1. Introduction

2. Literature Review

3. Problem Statement

4. Implementation

4.1. Allocation and Layout

The placement of equipment in satellite space involves two main parts: first, the allocation of equipment to floors and layers and then their placement in each layer which means that the problem is allocation and layout. Since the objective function involves minimizing mass moments of inertia, the equipment should be arranged as much as possible so that the moments of inertia occupy the minimum possible value in all directions of the coordinate axes (x, y, z). Since the problem consists of two floors and four layers, the placement of equipment in two ways can affect the moments of inertia.

The location of equipment at different levels affects the moment of inertia in the direction of the x and y axes, and the layout of the equipment at each layer can affect the moment of inertia in the direction of the z-axis (I_zz); In other words, to change the moment of inertia in the direction of the x and y axes, the distance of the equipment from the o_mx_m and o_my_m axes, respectively, plays a decisive role. Therefore, if a component is moved between layers, its distance from the mentioned axes changes and affects the moment of inertia in the x and y directions. Conversely, in this displacement, if the equipment distance from the o_mz_m axis remains constant, there will be no change in the moment of inertia in the direction of the z-axis. According to these explanations, proper allocation of equipment to different layers can play an important role in reducing the moment of inertia in the direction of the x and y axes (I_xx and I_yy).

4.1.1. Allocation of Components between Layers

At this stage, all components are assigned to 4 layers in the satellite so that the most optimal state is created for the target function. As previously explained, the assignment of equipment to surfaces can affect the moments of inertia along the x and y axes (I_xx and I_yy). Since Equations (12) and (13) are similar, the calculations are considered for one of them and the result is generalized to the other. As it is evident from Equation (26), To obtain the least possible value for this expression, the first 3 expressions must have the lowest value and the last expression must be at the maximum possible value.

(26)

Before considering the minimization of the above expression, we first discuss the value of z_i. This value indicates the final location of the equipment in terms of height (z dimension) after placement. The mentioned amount is calculated as follows:

$$z_i=\left\{\begin{array}{c}{H}_2+D_t+\frac{h_i}{2}\hspace{1em}if{l}_j=1\\ H_2-\frac{h_i}{2}\hspace{1em}\hspace{1em}if{l}_j=2\\ H_1+D_t+\frac{h_i}{2}\hspace{1em}if{l}_j=3\\ H_1-\frac{h_i}{2}\hspace{1em}\hspace{1em}if{l}_j=4\end{array}\right.$$

(27)

Since the length, width and height (a, b, h) for cuboid equipment and the radius and height (r, h) for cylindrical equipment are fixed, the first three expressions of Equation (26) cannot be altered. Therefore, to make a change in the equation, we come to the following statement:

$${\sum }_{i=1}^{\mathrm{n}}\left({\mathrm{m}}_{\mathrm{i}}\right({{\mathrm{y}}^2}_{\mathrm{i}}+{{\mathrm{z}}^2}_{\mathrm{i}})-{\sum }_{i=1}^{\mathrm{n}}\left({\mathrm{m}}_i\right({{\mathrm{y}}^2}_{\mathrm{m}}+{{\mathrm{z}}^2}_{\mathrm{m}})$$

(28)

First, we consider the first part of the above equation. Given that the allocation of equipment at distinct layers impacts their z-axis coordinates, it suffices to minimize the first half of the formula (28) to minimize the next value:

$${\sum }_{i=1}^{\mathrm{n}}\mathrm{m}\mathrm{i}.{{\mathrm{Z}}^2}_{\mathrm{i}}$$

(29)

We now turn to the second part of Equation (28). According to Equations (10) and (11), which are about the coordinates of the centre of mass in the direction of the axes y and z, the second expression of Equation (28) is written as follows:

$${\sum }_{i=1}^{\mathrm{n}}(m_i{\left(\frac{{\sum }_{i=1}^n{m}_i{y}_i}{{\sum }_{i=1}^n{m}_i}\right)}^2+m_i{\left(\frac{{\sum }_{i=1}^n{m}_i{z}_i}{{\sum }_{i=1}^n{m}_i}\right)}^2)$$

(30)

As can be observed, the denominator of both fractions in expression (30) is the sum of the mass of all the equipment, which is a constant and can be omitted from the maximizing computation. On the other hand, since in this part, the layout of equipment in each layer is not considered and only their location is important to the surfaces, we can omit the first part of term (30), which is related to the coordinates in the direction of the y-axis, so it is sufficient to maximize the following value to maximize the expression:

$${\sum }_{i=1}^{\mathrm{n}}\left(\mathrm{m}\mathrm{i}\right){\left({\sum }_{i=1}^n{m}_i{z}_i\right)}^2)$$

