# Linear Programming and Fuzzy Optimization to Substantiate Investment Decisions in Tangible Assets

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## Abstract

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## 1. Introduction

- linear programming, through which it is ensured that conditions are met for the tangible assets, namely: the fulfillment of the objective function condition that can be to minimize the cost of acquiring the tangible assets or to maximize the economic benefits generated by them; and at the same time the fulfillment of the restrictions that refer to the limited character of the resources. For example, limited investment budget, limited budget for maintenance expenses or limited areas destined to the production activities, etc.,
- the fuzzy optimization necessary for modeling the technical and economic criteria that are specific to the tangible assets. The fuzzy modeling for the acquisition criteria specific to the tangible assets has a number of advantages, such as ensuring the comparability between the acquisition criteria of the assets with different units of measure, ensuring the hierarchy of the tangible assets according to their economic performance, ensuring the value ranges stratification for the analysis of the acquisition criteria, etc.

## 2. State-of-the-Art

## 3. Fuzzy Modeling and the Company’s Investment Decision

#### 3.1. Fuzzy Numbers for Acquisition Criteria and Constraints

#### 3.2. Common Rules for Fuzzy Modeling

- (a)
- the economic criterion for the assets acquisition ${C}_{e}=\left\{{c}_{e},\frac{{\mu}_{{c}_{e}}}{{c}_{e}}\in {C}_{e}\right\}$, where ${\mu}_{{c}_{e}}:{C}_{e}\to \left[0,1\right]$;
- (b)
- the technical criterion for the assets acquisition ${C}_{t}=\left\{{c}_{t},\frac{{\mu}_{{c}_{t}}}{{c}_{t}}\in {C}_{t}\right\}$, where ${\mu}_{{c}_{t}}:{C}_{t}\to \left[0,1\right]$;
- (c)
- the limited character of the resources available to the company ${C}_{R}=\left\{{c}_{R},\frac{{\mu}_{{c}_{R}}}{{c}_{R}}\in {C}_{R}\right\}$, where ${\mu}_{{c}_{R}}:{C}_{R}\to \left[0,1\right]$;

- Fuzzy numbers addition$${C}_{1}+{C}_{2}=\left[{c}_{a1},{c}_{b1}\right]+\left[{c}_{a2},{c}_{b2}\right]=\left[{c}_{a1}+{c}_{a2},{c}_{b1}+{c}_{b2}\right]$$
- Fuzzy numbers subtraction$$C\_1-C\_2=\left[c\_a1,c\_b1]-[c\_a2,c\_b2]=[c\_a1-c\_a2,c\_b1-c\_b2\right]$$
- Fuzzy numbers multiplication$${C}_{1}\times {C}_{2}=\left[{c}_{a1},{c}_{b1}\right]\times \left[{c}_{a2},{c}_{b2}\right]=\phantom{\rule{0ex}{0ex}}\left[\mathrm{min}\left({c}_{a1}{c}_{a2},{c}_{a1}{c}_{b2},{c}_{b1}{c}_{a2},{c}_{b1}{c}_{b2}\right),\mathrm{max}\left({c}_{a1}{c}_{a2},{c}_{a1}{c}_{b2},{c}_{b1}{c}_{a2},{c}_{b1}{c}_{b2}\right)\right]$$
- Fuzzy numbers division$$\frac{{C}_{1}}{{C}_{2}}=\frac{\left[{c}_{a1},{c}_{b1}\right]}{\left[{c}_{a2},{c}_{b2}\right]}=\left[\mathrm{min}\left(\frac{{c}_{a1}}{{c}_{a2}},\frac{{c}_{a1}}{{c}_{b2}},\frac{{c}_{b1}}{{c}_{a2}},\frac{{c}_{b1}}{{c}_{b2}}\right),\mathrm{max}\left(\frac{{c}_{a1}}{{c}_{a2}},\frac{{c}_{a1}}{{c}_{b2}},\frac{{c}_{b1}}{{c}_{a2}},\frac{{c}_{b1}}{{c}_{b2}}\right)\right]$$
- The inverse of fuzzy numbers$$\frac{1}{{C}_{1}}=\frac{1}{{c}_{a1},{c}_{b1}}=\left[\mathrm{min}\left(\frac{1}{{c}_{a1}};\frac{1}{{c}_{b1}}\right),\mathrm{max}\left(\frac{1}{{c}_{a1}};\frac{1}{{c}_{b1}}\right)\right]$$$$\frac{1}{{C}_{2}}=\frac{1}{{c}_{a2},{c}_{b2}}=\left[\mathrm{min}\left(\frac{1}{{c}_{a2}};\frac{1}{{c}_{b2}}\right),\mathrm{max}\left(\frac{1}{{c}_{a2}};\frac{1}{{c}_{b2}}\right)\right]$$

