyle \sum _{j=1}^{n}{\gamma}_{ljk}{x}_{j}-{f}_{l}^{1}+{f}_{l}^{0}}}\left(\stackrel{\u25b5}{=}{g}_{lk}^{N}\left(\mathit{x}\right)\right)$$
- Case 3: If ${\mathit{d}}_{lk}\mathit{x}>{f}_{l}^{0}$, the value of $N\left({\tilde{\mathit{C}}}_{lk}\mathit{x}\stackrel{<}{\sim}{f}_{l}\right)$ is equal to 0, as shown in Figure 18.

**Augmented maximin problem for the NPE model (linear membership function case)**]

**[An algorithm for obtaining a (strong) Pareto optimal solution of NPE model (linear membership function case)]**

**Step 1:**- (Calculation of possible objective function values)By using a linear programming technique, solve individual minimization problems (34) for $l=1,2,\dots ,q$, namely$$\left.\begin{array}{cc}\hfill \mathrm{minimize}& {\displaystyle \sum _{j=1}^{n}\sum _{k=1}^{{r}_{l}}{p}_{lk}{d}_{ljk}{x}_{j}}\hfill \\ \hfill \mathrm{subject}\phantom{\rule{4pt}{0ex}}\mathrm{to}& \mathit{x}\in X\hfill \end{array}\right\}\mathrm{for}\phantom{\rule{4pt}{0ex}}l=1,2,\dots ,q,$$
**Step 2:**- (Setting of membership functions of fuzzy goals)Ask the DM to specify the values of ${f}_{l}^{0}$ and ${f}_{l}^{1}$, $l=1,2,\dots ,q$. If the decision has no idea of how ${f}_{l}^{0}$ and ${f}_{l}^{1}$, $l=1,2,\dots ,q$ are determined, the DM could set the values calculated by (33) as$$\left.\begin{array}{ccc}{f}_{l}^{1}\hfill & =\hfill & {\displaystyle \sum _{j=1}^{n}\sum _{k=1}^{{r}_{l}}{p}_{lk}{d}_{ljk}{\widehat{x}}_{j}^{l*}}\hfill \\ {f}_{l}^{0}\hfill & =\hfill & {\displaystyle \underset{r\in \{1,2,\dots ,q\}}{\mathrm{max}}\sum _{j=1}^{n}\sum _{k=1}^{{r}_{l}}{p}_{lk}{d}_{ljk}{\widehat{x}}_{j}^{r*}}\hfill \end{array}\right\}\mathrm{for}\phantom{\rule{4pt}{0ex}}l=1,2,\dots ,q,$$
**Step 3:**- (Derivation of a (strong) Pareto optimal solution of the NPE model)Solve the following augmented maximin problem (42) using a nonlinear programming technique:$$\begin{array}{cc}\hfill \mathrm{maximize}& {\displaystyle \underset{l\in \{1,2,\dots ,q\}}{\mathrm{min}}\sum _{k=1}^{{r}_{l}}{p}_{lk}\xb7\mathrm{min}\left[1,\phantom{\rule{4pt}{0ex}}\mathrm{max}\left\{0,\phantom{\rule{4pt}{0ex}}{g}_{lk}^{N}\left(\mathit{x}\right)\right\}\right]}\hfill \\ & {\displaystyle +\rho \sum _{l=1}^{q}\sum _{k=1}^{{r}_{l}}{p}_{lk}\xb7\mathrm{min}\left[1,\phantom{\rule{4pt}{0ex}}\mathrm{max}\left\{0,\phantom{\rule{4pt}{0ex}}{g}_{lk}^{N}\left(\mathit{x}\right)\right\}\right]}\hfill \\ \hfill \mathrm{subject}\phantom{\rule{4pt}{0ex}}\mathrm{to}& \mathit{x}\in X,\hfill \end{array}$$$${g}_{lk}^{N}\left(\mathit{x}\right)\stackrel{\u25b5}{=}\frac{{\displaystyle -\sum _{j=1}^{n}{d}_{ljk}{x}_{j}+{f}_{l}^{0}}}{{\displaystyle \sum _{j=1}^{n}{\gamma}_{ljk}{x}_{j}-{f}_{l}^{1}+{f}_{l}^{0}}},\phantom{\rule{4pt}{0ex}}l=1,2,\dots ,q,\phantom{\rule{4pt}{0ex}}k=1,2,\dots ,{r}_{l},$$