(31)

Since the total mass of all equipment is a fixed value, the only part that will need to be maximized is as follows:

$${\left({\sum }_{i=1}^n{m}_i{z}_i\right)}^2$$

(32)

Here there are two expressions (29) and (31) that one should take the maximum possible value and the other should be minimized:

Min $\sum _{i=1}^{\mathrm{n}}$(m_i.z²_i)

Max ${\left({\sum }_{i=1}^n{m}_i{z}_i\right)}^2$

As it is known, maximizing the expression ${\left({\sum }_{i=1}^n{m}_i{z}_i\right)}^2$ is equivalent to maximizing ${\sum }_{i=1}^n{m}_i{z}_i$ but in the first expression minimization of ${\sum }_{i=1}^n{m}_i{z}_i^2$ is considered; Therefore, according to the second power in the expression of minimization, this part has a higher priority than the part of maximization, and in principle, the heavier mass the equipment is at lower layers, the product of mass in their height is less and then the total moment of inertia in the direction of the x-axis ($I_{xx}$) assumes the least possible value.

On the other hand, according to Equation (11) and considering that the moment of inertia is calculated according to the coordinates of the centre of gravity, it can be concluded that the closer the equipment is to the centre of gravity of the satellite, the moment of inertia in the direction of the x-axis ($I_{xx}$) And y ($I_{yy}$) decreases. Therefore, it makes sense to place more equipment in the middle layers (layers $L_2$ and $L_3$) to minimize the moment of inertia. According to the above explanations, we conclude that to optimize the allocation of equipment at different levels of the satellite, it is best to place heavier equipment at lower layers ($L_3$ and $L_4$ layers) and to accumulate them more at the middle layers (layers $L_2$ and $L_3$). Now, to satisfy the mentioned cases, the following heuristic method is presented.

Heuristic Method to Allocate Equipment to Different Layers

Step 1: Arrange all the equipment at the same time based on height (h) and mass (m). Since all equipment is symmetrical and the mass distribution is assumed to be the same, the centre of mass of each equipment is located in the middle and has a height equal to half the height of the equipment ($\frac{h}{2}$). Therefore, equipment that has a lower height is in priority at the initial and final layer (layers $L_1$ and $L_4$). Conversely, if the height of the equipment is large, placing them in the middle layers (layers $L_2$ and $L_3$) will lower the distance between them and the centre of gravity of the satellite, hence decreasing their moment of inertia along the x-axis ($I_{xx}$) and y-axis ($I_{yy}$). Step 2: Suppose the space available in each layer is displayed as follows:

$$S_{j\hspace{1em}\hspace{1em}j\in 1.2.3.4}$$

Where j is the layer counter. $S_j^{\text{'}}$ is also a ratio of the layer j occupied by the equipment. Because if more than 70% of the area in each layer (0.7${\times S}_j$) is occupied by equipment, location is practically impossible and it is not possible to place the equipment without overlapping. Moreover, since the occupied area of the equipment at levels $L_2$ and $L_3$ must be at least two times the occupied area of the equipment at layers $L_1$ and $L_4$, the following ratio is formed between the surface areas:

$${2\times (S}_1^{\text{'}}+S_4^{\text{'}}){\le S}_2^{\text{'}}+S_3^{\text{'}}\le 1.4\times \left(\frac{1}{S^{\text{'}}-1.4}\right)\times (S_1^{\text{'}}+S_4^{\text{'}})$$

(33)

Where in

$$S^{\text{'}}={\sum }_{j=1}^4{S}_j^{\text{'}}$$

Step 3: After the equipment is arranged according to the height and mass, to minimize the moments of inertia, the equipment with lower height should be placed in $L_1$ and $L_4$ layers and also to satisfy Equation (22), the equipment from the lowest Height and mass are selected and placed in a list called A. This separation is necessary so that this amount of equipment can be assigned to two layers $L_1$ and $L_4$ and the rest of the equipment is naturally assigned to layers $L_2$ and $L_3$. An approximate value means that you can start from Equation (34) to satisfy Equation (33) and return to this step to add the next equipment to List A if the final assignment of Equation (33) is not met.

$${(S}_1^{\text{'}}+S_4^{\text{'}})\ge S^{\text{'}}-1.4$$

(34)

Where $S^{\text{'}}$ is the total occupied area of equipment available in 4 layers? There are only a few cases in which equations (33) and (34) are met. For example, if there are a total of 60 pieces of equipment and it is determined by the initial division that only in cases where $n_1+n_4=\mathrm{19,20,21,22}$ the ratio of equation (22) remains eligible, so for all these feasible cases the next steps are performed.

