Decision on the selection of the best height-diameter ratio for the optimal design of 13,000 m³ oil storage tank

Abstract

The decision on the selection of the best height-diameter ratio for the optimal design of oil storage tanks is a difficult task because there are so many criteria to be considered in selecting the most appropriate alternative. In this study, Multi-Criteria Decision Analysis (MCDA) was used by adopting Criteria Importance Through Inter-criteria Correlation (CRITIC) in determining the objective weight of the criteria. The six criteria considered in this study are: Area occupied by the tank (m²), Weight (kg), Wind Moment (Nm), Seismic Ringwall Moment RWM (Nm), Base shear (N) and Estimated Cost (₦) and the objective weights assigned to them by CRITIC are 0.2356, 0.2607, 0.1238, 0.1249, 0.1196 and 0.1354, respectively. The Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) was then used to rank each of the alternatives based on the objective weight obtained through CRITIC. The objective of obtaining an optimal design of a 13,000 m³ storage tank that will occupy less space, weighty to resist wind load and seismic overturning at a moderate (low) cost was achieved by Alternatives A₄ and A₅ (height-diameter ratios of 0.8 and 0.9). To validate the ranking result obtained by the TOPSIS method, another MCDA method, VIseKriterijumska Optimizacija I Kompromisno Resenje (VIKOR) was employed to compare the results. Validation of the TOPSIS method by VIKOR method revealed that there was high consistency in the results obtained, with a very high Spearman’s rank correlation coefficient of 0.8727. Sensitivity analysis was also done to ascertain the robustness of the results.

Keywords:

• Storage tank
• TOPSIS
• CRITIC
• design
• oil & gas
• VIKOR

PUBLIC INTEREST STATEMENT

Decision making involves identifying and choosing alternatives to get the best solution from a pool of available options based on different criteria and expectations of the decision-maker. Decisions are made in every facet of life, even in oil & gas. Multi-Criteria Decision Analysis (MCDA) is a potent tool used in decision making. The decision on the selection of the best height-diameter ratio for the optimal design of 13,000 m³ oil storage tanks is a difficult task because there are so many criteria to be considered in selecting the most appropriate alternative. MCDA adopted in this study was Criteria Importance Through Inter-criteria Correlation (CRITIC) to determine the objective weight of the criteria. Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) and VIKOR were then used to rank each alternative. The decision on designing a storage tank that will occupy less space, weighty to resist wind load and seismic overturning at a moderate cost was achieved by height-diameter ratio 0.8.

1. Introduction

Upright cylindrical tanks used in oil storage are designed to conform to standard and codes such as American Petroleum Institute (API) Standards, for example, API 650, API 620, and British Standard BS EN 14015. Tank design codes and standards are the documented facts and outcomes of the decades of work experience by some dedicated professionals. The full adoption of these standards helps in ensuring that the tanks can withstand the rigours of the stress conditions subjected to (Ammar et al., 2018; Okpala & Jombo, 2012).

Storage facility such as storage tanks is a key consideration in the oil sector because crude oil is stored temporarily before it is being exported or processed into refined products. Even in both the midstream and downstream sectors of oil and gas, the importance of storage tanks cannot be over-emphasized (Enarevba et al., 2016; Agboola et al., 2019; Agboola et al., 2017). For Nigeria to attain her desire of Vision 2020, all efforts should be channeled to the attainment of self-sufficiency in meeting her petroleum products’ demands through local refining. However, refineries require a high number of storage tanks for their operations. Samuel (2013) stated that this could be realized through proper and adequate maintenance of all her local refineries and by building new ones. As a result of expansion in the oil sector, there is a need for indigenous design and installation of bulk storage tanks in refineries and depots to adequately store these petroleum products instead of depending on the turnkey design from the foreign experts to save the country’s foreign exchange.

The choice of the type of upright cylindrical tanks depends on the number of factors. These factors are the safety requirement, environmental factor, nature of the fluid to be stored, available space, height-diameter ratio and operational cost (EEMUA Publication, 2003; John, 2004). In designing an upright cylindrical storage tank, the height-diameter ratio is an important factor that should be considered to know its effects on the design output parameters such as the Area occupied by the tank (m²), Weight (kg), Wind Moment (Nm), Seismic Ringwall Moment RWM (Nm), Base shear (N) and Estimated Cost (₦). Though API 650 does not give the actual value of height-diameter ratio but tank design experts stated that the height-diameter ratio ranges from 0.5 to 1.5 (Equipment Design Lecture 11 Tanks, 2019). The aim of this study is to select the best height-diameter for the optimum design of oil storage tank using Multi-Criteria Decision Analysis (MCDA). MCDA deals with decisions involving the choice of the best alternative from several available options, subject to several criteria or attributes which may be tangible or intangible (Stanojković & Radovanović, 2017). The objective weight of each criterion is defined based on the importance of each criterion relative to other criteria. Objective weights are numbers that are either subjectively selected or objectively derived for each criterion (Madić et al., 2015). Among the subjectively selected weights are Analytic Hierarchy Process and the Analytic Network Process. These are the two traditional MCDA methods developed by Saaty (Madić & Radovanović, 2015; Milić & Župac, 2012). However, the two methods possess inherent deficiencies that affect the rankings in the real-world application (Diakoulaki et al., 1995; Naga et al., 2016). This is because they failed to relate each criterion concurrently with other criteria (Balli & Korukoglu, 2009). To improve these deficiencies, another method of determining objective weight was developed called Criteria Importance Through Inter-criteria Correlation (CRITIC). CRITIC and Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) were employed by Stanojković and Radovanović (Stanojković & Radovanović, 2017) in the selection of solid carbide end mill for machining aluminum 6082-T6. An extensive review of the applications of the TOPSIS method in various fields of technical and managerial decision-making could be found in Babatunde and Ighravwe (2019), Bagheri et al. (2018), and Tlig and Rebai (2017); Behzadian et al. (2012) (Babatunde & Ighravwe, 2019; Bagheri et al., 2018; Behzadian et al., 2012; Tlig & Rebai, 2017).