- It is a monotonically increasing function, respectively $\forall x,y\in Randx\le y$, it follows that $f\left(x\right)\le f\left(y\right);$
- Checks the normality condition, namely: ${\int}_{0}^{1}f\left(\alpha \right)d\alpha ={\int}_{0}^{1}2\alpha d\alpha =2\frac{{\alpha}^{2}}{2}{\u2502}_{0}^{1}=1..$

- It is a monotonically increasing function. $\forall {\alpha}_{1},{\alpha}_{2}\in R$ with ${\alpha}_{1}\le {\alpha}_{2}$ results that $f\left({\alpha}_{1}\right)\le f\left({\alpha}_{2}\right)$. From this condition it appears that $2{\alpha}_{1}\le 2{\alpha}_{2}$, respectively ${\alpha}_{1}\le {\alpha}_{2}$.
- Checks the normality condition, namely: ${\int}_{0}^{1}f\left(\alpha \right)d\alpha ={\int}_{0}^{1}2\alpha d\alpha =2\frac{{\alpha}^{2}}{2}{\u2502}_{0}^{1}=1$.

## 4. Graphical Method for Solving Linear Programming Problem with Fuzzy Optimization: The Case of Two Tangible Assets

## 5. The Primal Simplex Algorithm with Fuzzy Variables for Minimization Problems: The Case with N-Tangible Assets

- A minimization problem is transformed into a maximization problem by changing the signs of the fuzzy coefficients from the respective objective function;$$max{C}_{a}^{T}x=-min\left[-{C}_{a}^{T}\right]x$$
- The sign of an inequality changes by multiplying it by (−1), respectively by multiplying the constraint with fuzzy variables and fuzzy coefficients by (−1).
- An inequality of the form ${C}_{i}^{T}x\le {B}_{i}$ with $\left\{{C}_{i},{B}_{i}\right\}$} specific fuzzy sets, is written as an equality of the form ${C}_{i}^{T}x+Y={B}_{i}$, with $Y\ge 0,$ by adding the offset fuzzy variable $Y=\left\{y,{\mu}_{y}/y\in Y\right\}$, where ${\mu}_{y}:Y\to \left[0,1\right]$. An inequality of the form ${C}_{i}^{T}x\ge {B}_{i},$ with $\left\{{C}_{i},{B}_{i}\right\}$ specific fuzzy sets is written as an equality of the form ${C}_{i}^{T}x+Y={B}_{i}$, with $Y\ge 0$, by subtracting the offset fuzzy $Y=\left\{y,{\mu}_{y}/y\in Y\right\}$, where ${\mu}_{y}:Y\to \left[0,1\right]$.
- An equality of the form ${C}_{i}^{T}x={B}_{i}$ is transformed into two fuzzy inequalities of the form ${C}_{i}^{T}x\le {B}_{i}$ and respectively ${C}_{i}^{T}x\ge {B}_{i}$.A negative fuzzy variable ${x}_{j}\le 0$ is transformed into a positive fuzzy variable ${x}_{j}\ge 0$ by replacing with $-{x}_{j}.$ An unsigned fuzzy variable is replaced by the difference of two fuzzy variables ${x}_{j}={x}_{j}^{\prime}-{x}_{j}^{\u2033},$ where ${x}_{j}^{\prime}\ge 0$ și ${x}_{j}^{\u2033}\ge 0$.