Step 4: The selected number of equipment (list A) that make up $n_1+n_4$ are sorted by mass from low to high.

Step 5: To allocate equipment at $L_1$ layer, start from the lowest height and the lowest amount of mass and work your way up until the following ratio for the total area of ​​the selected equipment remains feasible:

$$\le S_1^{\text{'}}\le 0.6(S_1^{\text{'}}+S_4^{\text{'}})(0.4(S_1^{\text{'}}+S_4^{\text{'}}$$

(35)

This division is to maintain the balance of equipment division between the first and fourth layers. Therefore, with the first choice that satisfies this ratio, the allocation of equipment to the first level is completed and the values of $n_1$ and $S_1^{\text{'}}$ are determined. Step 6: The rest of the equipment is assigned to layer $L_4$ and the values of $n_4$ and $S_4^{\text{'}}$ are determined.

Step 7: The rest of the equipment (remaining about $\frac{2}{3}$) that makes up $n_2+n_3$ should be assigned to two layers $L_2$ and $L_3$. So equipment is ordered concurrently by height and then mass, from greatest to least value.

Step 8: Since, according to the previous description, the equipment must occupy less than 70% of the area of ​​each layer, the remaining equipment is assigned to the two layers $L_2$ and $L_3$ in such a way that the components are again arranged just based from lowest to the highest value and selected for layer $L_2$until the following ratios are met, and then the remaining equipment is assigned to layer $L_3$.

$$S_2^{\text{'}}\le 0.7S_2$$

(36)

$$S_3^{\text{'}}\le 0.7S_3$$

(37)

$${0.9S}_3^{\text{'}}{\le S}_2^{\text{'}}{\le 1.1S}_3^{\text{'}}$$

(38)

According to the previously mentioned constraints such as equation (33) and the need for a balanced distribution of equipment between layers, with the fulfilment of equations (36) and (37), equation (38) must also be established for these two layers. Thus, all equipment was assigned to all layers, so that equipment with the heavier mass was at layers $L_3$ and $L_4$ and equipment with the lowest heights was at layers $L_1$ and $L_4$.

Step 9: Upon completion of the equipment allocation between layers, the information obtained is given to GAMS software the problem is solved initially and the optimal local solution is generated and stored. The initial solution means that the uncertainty discussion is not considered at the moment and the layout with the details of the equipment on each floor is done in the next section. By performing the presented heuristic method, all the cases that may produce a near-optimal answer are considered and the best answers are determined as inputs to the next stage.

4.1.2. Layout of Equipment in Each Layer

Since the satellite container’s components are cuboid or cylindrical, they can be viewed in two dimensions a rectangle or a circle. To satisfy the non-overlap constraint between equipment, the location of each equipment must be compared with the location of others and no overlap between all components is allowed. Here there are two types of surveys required: circular and rectangular cross-section equipment.

Satisfying the non-overlap constraint for equipment with a circular cross-section is easily achievable. To do this, it is enough to rewrite constraint number 2 for such equipment as follows.

$${(r_i+r_j)}^2\le {\left(x_{\mathrm{i}}-x_{\mathrm{j}}\right)}^2+{\left(y_{\mathrm{i}}-y_{\mathrm{j}}\right)}^2$$

(39)

As indicated by inequality (39), the Euclidean distance between the centres of both circles must be greater than or equal to the sum of the radii of the two circles.

For equipment with a rectangular cross-section, satisfying the non-overlap constraint is not as simple as circular equipment. In this article, we use flexible robust programming to solve this problem. It is assumed that each rectangle can be represented by a circle whose centre is the same as the centre of the rectangle, and after this, no overlap between these circles can be allowed.

Now, if the mentioned circle is considered as a circumscribed circle rectangular, satisfying the non-overlap constraint will increase the distance between the equipment more than necessary, and as a result, the objective function will degrade. If the circle is selected so small that it becomes an inscribed circle for the rectangle, then even though the objective function is greatly reduced and the components are positioned at shorter distances from each other, it leads to the overlap of part of the rectangular components and the non-overlap constraint will be violated.

Therefore, it is essential to find virtual circles which despite satisfying the non-overlap constraint, they can exploit optimal and minimal values for the objective function. To achieve the best hypothetical circle, a novel approach is applied. As shown in Figure 3, the value of sM is considered as a parameter to determine the optimal radius of the hypothetical circle.