2. Materials and methods

The Storage tank considered in this study is a 13,000 m³ PMS tank designed in accordance with API 650 and to be erected at Apapa, Nigeria in total compliance to the DPR guidelines. The values of design output parameters based on various height-diameter ratios (Alternatives) are tabulated and presented in Table 1. The criteria considered were space occupied, the overall weight of the tank, resistance to wind load, resistance to seismic ringwall moment, base shear and estimated fabrication cost. Then, the CRITIC was used to determine the objective weight of each criterion. Thereafter, the ranking of each alternative was achieved by both the TOPSIS and VIKOR (VIseKriterijumska Optimizacija I Kompromisno Resenje) methods. The two results (ranks) obtained from both the TOPSIS and VIKOR methods were compared. VIKOR and TOPSIS methods were used because they are based on an aggregating function representing “closeness to the ideal solution” which originates in the compromise programming method (San Cristóbal, 2011).

Table 1. Height-diameter ratios and design output parameters

Height-Diameter ratio	Criteria
	Area occupied (m^2)	Weight of tank (kg)	Wind moment (Nm)	Seismic Ringwall moment (Nm)	Base shear (N)	Estimated cost (₦)
0.5	773.61	271,213.25	7,398,007.96	138,941,492.16	20,392,184.87	171,271,170
0.6	687.66	271,748.34	8,778,819.38	173,832,828.30	22,877,103.97	171,609,076
0.7	618.89	272,649.99	10,236,850.57	209,646,707.96	24,986,591.72	172,178,470
0.8	562.63	276,305.28	11,768,181.85	251,334,864.56	26,299,403.95	174,486,783
0.9	562.63	276,305.28	11,768,181.85	251,334,864.56	26,299,403.95	174,486,783
1.0	515.74	284,786.45	13,369,439.65	293,779,079.78	27,365,058.39	179,842,644
1.1	476.07	287,172.24	15,037,679.73	335,676,349.98	28,216,805.15	181,349,272
1.2	442.06	295,095.48	16,770,302.75	377,968,944.07	28,940,257.82	186,352,795
1.3	412.59	306,716.21	18,564,991.54	420,561,372.12	29,562,710.73	193,691,284
1.4	412.59	306,716.21	18,564,991.54	420,561,372.12	29,562,710.73	193,691,284
1.5	386.81	317,033.32	20,419,663.25	462,973,356.98	30,091,816.42	200,206,539

The six (6) criteria used in evaluating the alternatives were determined from the standardized Equations (1(1) $\text{rm{Area occupied by the tank }}\left({\mathrm{m}}^2\right)=\frac{\pi D^2}{4}$(1) –6(6) $\text{hbox{rm{Estimated cost }}left({rm{N}} right) = left[{left({{{pi {D^2}{t\_b}{rho \_b}} over 4}} right) + left({pi D{rho \_s}mathop {mathop {{{mathop sum nolimits^ }^}^}limits^ }limits\_n^{i = 1} {h\_i}{t\_i}} right) + left({{{pi D{rho \_r}{t\_r}left({4r\_h^2 + {D^2}} right)} over 8}} right)} right] times {C\_u}}$(6) )

(1) $\text{rm{Area occupied by the tank }}\left({\mathrm{m}}^2\right)=\frac{\pi D^2}{4}$(1)

(2) $\text{hbox{rm{Weight of the tank }}left({{rm{kg}}} right){ } = left[{left({{{pi {D^2}{t\_b}{rho \_b}} over 4}} right) + left({pi D{rho \_s}mathop {mathop sum nolimits^ }limits\_n^{i = 1} {h\_i}{t\_i}} right) + left({{{pi D{rho \_r}{t\_r}left({4r\_h^2 + {D^2}} right)} over 8}} right)} right]}$(2)

(3) $\text{rm{Wind moment }}\left(\mathrm{N}\mathrm{m}\right)=\left(w_{r}H{r}_h\right)+\left(w_{s}H\right)$(3)

(4) $\text{rm{Seismic Ringwall moment }}\left(\mathrm{N}\mathrm{m}\right)=\sqrt{{\left[A_i\left(w_i{X}_i+w_s{l}_s+w_r{l}_r\right)\right]}^2+{\left[A_c\left(w_c{X}_c\right)\right]}^2}$(4)

(5) $\text{rm{Base Shear }}\left(\mathrm{N}\right)=\sqrt{V_i^2+V_c^2}$(5)

(6) $\text{hbox{rm{Estimated cost }}left({rm{N}} right) = left[{left({{{pi {D^2}{t\_b}{rho \_b}} over 4}} right) + left({pi D{rho \_s}mathop {mathop {{{mathop sum nolimits^ }^}^}limits^ }limits\_n^{i = 1} {h\_i}{t\_i}} right) + left({{{pi D{rho \_r}{t\_r}left({4r\_h^2 + {D^2}} right)} over 8}} right)} right] times {C\_u}}$(6)

where d is the nominal diameter of the tank, t_b is the thickness of bottom plate, ${\rho }_b$ is the density of bottom plate, ${\rho }_s$ is the density of shell plate, ${\rho }_r$ is the density of roof plate, h_i is the height of each course plate, t_i is the thickness of each plate, r_h is the roof height, H is the nominal tank height, w_r is the wind load on the roof, w_s is the wind load on the shell, V_i is the impulsive shear force, V_c is the convective shear force, A_i is the impulsive acceleration, A_c is the convective acceleration, w_i is the impulsive wind load, w_c is the impulsive wind load, w_s is the wind load on the shell, w_r is the wind load on the roof, x_i is the impulsive centre of action, x_c is the convective centre of action, l_s is the moment arm of the shell, l_r is the moment arm of the roof.