- Definition 1: A fuzzy vector $X={\left({x}_{1}{x}_{2}\dots {x}_{n}\right)}^{T}\in {R}^{n}$ whose components satisfy all the constraints of the linear programming problem is called an admissible program or admissible fuzzy solution or possible fuzzy solution.
- Definition 2: An admissible solution of the form $X={\left({x}_{1}{x}_{2}\dots {x}_{n}\right)}^{T}\in {R}^{n}$ whose components minimize the objective function or, as the case may be, satisfy the condition imposed for the objective function, is called an optimal fuzzy program or optimal fuzzy solution.
- Definition 3: An admissible solution of the form $X={\left({x}_{1}{x}_{2}\dots {x}_{n}\right)}^{T}\in {R}^{n},$ whose column vectors ${C}^{j}$ corresponding to the nonzero components ${x}_{j}$ are linearly independent, is called te fuzzy basis program or fuzzy base solution.
- Definition 4: If a base program has nonzero fuzzy m-components (rank C = m), then the base program is called undesirable fuzzy. If a base program does not have null m-components (rank C = m) then the base program is called a fuzzy degenerate.
- Definition 5: The matrix B formed by m × m columns corresponding to the nonzero components of the fuzzy matrix C, of a non-degenerate base program X is called the fuzzy base of the program X.

- Step 1: Determine an admissible primal base B and determine its specific sizes, namely: $\overline{{X}_{B}}$; $\overline{{Z}_{B}}$; $\overline{{C}_{ij}^{B}}$; ${Z}_{j}^{B}-{C}_{aj}$. Determining these specific sizes is necessary to determine the value of the objective function Z and the solution of problem ${X}_{B},$ as:$$\mathrm{Z}=\overline{{Z}^{B}}-{\sum}_{j\in {J}_{S}}^{n}\left({Z}_{j}^{B}-{C}_{aj}\right){x}_{j}$$$${X}_{B}=\overline{{X}_{B}}-{\displaystyle \sum _{j=1}^{n}}\overline{{C}_{ij}^{B}}{x}_{j}$$
- Step 2: Analyze all the fuzzy differences resulting from the objective function following the rules of the fuzzy differences operator ${Z}_{j}^{B}-{C}_{aj}$ (the entry criterion). There are the following situations:
- ○
- 2.1: If all fuzzy differences are negative ${Z}_{j}^{B}-{C}_{aj}\le 0,$ then the program is optimal. Subtracting the fuzzy numbers ${Z}_{j}^{B}$ and ${C}_{aj}$ is done as follows:$${Z}_{j}^{B}-{C}_{aj}=\left[{z}_{ja1}^{B},{z}_{jb1}^{B}\right]-\left[{c}_{aj1},{c}_{aj2}\right]=\left[{z}_{ja1}^{B}-{c}_{aj1},{z}_{jb1}^{B}-{c}_{aj2}\right]$$
- ○
- 2.2: If there is at least one index $j\in {J}_{S},$ for which the fuzzy difference is ${Z}_{j}^{B}-{C}_{aj}>0,$ then $k\in {J}_{S}$ is determined for which:$${Z}_{k}^{B}-{C}_{ak}=max\left({Z}_{j}^{B}-{C}_{aj}\right)$$

- Step 3: Establish the vector for the base exit (the base exit criterion), according to the following algorithm:
- ○
- 3.1: If all the fuzzy coefficients are negative ${C}_{ij}\le 0$ and the fuzzy difference is positive ${Z}_{j}^{B}-{C}_{aj}>0$ for any $j\in {J}_{B}$, then the linear programming problem has an infinite optimal according to Theorem 3 presented above.
- ○
- 3.2: If all the fuzzy coefficients are positive ${C}_{ij}>0$ and the fuzzy difference is positive ${Z}_{j}^{B}-{C}_{aj}>0$, for any $j\in {J}_{B}$, then the vector ${x}_{l}$ is chosen which replaces the vector ${x}_{k}$ and determines a new allowable basis $\tilde{B}$ with the value:$${x}_{l}=\underset{j\in {J}_{B}}{\mathrm{min}}\left\{\frac{{x}_{i}}{{C}_{ik}}\right\};{C}_{ik}0$$