If the hypothetical circle reaches its largest radius and converts to a circumscribed circle for the rectangle, the value of sM will be calculated as follows

$$s{M}_i=r\left(i\right)-\frac{\mathrm{m}\mathrm{i}\mathrm{n}(\mathrm{x}\left(\mathrm{i}\right).\mathrm{y}(\mathrm{i}\left)\right)}{2}$$

(40)

Where x and y are the length and width of the rectangle, respectively and r is the radius of its hypothetical circumscribed circle.

Conversely, if the radius of the hypothetical circle is at its lowest value as an inscribed circle for the rectangle, the value of sM tends to be zero. To reach a flexible constraint and utilize a robust concept, fuzzy numbers are an appropriate option used to illustrate such flexibility; Because the nature of fuzzy numbers is in close relationship with flexible robust programming. (Inuiguch and Sakawa 1998).

Robust Flexible Programming Model (RFPM)

To cope with the difficulties associated with overlapping issues for rectangle shapes, a robust flexible programming model is proposed.

Based on RFPM introduced in Pishvaee and Fazli Khalaf (2016), fuzzy numbers are represented as triangular numbers in this paper.

To solve the problem in a flexible programming way, the model is first written as follows:

$$\mathrm{Min}\mathrm{f}\left(\mathrm{x}\right)={\mathrm{I}}_{\mathrm{xx}}+{\mathrm{I}}_{\mathrm{yy}}+{\mathrm{I}}_{\mathrm{zz}}$$

(41)

s.t.

$${fr}_i\tilde{\ge }{r}_i\hspace{1em}\hspace{1em}\hspace{1em}i\in cubicequipments$$

(42)

$${fr}_i\ge r_i\hspace{1em}\hspace{1em}\hspace{1em}i\in cylinderequipments$$

(43)

$${({fr}_i+{fr}_{i+1})}^2\le {\left(x_i-x_{i+1}\right)}^2+{\left(y_i-y_{i+1}\right)}^2\hspace{1em}\hspace{1em}\hspace{1em}i\in cubicequipments$$

(44)

The sign of $\tilde{\ge }$represents the fuzzy version of $\ge$ sign and illustrates that the right-hand side value of the constraint is essentially smaller or similar to the left-hand side. The fuzzy number $\widetilde{sT}$ could be applied to depict the flexible condition of the fuzzy constraint.

Therefore, the model is rewritten as follows:

$$\mathrm{Min}\mathrm{f}\left(\mathrm{x}\right)={\mathrm{I}}_{\mathrm{xx}}+{\mathrm{I}}_{\mathrm{yy}}+{\mathrm{I}}_{\mathrm{zz}}$$

(45)

The

$${fr}_i\ge r_i-\widetilde{sT}\times \left(1-{\alpha }_i\right)i\in cubicequipments$$

(46)

$${fr}_i\ge r_{i}i\in cylinderequipments$$

(47)

$${({fr}_i+{fr}_{i+1})}^2\le {\left(x_i-x_{i+1}\right)}^2+{\left(y_i-y_{i+1}\right)}^{2}i\in cubicequipments$$

(48)

s.t.

The α parameter indicates the minimum level of satisfaction with the flexible constraint. Suppose that the fuzzy number$\widetilde{sT}$ is a triangular fuzzy number represented by three numbers ($\widetilde{sT}=(sP.sM.sO)$) and elucidated by the method demonstrated in Yager (1981) as follows:

$$\widetilde{sT}=\left(sM+\frac{\left(sO-sM\right)-\left(sM-sP\right)}{3}\right)$$

(49)

Based on constraint 2, the flexible programming model in non-fuzzy mode is rewritten as follows:

$$\mathrm{Min}\mathrm{f}\left(\mathrm{x}\right)={\mathrm{I}}_{\mathrm{xx}}+{\mathrm{I}}_{\mathrm{yy}}+{\mathrm{I}}_{\mathrm{zz}}$$

(50)

s.t.

$$\begin{gathered} {fr}_i\ge r_i-{sM}_i+\left(\frac{\left(sO-sM\right)-\left(sM-sP\right)}{3}\right)\times \left(1-{\alpha }_i\right) \\ i\in cubicequipments \end{gathered}$$

(51)

$${fr}_i\ge r_{i}i\in cylinderequipments$$

(52)

$${({fr}_i+{fr}_{i+1})}^2\le {\left(x_i-x_{i+1}\right)}^2+{\left(y_i-y_{i+1}\right)}^{2}i\in cubicequipments$$

(53)

The expression $\left(\frac{\left(sO-sM\right)-\left(sM-sP\right)}{3}\right)\times \left(1-{\alpha }_i\right)$ indicates the possible amount of violation of the flexible constraint. It should be noted that the use of this flexible fuzzy programming method allows other fuzzy numbers to be used in the fuzzy constraint, and other fuzzy ranking methods can also be used to de-fuzzy the existing uncertain parameters in soft constraints.