3. CRITIC

Determining the objective weights using the CRITIC method was done by adopting these six steps (Madić et al., 2015; Madić & Radovanović, 2015).

(7) ${\mathrm{r}}_{\mathrm{i}\mathrm{j}}=\frac{x_{ij}-x_j^{worst}}{x_j^{best}-x_j^{worst}}$(7)

1. Determination of the elements of normalized decision matrix r_ij using Equation 7(7) ${\mathrm{r}}_{\mathrm{i}\mathrm{j}}=\frac{x_{ij}-x_j^{worst}}{x_j^{best}-x_j^{worst}}$(7) .

where $x_j^{worst}$ = maximum of non-beneficial criteria and minimum of beneficial criteria (x_ij, i = 1, …,m) and $x_j^{best}$ = minimum of non-beneficial criteria and maximum of beneficial criteria (x_ij, i = 1, …, m)

(8) $\text{{{rm{sigma }}\_{rm{j}}}{rm{ = }}sqrt {{1 over n}{{left({mathop {mathop sum nolimits^ }limits\_m^{i = 1} {r\_{ij}} - bar r} right)}^2}}}$(8)

2. Based on the value r_ij a vector of criteria was formed, each vector has a standard deviation σ_j, which represents the degree of deviation of alternatives for a given criterion. Standard deviation is calculated using Equation (8)(8) $\text{{{rm{sigma }}\_{rm{j}}}{rm{ = }}sqrt {{1 over n}{{left({mathop {mathop sum nolimits^ }limits\_m^{i = 1} {r\_{ij}} - bar r} right)}^2}}}$(8)

where: n is a number of elements and $\bar{r}$ is an arithmetic mean.

3. Formation of a symmetric matrix n x n with elements R_ij, which represent linear correlation coefficients r_j, r_k, and n represents the number of alternatives, using Equation (9)(9) $\text{{{rm{R}}\_{{rm{ij}}}}{rm{ = }}{{nmathop sum nolimits^ {r\_j}{r\_k} - mathop sum nolimits^ {r\_j}mathop sum nolimits^ {r\_k}} over {sqrt {nmathop sum nolimits^ r\_j^2 } - {{left({mathop sum nolimits^ {r\_j}} right)}^2}. sqrt {nmathop sum nolimits^ r\_k^2 - {{left({mathop sum nolimits^ {r\_k}} right)}^2}} }}}$(9) . When there is a large discrepancy between the values of attributes for criteria j i k, it is the lower value of the coefficient R_ij that is considered (Madić & Radovanović, 2015; Milić & Župac, 2012).

(9) $\text{{{rm{R}}\_{{rm{ij}}}}{rm{ = }}{{nmathop sum nolimits^ {r\_j}{r\_k} - mathop sum nolimits^ {r\_j}mathop sum nolimits^ {r\_k}} over {sqrt {nmathop sum nolimits^ r\_j^2 } - {{left({mathop sum nolimits^ {r\_j}} right)}^2}. sqrt {nmathop sum nolimits^ r\_k^2 - {{left({mathop sum nolimits^ {r\_k}} right)}^2}} }}}$(9)

4. Determination of the rates of the conflict criteria (Diakoulaki et al., 1995) using Equation (10)(10) $\mathrm{R}=1-{\mathrm{r}}_{\mathrm{j}\mathrm{k}}$(10)

(10) $\mathrm{R}=1-{\mathrm{r}}_{\mathrm{j}\mathrm{k}}$(10)

5. Establishing the quantity of the information in relation to each criterion using Equation (11)(11) ${\mathrm{C}}_{\mathrm{j}}={\sigma }_j\sum _{k=1}^n\left(1-R_{jk}\right)$(11)

(11) ${\mathrm{C}}_{\mathrm{j}}={\sigma }_j\sum _{k=1}^n\left(1-R_{jk}\right)$(11)

6. Normalization of the value of C_j using Equation (12)(12) ${\mathrm{w}}_{\mathrm{j}}=\frac{C_j}{{\sum }_{j=1}^n{C}_j}$(12) to obtain the objective weight.

(12) ${\mathrm{w}}_{\mathrm{j}}=\frac{C_j}{{\sum }_{j=1}^n{C}_j}$(12)

4. TOPSIS

1. Determining the objective, alternatives and criteria.

The objective is to evaluate the eleven (11) alternatives (height-diameter ratio in designing oil storage tanks) based on the criteria [Area Occupied (m²), Weight (kg), Wind Moment (Nm), Seismic Ringwall moment (Nm), Base shear (N), Estimated Cost (₦)]

2. Based on the given values, the decision matrix X is defined by Equation (13)(13) $\mathrm{X}=\left[x_{ij}\right]=\left|\begin{array}{ccc}{x}_{11}& x_{12}& x_{1n}\\ x_{21}& x_{22}& x_{2n}\\ x_{m1}& x_{m2}& x_{mn}\end{array}\right|$(13)

(13) $\mathrm{X}=\left[x_{ij}\right]=\left|\begin{array}{ccc}{x}_{11}& x_{12}& x_{1n}\\ x_{21}& x_{22}& x_{2n}\\ x_{m1}& x_{m2}& x_{mn}\end{array}\right|$(13)

3. Normalization of the decision matrix is done by adopting Equation 14(14) ${\mathrm{r}}_{\mathrm{i}\mathrm{j}}=\frac{x_{ij}}{\sqrt{{\sum }_{n=1}^m{x}_{ij}^2}}$(14) (Balli & Korukoglu, 2009; Naga et al., 2016).