## 8. Numerical Experiments

#### 8.1. Crop Area Planning Problem Under a Fuzzy Random Environment

**Remark**

**1.**

#### 8.2. Computational Times for Different Size Problems

## 9. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 6.**Realized values ${\tilde{C}}_{ljk}$ for the kth event of a discrete L-R fuzzy random variable ${\tilde{\overline{C}}}_{lj}$.

**Figure 7.**Realized values ${\tilde{C}}_{ljk}$ for the kth event of a discrete L-R fuzzy random variable ${\tilde{\overline{C}}}_{lj}$.

**Figure 8.**Realized value ${\tilde{C}}_{ljk}$ for the kth event of a discrete triangular fuzzy random variable ${\tilde{\overline{C}}}_{lj}$.

**Figure 9.**Realized value ${\tilde{\mathit{C}}}_{lk}\mathit{x}$ for the kth event of a discrete triangular fuzzy random variable ${\tilde{\overline{\mathit{C}}}}_{l}\mathit{x}$.

**Figure 10.**Degree of possibility $\mathrm{\Pi}\left({\tilde{\mathit{C}}}_{lk}\mathit{x}\stackrel{<}{\sim}{f}_{l}\right)$.

**Figure 11.**Degree of necessity $N\left({\tilde{\mathit{C}}}_{lk}\mathit{x}\stackrel{<}{\sim}{f}_{l}\right)$.

Event | Probability | Situation |
---|---|---|

${\omega}_{11}$ | ${p}_{11}=0.50$ | average annual temperature is normal. |

${\omega}_{12}$ | ${p}_{12}=0.25$ | average annual temperature is high. |

${\omega}_{13}$ | ${p}_{13}=0.15$ | average annual temperature is low. |

${\omega}_{14}$ | ${p}_{14}=0.06$ | it happens an epidemic disease for cucurbitaceous vegetables such |

as cucumber and watermelon, due to a very high-temperature. | ||

${\omega}_{15}$ | ${p}_{15}=0.04$ | it happens an epidemic disease for solanaceae vegetables |

such as bell pepper, eggplant and tomato, due to a very low-temperature. | ||

${\omega}_{21}$ | ${p}_{21}=0.50$ | average annual humidity is normal. |

${\omega}_{22}$ | ${p}_{22}=0.20$ | average annual humidity is high. |

${\omega}_{23}$ | ${p}_{23}=0.16$ | average annual humidity is low. |

${\omega}_{24}$ | ${p}_{24}=0.08$ | it happens an epidemic disease for cucurbitaceous vegetables such |

as cucumber and watermelon, caused by very low-humidity. | ||

${\omega}_{25}$ | ${p}_{25}=0.06$ | it happens an epidemic disease for solanaceae vegetables such |

as bell pepper, eggplant and tomato, due to a very low-temperature. |

Parameter | $\mathit{k}=\mathbf{1}$ | $\mathit{k}=\mathbf{2}$ | $\mathit{k}=\mathbf{3}$ | $\mathit{k}=\mathbf{4}$ | $\mathit{k}=\mathbf{5}$ |
---|---|---|---|---|---|