Also, using α-cuts as the degree of violation in soft constraints can lead us to different fuzzy solutions that help decision-makers to compare different outputs and to achieve better solutions with sensitivity analysis.

In the fuzzy flexible planning model, the allocation of the minimum level of satisfaction to the flexible constraints (0≤α≤1) must be determined by the decision maker. In other words, this method should be reactive and the decision maker achieves different results by manually changing the minimum level of satisfaction and extracting the best solutions for fuzzy parameters.

The desirability of the choices at each stage is determined by the output of the model. The drawback of this method is that there is no guarantee to achieve the optimal level of satisfaction. For this purpose, the RFPM is represented as follows.

$$\mathrm{Min}\mathrm{f}\left(\mathrm{x}\right)={\mathrm{I}}_{\mathrm{xx}}+{\mathrm{I}}_{\mathrm{yy}}+{\mathrm{I}}_{\mathrm{zz}}+\gamma \times \left[{sM}_i+\left(\frac{\left(sO-sM\right)-\left(sM-sP\right)}{3}\right)\right]\times (1-\alpha )$$

(54)

s.t.

$$\begin{gathered} {fr}_i\ge r_i-\left({sM}_i+\left(\frac{\left(sO-sM\right)-\left(sM-sP\right)}{3}\right)\right)\times \left(1-{\alpha }_i\right) \\ i\in cubic\hspace{1em}\hspace{1em}\hspace{1em}equipments \end{gathered}$$

(55)

$${fr}_i\ge r_i\hspace{1em}i\in cylinder\hspace{1em}equipments$$

(56)

$${({fr}_i+{fr}_{i+1})}^2\le {\left(x_i-x_{i+1}\right)}^2+{\left(y_i-y_{i+1}\right)}^2\hspace{1em}\hspace{1em}\hspace{1em}i\in cubic\hspace{1em}equipments$$

(57)

$$0\le \alpha \le 1\hspace{1em}\hspace{1em}\hspace{1em}.\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{I}\mathrm{x}\mathrm{x}.\mathrm{I}\mathrm{y}\mathrm{y}.\mathrm{I}\mathrm{z}\mathrm{z}.fr\ge 0$$

(58)

In this model, in addition to minimizing moments of inertia, a new section has been added to the objective function, which depicts the total penalty cost of possible non-compliance with the flexible constraints. In essence, this phase controls the feasibility and robustness of flexible constraints.

In other words, this expression shows the difference between the minimum and maximum possible value for flexible constraint as follows:

$$\left({sM}_i+\left(\frac{\left(sO-sM\right)-\left(sM-sP\right)}{3}\right)\right)\times \left(1-{\alpha }_i\right)=r_i-\left[r_i-\left({sM}_i+\left(\frac{\left(sO-sM\right)-\left(sM-sP\right)}{3}\right)\right)\times \left(1-{\alpha }_i\right)\right]\hspace{1em}\hspace{1em}\hspace{1em}i\in cubic\hspace{1em}equipments$$

(59)

In this model, the penalty cost for each unit of violation of the soft constraint is also considered, which is represented by the parameter γ. In the RFPM, unlike the initial flexible programming model, the minimum level of satisfaction (α) is no longer a parameter and is determined by the model as a variable.

Therefore, by solving the model in one time, the optimal value of this variable can be achieved and there is no need to repeat the experiments. Because the objective function of the model tries to achieve a balance between the robustness cost (the last expression of the objective function) and the overall performance of the system including other expressions of the objective function (such as moments of inertia), the proposed model is called a realistic RFPM.

It should be noted that the parameter γ is an important value and its value is determined based on the application and the subject of discussion. Here, for example, for cuboid components that need to be more distant from other equipment, the value of the penalty parameter can be assigned much more, which in this case the variable α tends to increase near 1, and therefore the soft constraint for this equipment becomes like cylindrical components.

It should be noted that the use of the penalty parameter in the objective function helps to optimize the variable of minimum satisfaction level (α) and prevents dirthe ect involvement of decision-makers in quantifying this variable.

5. Implication and Sensitivity Analysis

6. Conclusion and Future Work

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