(14) ${\mathrm{r}}_{\mathrm{i}\mathrm{j}}=\frac{x_{ij}}{\sqrt{{\sum }_{n=1}^m{x}_{ij}^2}}$(14)

4. Calculate the values of the objective weight coefficients with Equation (15)(15) $\sum _j^n{w}_j=1$(15)

(15) $\sum _j^n{w}_j=1$(15)

5. Determine the weighted decision-making matrix using Equation (16)(16) ${\mathrm{v}}_{\mathrm{i}\mathrm{j}}={\mathrm{r}}_{\mathrm{i}\mathrm{j}}.{\mathrm{w}}_{\mathrm{j}}$(16) , which represents the multiplication of elements of a column of the normalized matrix with appropriate objective weight coefficients obtained from Equation (12)(12) ${\mathrm{w}}_{\mathrm{j}}=\frac{C_j}{{\sum }_{j=1}^n{C}_j}$(12) .

(16) ${\mathrm{v}}_{\mathrm{i}\mathrm{j}}={\mathrm{r}}_{\mathrm{i}\mathrm{j}}.{\mathrm{w}}_{\mathrm{j}}$(16)

6. Identify the positive and negative ideal solution based on Equations (17)(17) ${\mathrm{V}}^{+}=\left\{max\left(v_{ij}\right),jϵJ,min\left(v_{ij}\right),jϵJ,i=1,\text{ldotsm}\right\}=\left\{V_1^{+},V_2^{+},\dots V_n^{+}\right\}$(17) and (18(18) ${\mathrm{V}}^{-}=\left\{min\left(v_{ij}\right),jϵJ,max\left(v_{ij}\right),jϵJi=1,\text{ldotsm}\right\}=\left\{V_1^{+},V_2^{+},\dots V_n^{+}\right\}$(18) )

(17) ${\mathrm{V}}^{+}=\left\{max\left(v_{ij}\right),jϵJ,min\left(v_{ij}\right),jϵJ,i=1,\text{ldotsm}\right\}=\left\{V_1^{+},V_2^{+},\dots V_n^{+}\right\}$(17)

(18) ${\mathrm{V}}^{-}=\left\{min\left(v_{ij}\right),jϵJ,max\left(v_{ij}\right),jϵJi=1,\text{ldotsm}\right\}=\left\{V_1^{+},V_2^{+},\dots V_n^{+}\right\}$(18)

7. Calculate the Euclidean separation distance of each competitive alternative from the positive and negative solution using Equations (19)(19) $S_i^{+}=\sqrt{\sum _{j=1}^n{\left(v_{ij}-V_j^{+}\right)}^2}$(19) and (20(20) $\text{S\_i^ - = sqrt {mathop {mathop sum nolimits^ }limits\_n^{j = 1} {{left({{v\_{ij}} - V\_j^ - } right)}^2}}}$(20) )

(19) $S_i^{+}=\sqrt{\sum _{j=1}^n{\left(v_{ij}-V_j^{+}\right)}^2}$(19)

(20) $\text{S\_i^ - = sqrt {mathop {mathop sum nolimits^ }limits\_n^{j = 1} {{left({{v\_{ij}} - V\_j^ - } right)}^2}}}$(20)

8. Measure the relative closeness of each location of the ideal solution P_i. For each competitive alternative the relative closeness of the potential location with respect to the ideal solution is calculated by using Equation (21)(21) ${\mathrm{P}}_{\mathrm{i}}=\frac{S_i^{-}}{S_i^{+}+S_i^{-}}$(21)

(21) ${\mathrm{P}}_{\mathrm{i}}=\frac{S_i^{-}}{S_i^{+}+S_i^{-}}$(21)

9. The order of the alternatives is done according to the value of P_i obtained in Equation (21)(21) ${\mathrm{P}}_{\mathrm{i}}=\frac{S_i^{-}}{S_i^{+}+S_i^{-}}$(21)

5. VIKOR

VIKOR (VIseKriterijumska Optimizacija I Kompromisno Resenje) is a Serbian language which means Multicriteria Optimization and Compromise Solution was practically applied in 1998 (Opricovic, 1998). The procedures for VIKOR are as itemized in Equations (22(22) $\mathrm{I}=\left[I_{ij}\right]=\left|\begin{array}{ccc}{I}_{11}& I_{12}& I_{1n}\\ I_{21}& I_{22}& I_{2n}\\ I_{m1}& I_{m2}& I_{mn}\end{array}\right|$(22) –26(26) $Q_i=v\left[\frac{S_i-{\left(S_i\right)}_{min}}{{\left(S_i\right)}_{max}-{\left(S_i\right)}_{min}}\right]+\left(1-v\right)\left[\frac{R_i-{\left(R_i\right)}_{min}}{{\left(R_i\right)}_{max}-{\left(R_i\right)}_{min}}\right]$(26) )

1. Establishment of a decision matrix as shown in Equation (22)(22) $\mathrm{I}=\left[I_{ij}\right]=\left|\begin{array}{ccc}{I}_{11}& I_{12}& I_{1n}\\ I_{21}& I_{22}& I_{2n}\\ I_{m1}& I_{m2}& I_{mn}\end{array}\right|$(22)

(22) $\mathrm{I}=\left[I_{ij}\right]=\left|\begin{array}{ccc}{I}_{11}& I_{12}& I_{1n}\\ I_{21}& I_{22}& I_{2n}\\ I_{m1}& I_{m2}& I_{mn}\end{array}\right|$(22)

2. Normalization of decision matrix using Equation (23)(23) $f_{ij}=\frac{I_i^j}{\sqrt{{\sum }_{i=1}^m{\left(I_i^j\right)}^2}}$(23)

(23) $f_{ij}=\frac{I_i^j}{\sqrt{{\sum }_{i=1}^m{\left(I_i^j\right)}^2}}$(23)

