*3.1. Multi-Objective Function Model—Scenario 1 (Minimizing Gas Consumption in Different Sectors)*

Simplex in linear programming is standard method for solving an optimization problem, typically one involving a function and several constraints expressed as inequalities. The inequalities define a polygonal region, and the solution is typically at one of the vertices. The simplex method is a systematic procedure for testing the vertices as possible solutions. In this research the simplex method was used for model formulation based on goal programming.

Let us take X*ij* the amount of gas which must be allocated in year *t* to the sectors *i =* 1, 2, . . . , 6, which represent the household business, industry, petrochemicals, power plants, injection into oil fields, and exports, respectively, and *j* is the number of the year (2018–2025—*j =* 1, 2, . . . ,8).

Definition of indexes and parameters are as follows:

*j*: time (year);

*i*: The number of the consumer sector;

*Zi*: Objective *I*;

*Pi*: The priority of objective *I*;

*di <sup>+</sup>*: Positive deviation from objective *I*;

*di* −: Negative deviation from objective *I*;

*Pij*: Price of gas consumed in sector *i* in year *j*;

*ACij*: Minimum natural gas consumption of each sector; and

*FCij*: The goal portion of various gas sectors.

The main proposed model is the multi-objective function based on goal programming, as follows:

$$\text{Min } P\_k d\_k^+ ; \quad \forall \ k \in i \tag{1}$$

Subject to:

$$\sum\_{k=1}^{j} \sum\_{l=1}^{j} P\_{kl} \cdot X\_{kl} = Z\_k; \quad \forall \ k \in i \text{ and } \forall \ l \in j \tag{2}$$

$$\sum\_{k=1}^{i} \sum\_{l=1}^{j} P\_{kl} \cdot X\_{kl} \le E\_k; \qquad \forall \ k \in i \text{ and } \forall \ l \in j \tag{3}$$

$$AC\_{ij} \le X\_{ij} \le FC\_{ij}; \qquad \forall \ i = 1, \dots, 6 \text{ and } \forall \ j = 1, \dots, 8$$

$$d\_{k}^{+}\,d\_{k}^{-}=0; \qquad \forall\,k\in i\tag{4}$$

$$X\_{ij\prime} \; d\_k^+ \; d\_k^- \; \geq 0 \; ; \quad \forall \; k \in i \tag{5}$$

As it is clear, this model consists of six objectives. The prior objective is to allocate gas to the export sector in different production years. Therefore, in the equation of the multi-objective model, the related deviations are considered as positive and so for the other sectors. Also, each of the objective functions are assigned and continued to minimize deviations from the goal that are included in the final objective function constraints. Also, since a target function cannot have values greater than or less than the goal, six final constraints are used in this regard. Discussed target functions are presented as below:

*3.2. Objective Function Model Allocation of Household-Commercial Gas Surplus to Other Sectors*

*Fj*: Gas deficit rate in year *j*

*Ej*: Gas household surplus in *j* year

$$\dim Z = \sum\_{i=2}^{6} \sum\_{j=1}^{8} P\_{ij} \cdot X\_{ij} \tag{6}$$

$$\sum\_{i=2}^{6} X\_{ik} \le E\_{k\prime} \ k = 1, \ldots, 8 \tag{7}$$

$$X\_{ij} \ge 0;\tag{8}$$