- Step 4: The vector ${x}_{l}$ is replaced by the vector ${x}_{k}$ in base B determined as the new allowable base $\tilde{B}$ for which $\overline{{X}_{\tilde{B}}}$; $\overline{{Z}_{\tilde{B}}}$; $\overline{{C}_{ij}^{\tilde{B}}}$; ${Z}_{j}^{\tilde{B}}-{C}_{aj}$. The determination of specific values $\overline{{X}_{\tilde{B}}}$; $\overline{{Z}_{\tilde{B}}}$; $\overline{{C}_{ij}^{\tilde{B}}}$; ${Z}_{j}^{\tilde{B}}-{C}_{aj}$ for the new allowable base $\tilde{B}$ is made according to the values of the allowable base B as follows. We know that the components of the main variable are written as follows:$${x}_{i}=\overline{{x}_{j}^{B}}-{\displaystyle \sum}_{j=1}^{n}\overline{{C}_{ij}^{B}}{x}_{j}$$

- Rule 1: All the elements in the line with the pivot ${C}_{kl}$ are divided by the pivot;
- Rule 2: All the elements in the pivot column become zero, except the pivot whose value becomes 1;
- Rule 3: All the other elements of the table are obtained according to the rule of the rectangle;$$\tilde{{C}_{ij}}={C}_{ij}-\frac{{C}_{lj}{C}_{ik}}{{C}_{kl}}$$

## 6. Case Study on the Application of the Primal Simplex Algorithm with Fuzzy Coefficients

- The objective function with fuzzy coefficients and variables is of the form:$$f\left(x\right)=\left(1000,1500\right){x}_{1}+\left(2000,3000\right){x}_{2}+\left(5000,10,000\right){x}_{3}\to min$$
- The constraints of the linear programming problem with fuzzy coefficients are:$$\{\begin{array}{c}\left(1000,1500\right){x}_{1}+\left(2000,3000\right){x}_{2}+\left(5000,\mathrm{10,000}\right){x}_{3}\le \left(\mathrm{100,000},\mathrm{200,000}\right)\\ 10{x}_{1}+50{x}_{2}+100{x}_{3}\le \left(5000,\mathrm{10,000}\right)\\ \left(100,150\right){x}_{1}+\left(200,300\right){x}_{2}+\left(500,1000\right){x}_{3}\le \left(\mathrm{25,000},\mathrm{50,000}\right)\end{array}$$
- The non-negativity conditions of the linear programming problem are of the form:$${x}_{1}\ge 0,{x}_{2}\ge 0,{x}_{3}\ge 0;$$

- The objective function with fuzzy deviation variables is set as follows:$$f\left(x\right)=\left(1000,1500\right){x}_{1}+\left(2000,3000\right){x}_{2}+\left(5000,\mathrm{10,000}\right){x}_{3}+0{x}_{4}+0{x}_{5}+0{x}_{6}\to min$$
- The constraints of the linear programming problem with the help of the fuzzy offset variables are established as follows:$$\{\begin{array}{c}\left(1000,1500\right){x}_{1}+\left(2000,3000\right){x}_{2}+\left(5000,\mathrm{10,000}\right){x}_{3}+{x}_{4}=\left(\mathrm{100,000},\mathrm{200,000}\right)\\ 10{x}_{1}+50{x}_{2}+100{x}_{3}+{x}_{5}=\left(5000,\mathrm{10,000}\right)\\ \left(100,150\right){x}_{1}+\left(200,300\right){x}_{2}+\left(500,1000\right){x}_{3}+{x}_{6}=\left(\mathrm{25,000},\mathrm{50,000}\right)\end{array}$$
- The non-negativity conditions for the mathematical model, completed with the help of the offset variables are of the form: ${x}_{1}\ge 0,$ ${x}_{2}\ge 0,{x}_{3}\ge 0,{x}_{4}\ge 0,$ ${x}_{5}\ge 0,{x}_{6}\ge 0$.