3. Calculate utility measure (S_i) and Regret measure (R_i) using Equations (24a(24a) $s_i=\sum _{i=1}^n{w}_i\left[\frac{{\left(f_{ij}\right)}_{max}-\left(f_{ij}\right)}{{\left(f_{ij}\right)}_{max}-{\left(f_{ij}\right)}_{min}}\right]\text{rm{Beneficial criteria}}$(24a) ,b(24b) $s_i=\sum _{i=1}^n{w}_i\left[\frac{\left(f_{ij}\right)-{\left(f_{ij}\right)}_{min}}{{\left(f_{ij}\right)}_{max}-{\left(f_{ij}\right)}_{min}}\right]\text{rm{Non beneficial criteria}}$(24b) ), (25a,b(25a) $R_i=maximum\left\{w_i\left[\frac{{\left(f_{ij}\right)}_{max}-\left(f_{ij}\right)}{{\left(f_{ij}\right)}_{max}-{\left(f_{ij}\right)}_{min}}\right]\right\}\mathrm{B}\mathrm{e}\mathrm{n}\mathrm{e}\mathrm{f}\mathrm{i}\mathrm{c}\mathrm{i}\mathrm{a}\mathrm{l}$(25a) )

(24a) $s_i=\sum _{i=1}^n{w}_i\left[\frac{{\left(f_{ij}\right)}_{max}-\left(f_{ij}\right)}{{\left(f_{ij}\right)}_{max}-{\left(f_{ij}\right)}_{min}}\right]\text{rm{Beneficial criteria}}$(24a)

(24b) $s_i=\sum _{i=1}^n{w}_i\left[\frac{\left(f_{ij}\right)-{\left(f_{ij}\right)}_{min}}{{\left(f_{ij}\right)}_{max}-{\left(f_{ij}\right)}_{min}}\right]\text{rm{Non beneficial criteria}}$(24b)

(25a) $R_i=maximum\left\{w_i\left[\frac{{\left(f_{ij}\right)}_{max}-\left(f_{ij}\right)}{{\left(f_{ij}\right)}_{max}-{\left(f_{ij}\right)}_{min}}\right]\right\}\mathrm{B}\mathrm{e}\mathrm{n}\mathrm{e}\mathrm{f}\mathrm{i}\mathrm{c}\mathrm{i}\mathrm{a}\mathrm{l}$(25a)

(25a) $R_i=maximum\left\{w_i\left[\frac{\left(f_{ij}\right)-{\left(f_{ij}\right)}_{min}}{{\left(f_{ij}\right)}_{max}-{\left(f_{ij}\right)}_{min}}\right]\right\}\text{rm{Non beneficial}}$(25a)

4. Calculate the value of Q_i () using Equation 26(26) $Q_i=v\left[\frac{S_i-{\left(S_i\right)}_{min}}{{\left(S_i\right)}_{max}-{\left(S_i\right)}_{min}}\right]+\left(1-v\right)\left[\frac{R_i-{\left(R_i\right)}_{min}}{{\left(R_i\right)}_{max}-{\left(R_i\right)}_{min}}\right]$(26)

(26) $Q_i=v\left[\frac{S_i-{\left(S_i\right)}_{min}}{{\left(S_i\right)}_{max}-{\left(S_i\right)}_{min}}\right]+\left(1-v\right)\left[\frac{R_i-{\left(R_i\right)}_{min}}{{\left(R_i\right)}_{max}-{\left(R_i\right)}_{min}}\right]$(26)

where f_ij is the normalized decision matrix; S_i is the utility measure, R_i is the regret measure, w_i is the objective weight of each criterion as obtained from CRITIC, v (compromised strategy) = 0.6; Q_i is the ranking index.

6. Results and discussion

Table 1 shows that as the value of the Height-Diameter ratio increases, the area occupied by the storage tank decreases while other output parameters such as weight, wind moment, seismic RWM (Ringwall moment), base shear and cost increase. The increase in cost is attributed to the increase in the weight because the cost of tank fabrication depends majorly on the weight of the metal plate used (API Standard 650, 2012; Enarevba et al., 2016).

Tables 2–5 are the Element of Normalized decision matrix and Standard deviation, 6 × 6 Matrix for the Criteria, Quantity of Information in relation to each criterion and Coefficient of Objective weight, respectively.

Table 2. Element of normalized decision matrix and standard deviation

Alternatives	C1	C2	C3	C4	C5	C6
A1	0.0000	0.0000	1.0000	1.0000	1.0000	1.00
A2	0.2222	0.0117	0.8940	0.8923	0.7438	0.99
A3	0.4000	0.0314	0.7820	0.7818	0.5263	0.97
A4	0.5454	0.1111	0.6644	0.6531	0.3910	0.89
A5	0.5454	0.1111	0.6644	0.6531	0.3910	0.89
A6	0.6667	0.2962	0.5414	0.5222	0.2811	0.70
A7	0.7692	0.3483	0.4133	0.3929	0.1933	0.65
A8	0.8572	0.5212	0.2803	0.2623	0.1187	0.48
A9	0.9334	0.7748	0.1424	0.1309	0.0545	0.23
A10	0.9334	0.7748	0.1424	0.1309	0.0545	0.23
A11	1.0000	1.0000	0.0000	0.0000	0.0000	0.00
Standard deviation	0.3199	0.3559	0.3320	0.3348	0.3148	0.3559

Table 3. 6 × 6 matrix for the criteria (Design output parameters)

Criteria	C1	C2	C3	C4	C5	C6
C1	1.0000	0.8851	−0.9705	−0.9741	−0.9966	−0.8851
C2	0.8851	1.0000	−0.9695	−0.9653	−0.8474	−1.0000
C3	−0.9705	−0.9695	1.0000	0.9998	0.9487	0.9695
C4	−0.9741	−0.9653	0.9998	1.0000	0.9531	0.9653
C5	−0.9966	−0.8474	0.9487	0.9531	1.0000	0.8474
C6	−0.8851	−1.0000	0.9695	0.9653	0.8474	1.0000