${d}_{11k}$ | 89.50 | 95.10 | 83.80 | 97.50 | 61.80 |

${d}_{12k}$ | 118.50 | 118.80 | 117.90 | 79.60 | 117.00 |

${d}_{13k}$ | 122.60 | 123.60 | 125.60 | 113.30 | 83.40 |

${d}_{14k}$ | 90.30 | 82.60 | 93.10 | 85.10 | 66.10 |

${d}_{15k}$ | 25.80 | 28.90 | 24.40 | 21.50 | 23.70 |

${\gamma}_{11k}$ | 8.20 | 8.60 | 7.40 | 10.20 | 5.70 |

${\gamma}_{12k}$ | 10.70 | 10.90 | 10.60 | 8.50 | 11.20 |

${\gamma}_{13k}$ | 9.00 | 8.70 | 8.80 | 8.70 | 5.90 |

${\gamma}_{14k}$ | 8.10 | 7.60 | 8.40 | 7.30 | 5.80 |

${\gamma}_{15k}$ | 2.60 | 3.20 | 2.50 | 2.10 | 2.20 |

${\beta}_{11k}$ | 11.40 | 11.80 | 11.30 | 12.20 | 8.60 |

${\beta}_{12k}$ | 10.70 | 10.30 | 10.20 | 7.10 | 9.70 |

${\beta}_{13k}$ | 9.70 | 9.10 | 9.80 | 8.60 | 5.10 |

${\beta}_{14k}$ | 6.40 | 6.20 | 6.40 | 5.90 | 5.00 |

${\beta}_{15k}$ | 3.90 | 4.20 | 3.60 | 3.30 | 3.60 |

Parameter | $\mathit{k}=\mathbf{1}$ | $\mathit{k}=\mathbf{2}$ | $\mathit{k}=\mathbf{3}$ | $\mathit{k}=\mathbf{4}$ | $\mathit{k}=\mathbf{5}$ |
---|---|---|---|---|---|

${d}_{21k}$ | 97.00 | 100.20 | 90.30 | 102.50 | 124.50 |

${d}_{22k}$ | 116.50 | 114.60 | 119.50 | 172.50 | 121.90 |

${d}_{23k}$ | 131.10 | 133.80 | 128.10 | 146.40 | 172.50 |

${d}_{24k}$ | 88.60 | 86.10 | 93.50 | 89.90 | 139.70 |

${d}_{25k}$ | 27.60 | 23.10 | 28.10 | 31.40 | 29.50 |

${\beta}_{21k}$ | 18.40 | 18.80 | 16.90 | 20.20 | 23.60 |

${\beta}_{22k}$ | 11.70 | 11.40 | 12.20 | 17.90 | 13.20 |

${\beta}_{23k}$ | 14.70 | 15.60 | 12.90 | 15.90 | 21.80 |

${\beta}_{24k}$ | 5.40 | 5.20 | 5.80 | 5.30 | 7.20 |

${\beta}_{25k}$ | 5.10 | 4.80 | 5.40 | 6.30 | 5.70 |

${\gamma}_{21k}$ | 6.80 | 7.10 | 6.80 | 8.10 | 11.70 |

${\gamma}_{22k}$ | 19.10 | 19.20 | 19.90 | 27.80 | 20.50 |

${\gamma}_{23k}$ | 6.60 | 7.20 | 6.60 | 7.00 | 8.80 |

${\gamma}_{24k}$ | 12.30 | 12.70 | 12.10 | 12.60 | 26.70 |

${\gamma}_{25k}$ | 3.30 | 2.90 | 3.70 | 3.80 | 3.50 |

LHS Value | $\mathit{j}=\mathbf{1}$ | $\mathit{j}=\mathbf{2}$ | $\mathit{j}=\mathbf{3}$ | $\mathit{j}=\mathbf{4}$ | $\mathit{j}=\mathbf{5}$ |
---|---|---|---|---|---|

${a}_{1j}$ | 53.20 | 58.80 | 57.70 | 63.70 | 33.00 |

${a}_{2j}$ | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |

${a}_{3j}$ | −1.00 | −1.00 | −1.00 | −1.00 | −1.00 |

${a}_{4j}$ | −53.90 | −80.50 | −75.30 | 0.00 | 0.00 |

${a}_{5j}$ | 0.00 | 0.00 | 0.00 | −75.00 | −48.40 |

RHS Value | $\mathit{i}=\mathbf{1}$ | $\mathit{i}=\mathbf{2}$ | $\mathit{i}=\mathbf{3}$ | $\mathit{i}=\mathbf{4}$ | $\mathit{i}=\mathbf{5}$ |
---|---|---|---|---|---|

${b}_{i}$ | 30,000.00 | 500.00 | −300.00 | −10,500.00 | −7000.00 |