## 7. Concluding Remarks

## Author Contributions

## Funding

## Conflicts of Interest

## References

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The Start Admissible Base (B) | The Fuzzy Coefficients from Base B | The Objective Function Coefficients | ${\mathit{C}}_{\mathit{a}\mathbf{1}}$ | ${\mathit{C}}_{\mathit{a}\mathbf{2}}$ | ${\mathit{C}}_{\mathit{a}\mathit{n}}$ | |
---|---|---|---|---|---|---|

Variable Values ${\mathit{B}}_{\mathit{F}}$ | ${\mathit{x}}_{\mathbf{1}}$ | ${\mathit{x}}_{\mathbf{2}}$ | … | ${\mathit{x}}_{\mathit{n}}$ | ||

${\mathit{x}}_{\mathbf{1}}^{\mathit{B}}$ | ${C}_{1}^{B}$ | ${B}_{1}$ | ${C}_{11}$ | ${C}_{21}$ | … | ${C}_{n1}$ |

${\mathit{x}}_{\mathbf{2}}^{\mathit{B}}$ | ${C}_{2}^{B}$ | ${B}_{2}$ | ${C}_{21}$ | ${C}_{22}$ | … | ${C}_{2n}$ |

… | … | … | … | … | … | … |

${\mathit{x}}_{\mathit{n}}^{\mathit{B}}$ | ${C}_{n}^{B}$ | ${B}_{n}$ | ${C}_{n1}$ | ${C}_{n2}$ | … | ${C}_{nn}$ |

${\mathit{Z}}^{\mathit{B}}$ | *** | ${Z}^{B}\left({B}_{F}\right)$ | ${Z}^{B}\left({x}_{1}\right)$ | ${Z}^{B}\left({x}_{2}\right)$ | … | ${Z}^{B}\left({x}_{n}\right)$ |

${\mathit{Z}}_{\mathit{j}}^{\mathit{B}}\mathbf{-}{\mathit{C}}_{\mathit{j}}$ | *** | *** | ${Z}^{B}\left({x}_{1}\right)-{C}_{a1}$ | ${Z}^{B}\left({x}_{2}\right)-{C}_{a2}$ | … | ${Z}^{B}\left({x}_{n}\right)-{C}_{an}$ |

Criterion | Criterion Type: Acquisition/Constraint Resulting from Company’s Activity | Notation | The Value for Asset $\left({\mathit{A}}_{1}\right)$ | The Value for Asset $\left({\mathit{A}}_{2}\right)$ | The Value for Asset $\left({\mathit{A}}_{3}\right)$ |
---|---|---|---|---|---|

Acquisition cost | Acquisition criteria | ${C}_{a}\left({A}_{i}\right)$ | $\left(\begin{array}{c}\$1000,\\ \$1500\end{array}\right)$ | $\left(\begin{array}{c}\$2000,\\ \$3000\end{array}\right)$ | $\left(\begin{array}{c}\$5000\\ \$10,000\end{array}\right)$ |

The allocated budget for the tangible assets acquisition | Activity constraint | ${B}_{inv}$ | $\left(\$\mathrm{100,000},\$\mathrm{200,000}\right)$ | ||

The mounting surface/asset | Activity constraint | ${S}_{m}\left({A}_{i}\right)$ | 10 m^{2} | 50 m^{2} | 100 m^{2} |

The total surface for mounting | Activity constraint | $S{t}_{m}$ | $\left(\mathrm{5,000}{\mathrm{m}}^{2},\mathrm{10,000}{\mathrm{m}}^{2}\right)$ | ||

The operating expenses | Acquisition criteria | $C{h}_{f}\left({A}_{i}\right)$ | $\left(\begin{array}{c}\$100,\\ \$150\end{array}\right)$ | $\left(\begin{array}{c}\$200,\\ \$300\end{array}\right)$ | $\left(\begin{array}{c}\$500,\\ \$1000\end{array}\right)$ |

The allocated budget for the operating expenses | Activity constraint | $BC{h}_{f}$ | $\left(\$\mathrm{25,000},\$\mathrm{50,000}\right)$ |