Table 4. Quantity of information in relation to each criterion

Criteria	C1	C2	C3	C4	C5	C6
C1	0.0000	0.1149	1.9705	1.9741	1.9966	1.8851
C2	0.1149	0.0000	1.9695	1.9653	1.8474	2.0000
C3	1.9705	1.9695	0.0000	0.0002	0.0513	0.0305
C4	1.9741	1.9653	0.0002	0.0000	0.0469	0.0347
C5	1.9966	1.8474	0.0513	0.0469	0.0000	0.1526
C6	1.8851	2.0000	0.0305	0.0347	0.1526	0.0000

Table 5. Objective weight coefficient of each criterion

C1	C2	C3	C4	C5	C6
0.2356	0.2607	0.1238	0.1249	0.1196	0.1354

where, A₁ = 0.5, A₂ = 0.6, A₃ = 0.7, A₄ = 0.8, A₅ =0.9, A₆ = 1.0, A₇ = 1.1, A₈ = 1.2, A₉ = 1.3, A₁₀ = 1.4, A₁₁ = 1.5, C₁ = Area, C₂ = Weight, C₃ = Wind Moment, C₄ = Seismic RWM, C₅ = Base Shear, C₆ = Estimated Cost

The weight (kg) of the tank has the highest coefficient of objective weight among other criteria while the Base shear (N) has the least coefficient as could be seen in Table 5. This implies that the weight of the tank (kg) has more contribution to the overall objective weight while the base shear has the least contribution.

Having determined the coefficient of the objective weight of each criterion, then the ranking of each alternative (Height-Diameter) was done using TOPSIS.

7. TOPSIS (Technique for Order Preference by Similarity to Ideal Solution)

The result of normalizing Table 1 using TOPSIS is shown in Table 6. Thereafter, Table 6 is multiplied by the coefficient of objective weight of each criterion to obtain the weightage normalized decision matrix shown in Table 7

Table 6. Normalized decision using TOPSIS

Alternatives	C1	C2	C3	C4	C5	C6
A1	0.4281	0.2837	0.1541	0.1307	0.2282	0.2837
A2	0.3805	0.2843	0.1828	0.1635	0.2560	0.2843
A3	0.3425	0.2852	0.2132	0.1972	0.2797	0.2852
A4	0.3113	0.2891	0.2451	0.2365	0.2944	0.2891
A5	0.3113	0.2891	0.2451	0.2365	0.2944	0.2891
A6	0.2854	0.2979	0.2784	0.2764	0.3063	0.2979
A7	0.2634	0.3004	0.3132	0.3158	0.3158	0.3004
A8	0.2446	0.3087	0.3492	0.3556	0.3239	0.3087
A9	0.2283	0.3209	0.3866	0.3957	0.3309	0.3209
A10	0.2283	0.3209	0.3866	0.3957	0.3309	0.3209
A11	0.2141	0.3317	0.4252	0.4356	0.3368	0.3317

Table 7. Weightage normalized decision matrix for TOPSIS

Alternatives	C1	C2	C3	C4	C5	C6
A1	0.1009	0.0740	0.0191	0.0163	0.0273	0.0384
A2	0.0897	0.0741	0.0226	0.0204	0.0306	0.0385
A3	0.0807	0.0744	0.0264	0.0246	0.0334	0.0386
A4	0.0734	0.0754	0.0303	0.0295	0.0352	0.0391
A5	0.0734	0.0754	0.0303	0.0295	0.0352	0.0391
A6	0.0672	0.0777	0.0345	0.0345	0.0366	0.0404
A7	0.0621	0.0783	0.0388	0.0394	0.0378	0.0407
A8	0.0576	0.0805	0.0432	0.0444	0.0387	0.0418
A9	0.0538	0.0836	0.0479	0.0494	0.0396	0.0435
A10	0.0538	0.0836	0.0479	0.0494	0.0396	0.0435
A11	0.0504	0.0865	0.0527	0.0544	0.0403	0.0449
Ideal Best	0.0504	0.0865	0.0191	0.0163	0.0273	0.0384
Ideal Worst	0.1009	0.0740	0.0527	0.0544	0.0403	0.0449

From the ideal best and ideal worst values obtained in Table 7, Euclidean distances of each competitive alternative from the positive S⁺ and negative S⁻ solutions were calculated for ranking as displayed in Table 8.

Table 8. Euclidean distances and ranking

Alternatives	S^+	S^−	S^++S^−	Pi	Rank position
A1	0.0520	0.0528	0.1048	0.5040	10
A2	0.0416	0.0481	0.0897	0.5362	6
A3	0.0350	0.0455	0.0804	0.5653	4
A4	0.0318	0.0440	0.0758	0.5801	1
A5	0.0318	0.0440	0.0758	0.5801	1
A6	0.0319	0.0436	0.0755	0.5775	3
A7	0.0352	0.0443	0.0795	0.5574	5
A8	0.0400	0.0460	0.0860	0.5344	7
A9	0.0460	0.0486	0.0946	0.5135	8
A10	0.0460	0.0486	0.0946	0.5135	8
A11	0.0528	0.0520	0.1048	0.4960	11

8. VIKOR (VIseKriterijumska Optimizacija I Kompromisno Resenje)

The result of normalizing Table 1 using VIKOR is shown in Table 9. Then, Table 9 is multiplied by the coefficient of objective weight of each criterion as obtained from CRITIC to get the weightage normalized decision matrix shown in Table 10

