**Appendix A. Procedure for Determining the Matrix of Acceptable Variants of the HVAC System**

The matrix of normalized constant parameters is defined as:


(A1)

where:

1/—correlation with cleanliness class CS applies to special cases (DIN 1946-4:2018- 09 [43]).

**X**I\*—binary matrix of normalized constant parameters xi of HVAC system,

i = 1 . . . I\*—number of normalized constant parameter,

I\*—number of normalized constant parameters of HVAC system,

xi—constant parameter of HVAC system,

xi—vector of normalized constant parameters of HVAC system.

Matrix **W** of all possible variants of combinations of decision variables for normalizing all constant parameters of HVAC system is defined as:

where:

**W**—binary matrix of all possible variants of combinations of decision variables values xj for normalizing all constant parameters xi of HVAC system,

xj—decision variable of HVAC system,

j—index of decision variable of HVAC system,

j\*—index of decision variable of HVAC system from a universal set in Tables A2 and A3 [37],

mi or r(i,mi)—index of mi- or r(i,mi)-variant of combinations of decision variables values xj for normalizing xi-constant parameter of HVAC system,

Mi—number of all possible variants of combinations of decision variables values xj for normalizing xi-constant parameter of HVAC system,

wmi or wr(i,mi)—binary vector defining mi- or r(i, mi)-variant of combinations of decision variables values xj for normalizing xi-constant parameter of HVAC system (matrix element **W**).

$$\mathbf{m\_i} = 1 \; \dots \; \mathbf{M\_i}$$

$$\mathbf{r}(\mathbf{i}, \mathbf{m}\_{\mathbf{i}}) = \begin{cases} \mathbf{m}\_{\mathbf{i}\prime} \text{ i} = 1\\ \sum\_{\mathbf{k}=1}^{\mathbf{i}=1} (\mathbf{M}\_{\mathbf{k}}) + \mathbf{m}\_{\mathbf{i}\prime} \text{ i} = 2 \dots \text{I}^{\*} \end{cases} \tag{A3}$$

whereby:

r(i, mi)=1... M

$$\mathbf{M} = \mathbf{r}(\mathbf{I}^\*, \mathbf{M}\_i) = \sum\_{i=1}^{\mathbf{I}^\*} \mathbf{M}\_i \tag{A4}$$

(A2)

The rows of the **W** matrix are variants (mi or r(i,mi)) of combinations of decision variables for the normalization of successive of the constants parameters included in the matrix **X**I\*:

(i = 1, x1 ≡ tR)—two variants including the CAV (Constant Air Volume) system with AHU with heat recovery (1.51—here cumulative variable), primary heater (option), cooler and secondary heater.

(i = 2, x2 ≡ ϕR)—two variants including AHU with cooler, secondary heater and steam humidifier in the unit or in the duct (option),

(i = 3, x3 ≡ kd)—one variant with the hygienic design of AHU, three stages of filtration at the supply, 3rd stage filter integrated with a supply diffuser,

(i = 4, x4 ≡ Cs)—two variants as for (i = 3, x3 ≡ kd) and a steam humidifier in the AHU unit or in the duct,

(i = 5, x5 ≡ α)—five variants depending on the number of units in the cascade (one, two or three) and the recirculation option:

x1.57a—internal (room) recirculation, cumulative variable,

x1.57b—external recirculation (in front of the AHU),

x1.57c—without recirculation,

Matrix **W** is of type (15,27). Matrix with limitations **G**<sup>i</sup> for matrix **W** is of type (15,15) and is defined as:


(A5)

where:

**G**i —binary matrix of limitation conditions for matrix **W**,

gr(i,mi)r(i,mi)—binary value in matrix of elimination of unnecessary decision variables **G**<sup>i</sup> for r(i,mi)—variant of combinations of decision variables values xj for normalizing xi-constant parameter defined by vector wr(i,mi) in matrix **W**.

The words gr(i,mi)r(i,mi) correspond to the eliminated variants in matrix W and result from limitations. Taking into account the notations in the table of restrictions A9 [37], the assignment here is as follows: r(i,mi)=2 − gT6, gT19, r(i,mi)=4 − gT11, gT28, r(i,mi)=7 − gT11, gT28.

Interpretation of elements gr(i,mi)r(i,mi) = 0—of eliminated variants in matrix W is shown in Table A1.

After taking into account the constraints, redundant decision variables can be identified, which in the adopted methodology are eliminated in order to reduce the description of the mathematical problem.


**Table A1.** Interpretation of elements gr(i,mi)r(i,mi ) = 0 in matrix **G**.

\*/ Designation of restrictions according to [37].