The Admissible Starting Base (B) | The Fuzzy Coefficients from Base B | The Coefficients of the Objective Function | $\left(\begin{array}{c}1000,\\ 1500\end{array}\right)$ | $\left(\begin{array}{c}2000,\\ 3000\end{array}\right)$ | $\left(\begin{array}{c}5000,\\ \mathrm{10,000}\end{array}\right)$ | $\left(0,0\right)$ | $\left(0,0\right)$ | $\left(0,0\right)$ |
---|---|---|---|---|---|---|---|---|

${\mathit{B}}_{\mathit{F}}$ | ${\mathit{x}}_{1}$ | ${\mathit{x}}_{2}$ | ${\mathit{x}}_{3}$ | ${\mathit{x}}_{4}$ | ${\mathit{x}}_{5}$ | ${\mathit{x}}_{6}$ | ||

${\mathit{x}}_{\mathbf{4}}^{\mathit{B}}$ | $\left(0,0\right)$ | $\left(\begin{array}{c}\mathrm{100,000},\\ \mathrm{200,000}\end{array}\right)$ | $\left(\begin{array}{c}1000,\\ 1500\end{array}\right)$ | $\left(\begin{array}{c}2000,\\ 3000\end{array}\right)$ | $\left(\begin{array}{c}5000,\\ \mathrm{10,000}\end{array}\right)$ | $\left(1,1\right)$ | $\left(0,0\right)$ | $\left(0,0\right)$ |

${\mathit{x}}_{\mathbf{5}}^{\mathit{B}}$ | $\left(0,0\right)$ | $\left(\begin{array}{c}5000,\\ \mathrm{10,000}\end{array}\right)$ | $\left(10\right)$ | $\left(50\right)$ | $\left(100\right)$ | $\left(0,0\right)$ | $\left(1,1\right)$ | $\left(0,0\right)$ |

${\mathit{x}}_{\mathbf{6}}^{\mathit{B}}$ | $\left(0,0\right)$ | $\left(\begin{array}{c}\mathrm{25,000},\\ \mathrm{50,000}\end{array}\right)$ | $\left(\begin{array}{c}100,\\ 150\end{array}\right)$ | $\left(\begin{array}{c}200,\\ 300\end{array}\right)$ | $\left(\begin{array}{c}500,\\ 1000\end{array}\right)$ | $\left(0,0\right)$ | $\left(0,0\right)$ | $\left(1,1\right)$ |

${\mathit{Z}}^{\mathit{B}}$ | *** | $\left(0,0\right)$ | $\left(0,0\right)$ | $\left(0,0\right)$ | *** | *** | *** | |

${\mathit{Z}}_{\mathit{j}}^{\mathit{B}}\mathbf{-}{\mathit{C}}_{\mathit{j}}$ | *** | *** | $\left(\begin{array}{c}-1000,\\ -1500\end{array}\right)$ | $\left(\begin{array}{c}-2000,\\ -3000\end{array}\right)$ | $\left(\begin{array}{c}-5000,\\ -\mathrm{10,000}\end{array}\right)$ | *** | *** | *** |

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## Share and Cite

**MDPI and ACS Style**

Boloș, M.-I.; Bradea, I.-A.; Delcea, C.
Linear Programming and Fuzzy Optimization to Substantiate Investment Decisions in Tangible Assets. *Entropy* **2020**, *22*, 121.
https://doi.org/10.3390/e22010121

**AMA Style**

Boloș M-I, Bradea I-A, Delcea C.
Linear Programming and Fuzzy Optimization to Substantiate Investment Decisions in Tangible Assets. *Entropy*. 2020; 22(1):121.
https://doi.org/10.3390/e22010121

**Chicago/Turabian Style**

Boloș, Marcel-Ioan, Ioana-Alexandra Bradea, and Camelia Delcea.
2020. "Linear Programming and Fuzzy Optimization to Substantiate Investment Decisions in Tangible Assets" *Entropy* 22, no. 1: 121.
https://doi.org/10.3390/e22010121