Table 9. Normalized decision matrix using VIKOR

Alternatives	C1	C2	C3	C4	C5	C6
A1	0.4281	0.2837	0.1541	0.1307	0.2282	0.2837
A2	0.3805	0.2843	0.1828	0.1635	0.2560	0.2843
A3	0.3425	0.2852	0.2132	0.1972	0.2797	0.2852
A4	0.3113	0.2891	0.2451	0.2365	0.2944	0.2891
A5	0.3113	0.2891	0.2451	0.2365	0.2944	0.2891
A6	0.2854	0.2979	0.2784	0.2764	0.3063	0.2979
A7	0.2634	0.3004	0.3132	0.3158	0.3158	0.3004
A8	0.2446	0.3087	0.3492	0.3556	0.3239	0.3087
A9	0.2283	0.3209	0.3866	0.3957	0.3309	0.3209
A10	0.2283	0.3209	0.3866	0.3957	0.3309	0.3209
A11	0.2141	0.3317	0.4252	0.4356	0.3368	0.3317
Max	0.4281	0.3317	0.4252	0.4356	0.3368	0.3317
Min	0.2141	0.2837	0.1541	0.1307	0.2282	0.2837

Table 10. Weightage normalized decision matrix for VIKOR method

Alternatives	C1	C2	C3	C4	C5	C6
A1	0.2356	0.2607	0.0000	0.0000	0.0000	0.0000
A2	0.1832	0.2577	0.0131	0.0134	0.0306	0.0016
A3	0.1414	0.2525	0.0270	0.0273	0.0567	0.0042
A4	0.1071	0.2317	0.0415	0.0433	0.0728	0.0150
A5	0.1071	0.2317	0.0415	0.0433	0.0728	0.0150
A6	0.0785	0.1835	0.0568	0.0597	0.0860	0.0401
A7	0.0544	0.1699	0.0726	0.0758	0.0965	0.0472
A8	0.0337	0.1248	0.0891	0.0921	0.1054	0.0706
A9	0.0157	0.0587	0.1062	0.1086	0.1131	0.1049
A10	0.0157	0.0587	0.1062	0.1086	0.1131	0.1049
A11	0.0000	0.0000	0.1238	0.1249	0.1196	0.1354

From Table 10, Measure of Utility S_i and Measure of Regret R_i and final Ranking index Q_i were computed as shown in Table 11.

Table 11. Values of measure of utility, measure of regret and ranking index

Alternatives	Si	Ri		Qi	Rank
A1	0.4963	0.2607	0.5000	0.4000	8
A2	0.4997	0.2577	0.6000	0.4934	6
A3	0.5090	0.2525	0.7000	0.7582	3
A4	0.5116	0.2317	0.8000	0.7781	1
A5	0.5116	0.2317	0.9000	0.7781	1
A6	0.5045	0.1835	1.0000	0.4373	7
A7	0.5164	0.1699	1.1000	0.7540	4
A8	0.5157	0.1248	1.2000	0.6113	5
A9	0.5071	0.1131	1.3000	0.3232	9
A10	0.5071	0.1131	1.4000	0.3232	9
A11	0.5037	0.1354	1.5000	0.2817	11

Spearman’s rank correlation coefficient r for Table 12 is 0.8727

Table 12. Comparison of ranking by TOPSIS and VIKOR

Alternatives	TOPSIS	VIKOR
A1	10	8
A2	6	6
A3	4	3
A4	1	1
A5	1	1
A6	3	7
A7	5	4
A8	7	5
A9	8	9
A10	8	11
A11	11	11

The alternatives A₄ and A₅ (height-diameter ratios, 0.8 and 0.9) were considered to be the ideal best alternatives because they have the highest ranking with the value of 0.5801. However, the alternative A₁₁ (height-diameter ratio 1.5) was considered as the worst alternative with a value of 0.4960. Since all the alternatives (0.5–1.4) have their ranks above 0.4999, these alternatives could also be chosen but the alternative(s) with the highest-rank was termed the best alternative. Terming the highest ranked alternative(s) as the best alternative aligns with the study of Stanojković and Radovanović (Stanojković & Radovanović, 2017). Any rank value from 0.50 and above could be regarded as a good alternative. However, the best choice of alternatives is the one with the highest-rank value. Based on this statement, alternatives 0.8 and 0.9 were given priority over the other alternatives in this study because they have the highest ranking value of 0.5801.

More so, comparing the ranking results of alternatives using the TOPSIS method against the VIKOR method, it could be deduced that both methods ranked alternatives A₄ and A₅ as the best alternatives, while alternative A₁₁ was least ranked. Few discrepancies observed when comparing the ranking value in Table 12 were due to the value of v chosen (Manoj & Sagar, 2018). The effect of these discrepancies was quantified using Spearman’s correlation coefficient. Spearman’s correlation coefficient of 0.8727 shows that there is a strong positive correlation between the two MCDA methods (Prasenjit & Shankar, 2016).

9. Sensitivity analysis

The reason for conducting sensitivity analysis is to evaluate the effect of changing the objective weights of criteria on the ranking of alternatives (Ali et al., 2011; Saeed et al., 2017). These changes create different scenarios that may alter the ranking of alternatives. According to Nazari-Shirkouhi et al. (2017), the results are said to be sensitive when original ranking is changed by changing the objective weights of the criteria, otherwise results are termed robust.

Sensitivity analysis was carried out by using nine (9) scenarios. Objective weights were first re-assigned using Entropy method instead of initial CRITIC method used. The second scenario was to assign equal weights to all the criteria. All the nine scenarios are presented in Table 13. Then, ranking based on sensitivity analysis for TOPSIS and VIKOR ispresented in Tables 14 and 15, respectively.