Matrix G<sup>j</sup> of type (27,27) of elimination of unnecessary decision variables is defined as:

where:

**G**j —binary matrix of elimination of unnecessary decision variables for matrix (**G**<sup>i</sup> × **<sup>W</sup>**),

Matrix Wg—after considering limitations and eliminating unnecessary decision variables is obtained as the product of matrices defined as:

$$\mathbf{W^{\overline{g}} = G^{\mathrm{i}} \times W \times G^{\mathrm{j}}} \tag{A7}$$

After eliminating zero rows and columns—matrix **W**<sup>g</sup> is defined as:


where:

**W**g—binary matrix of acceptable variants of combinations of decision variables values xj for normalizing all constant parameters xi of HVAC system,

wmg i or wr(i,m<sup>g</sup> i ) —binary vector defining mi- or r(i, mi)-variant r(i,m<sup>g</sup> <sup>i</sup> )-acceptable variant) of combinations of decision variables values xj for normalizing xi-constant parameter of HVAC system (matrix element **W**g),

mg <sup>i</sup> or r(i,m<sup>g</sup> <sup>i</sup> )—index of <sup>m</sup><sup>g</sup> <sup>i</sup> - or r(i,m<sup>g</sup> <sup>i</sup> )-variant of combinations of decision variables values xj for normalizing xi-constant parameter of HVAC system, after considering limitations.

Matrix **X**<sup>J</sup> of all possible variants of HVAC system is defined as:


(A9)

(A8)

where:

**XJ** —binary matrix of acceptable variants of HVAC system for normalizing constant parameters,

xn—binary vector defining the n-variant of structure of HVAC system used for normalizing all constant parameters,

xmg <sup>i</sup> <sup>n</sup> or xr(i,m<sup>g</sup> <sup>i</sup> )n—binary value in the n-variant structures of HVAC system for <sup>m</sup><sup>g</sup> i or r(i,m<sup>g</sup> <sup>i</sup> )-variant of combinations of decision variables values for normalizing xiconstant parameter defined by vector wmg i or wr(i,mg <sup>i</sup> ) in matrix **<sup>W</sup>**g, matrix element **<sup>X</sup>**<sup>J</sup> , Mg <sup>i</sup> —number of all possible variants of combinations of decision variables values xj for normalizing xi-constant parameter of HVAC system, after considering limitations.

Vector xn is created by selecting from matrix **W**<sup>g</sup> a single variant of combinations of decision variables wr(i,mg <sup>i</sup> ) <sup>=</sup> wmg i from ranges mg <sup>i</sup> <sup>=</sup> <sup>1</sup> ...Mg <sup>i</sup> for each of normalized constant parameters xi,I=1 ... I \* and by assigning this variant value xr(i,mg <sup>i</sup> )<sup>=</sup> 1, and by assigning the remaining variants from range m<sup>g</sup> <sup>i</sup> <sup>=</sup> <sup>1</sup> ...M<sup>g</sup> <sup>i</sup> each of normalized constant parameters xi value xr(i,m<sup>g</sup> <sup>i</sup> ) = 0.

The number of all, defined in this particular way, possible vectors xn—of all possible variants of HVAC system for cleanroom equals:

$$\mathbf{N} = \prod\_{i=1}^{8} \mathbf{M}\_i^{\mathbb{G}} = 1 \cdot 1 \cdot 1 \cdot 1 \cdot 3 \cdot 5 = 15 \tag{A10}$$

By identifying the values of the words of the vector xn (column of the **X**<sup>J</sup> matrix), the structure of the HVAC system is identified for the normalization of successive constant parameters.

### *Limiting Conditions, Set of Acceptable Variants*

Matrix **G** of type (15,15) of limiting conditions for matrix **X**<sup>J</sup> of all possible variants of HVAC system for cleanroom is defined as:

(A11)

where:

**G**—binary matrix of limitation conditions for matrix **X**<sup>J</sup> ,

gnn—binary value in matrix with limitations **G** for the n-variant of structure of HVAC system defined by vector xn in matrix **X**<sup>J</sup> ,

The words gn,n = 0 correspond to eliminating variants (vectors xn) in the matrix **X**<sup>J</sup> . Variants that are internally inconsistent or identical to other variants are eliminated.

Interpretation of elements gn,n = 0—of eliminated vectors xn in matrix **<sup>X</sup>**<sup>J</sup> —is shown in Table A2.

**Table A2.** Interpretation of vectors gn,n = 0 in matrix **<sup>X</sup>**<sup>J</sup> .


Matrix **X** of acceptable variants of the structure of HVAC system xng for cleanroom is the product of:

$$\mathbf{X} = \mathbf{X}^{\dagger} \times \mathbf{G} \tag{A12}$$

after eliminating zero columns. Matrix **X** is defined as:


(A13)

where:

X—binary matrix of acceptable variants of HVAC system for normalizing constant parameters,

(xmg <sup>i</sup> ng or xr(i,m<sup>g</sup> <sup>i</sup> )n<sup>g</sup> )—binary value in the ng-acceptable variant structures of HVAC system for m<sup>g</sup> i, mg —variant of combinations of decision variables values for

<sup>i</sup> - or r i nor-malizing xi-constant parameter defined by vector wmg i or wr(i,mg <sup>i</sup> ) in matrix Wg, matrix element XJ ,

xng—binary vector defining the ng-acceptable variant of structure of HVAC system used for normalizing all constant parameters,

n(ng)—index of the n-variant (of the ng-acceptable variant) of HVAC system.

By analyzing and interpreting backwards values of elements of subsequent matrices X, XJ and Wg defining a function of constant parameters and decision variables, it is possible to identify the full structure of acceptable variants of HVAC system.