Table 13. Scenarios with varying criteria weights

Scenarios	Criteria weight
	C1	C2	C3	C4	C5	C6
Entropy method (SE)	0.1727	0.0105	0.3260	0.4361	0.0443	0.0105
Equal weight (Seq)	0.1667	0.1667	0.1667	0.1667	0.1667	0.1667
Priority to criterium 1 (S1)	0.2500	0.1500	0.1500	0.1500	0.1500	0.1500
Priority to criterium 2 (S2)	0.1500	0.2500	0.1500	0.1500	0.1500	0.1500
Priority to criterium 3 (S3)	0.1500	0.1500	0.2500	0.1500	0.1500	0.1500
Priority to criterium 4 (S4)	0.1500	0.1500	0.1500	0.2500	0.1500	0.1500
Priority to criterium 5 (S5)	0.1500	0.1500	0.1500	0.1500	0.2500	0.1500
Priority to criterium 6 (S6)	0.1500	0.1500	0.1500	0.1500	0.1500	0.2500
Priority to criteria 2 and 6 (S2,6)	0.1250	0.2000	0.1250	0.1250	0.1250	0.3000

Table 14. Ranking of alternatives for different scenarios using TOPSIS

Alternatives	Ranking based on sensitivity analysis
	Entropy method (SE)	Equal weight (Seq)	Priority to criterium 1 (S1)	Priority to criterium 2 (S2)	Priority to criterium 3 (S3)	Priority to criterium 4 (S4)	Priority to criterium 5 (S5)	Priority to criterium 6 (S6)	Priority to criteria 2 and 6 (S2,6)
A1	1	3	6	3	2	2	2	3	3
A2	2	1	5	1	1	1	1	1	1
A3	3	2	3	2	3	3	3	2	2
A4	4	4	1	4	4	4	4	4	4
A5	4	4	1	4	4	4	4	4	4
A6	6	6	4	6	6	6	6	6	6
A7	7	7	7	7	7	7	7	7	7
A8	8	8	8	8	8	8	8	8	8
A9	9	9	9	9	9	9	9	9	9
A10	9	9	9	9	9	9	9	9	9
A11	11	11	11	11	11	11	11	11	11

Table 15. Ranking of alternatives for different scenarios using VIKOR

Alternatives	Ranking based on sensitivity analysis
	Entropy method (SE)	Equal weight (Seq)	Priority to criterium 1 (S1)	Priority to criterium 2 (S2)	Priority to criterium 3 (S3)	Priority to criterium 4 (S4)	Priority to criterium 5 (S5)	Priority to criterium 6 (S6)	Priority to criteria 2 and 6 (S2,6)
A1	11	10	5	11	11	11	11	11	11
A2	10	9	7	10	10	10	10	10	10
A3	9	5	11	4	9	9	9	9	9
A4	7	7	9	2	7	7	7	7	6
A5	7	7	9	2	7	7	7	7	6
A6	6	11	8	9	6	6	6	6	8
A7	5	6	6	7	5	5	5	5	5
A8	4	4	4	8	4	4	4	4	4
A9	2	2	2	5	2	2	2	2	2
A10	2	2	2	5	2	2	2	2	2
A11	1	1	1	1	1	1	1	1	1

Tables 14 and 15 show a considerable difference in the ranking of alternatives when compared with the initial ranking by TOPSIS (Table 8) and VIKOR (Table 11). This shows that ranking results based on the two methods are sensitive to a change in the objective weights of the criteria. In all the nine scenarios considered for sensitivity analysis for the TOPSIS method, alternative 2 is adjudged to be the best option because it was ranked 1^st seven times while alternative 11 was the worst alternative because it was ranked least in all the nine scenarios. In sharp contrast to sensitivity analysis for TOPSIS, sensitivity for VIKOR rated alternative 11 as the best due to the fact that it was ranked first in all the scenarios. As could be seen in Table 15, alternatives 7–11 maintain a higher degree of consistency because they retain their positions virtually in all the scenarios. Also, in Table 14, there is consistency in the alternatives 1–5.

10. Conclusion

CRITIC and TOPSIS as MCDA tools have been successfully adopted for the selection of the Height-Diameter ratio for the optimal design of a 13,000 m³ oil storage tank. Objective weight of each criterion [Area occupied by the tank (m²), Weight (kg), Wind Moment (Nm), Seismic Ringwall Moment RWM (Nm), Base shear (N) and Estimated Cost (₦)] was determined by CRITIC method while the ranking and final selection of Alternatives (Height-Diameter ratios) were done by using TOPSIS. The ranking results obtained through TOPSIS were compared with the VIKOR method as a confirmatory test. VIKOR method also ranked alternatives A₄ and A₅ as the best alternatives. This study reveals that the best alternatives to achieve the optimal design of 13,000 m³ oil storage tank are to select Height-Diameter ratios of 0.8 or 0.9. To further justify the effectiveness of the MCDAs used, Spearman’s rank correlation coefficient r was determined for TOPSIS and VIKOR and the result showed that there is a high correlation of 0.8727 between them. Sensitivity analysis conducted reveals that both TOPSIS and VIKOR used in the ranking of the alternatives are sensitive to the changes in objective weight of the criteria.

Further study shall be the detailed design of oil storage tank using the height-diameter ratio of 0.8. This is expected to include material selection and other design considerations, simulation, etc.

Cover image

Source: Author.

Additional information

Funding

The authors received no direct funding for this research.

Notes on contributors

O. O. Agboola

Agboola O. O. has a wide experience in Oil and Gas industry spanning Oil storage tank design, calibration and Non Destructive Testing (NDT). His research interests revolve round the Industrial and Production Engineering.

B. O. Akinnuli

Akinnuli B. O. is currently an Associate Professor in the Department Industrial and Production Engineering, Federal University of Technology Akure, Nigeria. His research interests include design and development of machines, Artificial Intelligence and Decision science.

B. Kareem

Kareem B. is a professor and erudite scholar in the Department of Industrial and Production Engineering, Federal University of Technology Akure, Nigeria. His research interests include industrial and production engineering, Decision science and reliability of systems.

M. A. Akintunde

Akintunde M. A is a professor and erudite scholar in the Department of Mechanical Engineering, Federal University of Technology Akure, Nigeria. He is an expert in the thermofluid option of Mechanical Engineering.