Application of fuzzy TOPSIS in assessing performance of agro-processed organic crops: A case of Sustainable Agriculture Tanzania (SAT) in Morogoro

Abstract

It is a common practice for organisation to carry out assessment exercises regarding performance of their organisational activities and processes. This paper assesses the market performance of five agro-processed crops at Sustainable Agriculture Tanzania (SAT) against some criteria. Experts at SAT supplied useful information by responding through a questionnaire. The fuzzy TOPSIS model was applied in the methodology to rank the processed products. For the sake of comparing results, the fuzzy analytical hierarchy process (AHP) model was also applied to rank the products. It was found that three products maintained their positions in the two models while the other two products (alternatives) exchanged their positions. It was further suggested that more efforts have to go for lower market performing products by looking upon means to improve on their corresponding low weight criteria/sub-criteria.

Keywords

Agro-processing fuzzy TOPSIS fuzzy AHP linguistic variables fuzzy aggregation multi-criteria decision making

1 Introduction

Decision making is the process of obtaining or identifying the best alternative from a set of feasible alternatives [1]. It is pointed out that decisions include judgements on the available information [2]. It involves scientific approaches [3]. Identifying the best performing option from different alternatives each on several criteria is termed as multi-criteria decision making (MCDM) problem. “MCDM is considered as a sophisticated decision making tool involving both quantitative and qualitative factors” [4]. It is widely applied for mathematical optimisation in various fields requiring decision making [5]. Besides making choices of the best performing alternatives from among a finite set of decision alternatives, MCDM problems are usually in terms of multiple and so often with conflicting criteria [6, 7]. Also [8] pointed out that it is not easy to get a better alternative from a set of alternatives. MCDM methods are very much effective in many fields which require decision making [9].

Fuzzy technique for order preference by similarity to ideal solutions (TOPSIS) is a MCDM method that considers both the best and the worst choice of alternatives [10, 11]. It is by considering an alternative closest to the fuzzy positive ideal solution (FPIS) and the alternative furthest to the fuzzy negative ideal solution (FNIS) respectively [12, 13]. FPIS maximises the benefit criteria and minimises the cost criteria while FNIS does the opposite [14]. Fuzzy TOPSIS method is almost equivalent to human thinking in the real word [15]. According to [16], fuzzy TOPSIS (FTOPSIS) has some simple steps to follow and it takes same steps regardless of the number of criteria or alternatives. It is easily programmable but the use of Euclidian distance does not consider the correlation of criteria nor consistency of the experts’ judgement.

In fuzzy TOPSIS, all ratings and importance weights are defined by means of linguistic variables which in turn each variable is associated with a fuzzy number [17] as opposed to classical TOPSIS that uses known crisp values [18]. MCDM with crisp values does not capture the imprecision and vagueness particularly from the human judgement and preferences [19]. [20] added that linguistic terms evaluation often have intangible and inherently imprecise data when crisp values are used. So, MCDM method in the fuzzy environment was used. Also, [9] added that MCDM methods have been used by researchers in agriculture to support decision making. As part of methodology, the fuzzy TOPSIS method and triangular fuzzy numbers were used because of the ideal solutions it gives [21]. The results found by fuzzy TOPSIS method were compared with results by fuzzy analytic hierarchy process (FAHP). As opposed to AHP that uses crisp values in assigning weights to criteria, FAHP uses fuzzy numbers. In both cases, pairwise comparisons of criteria and alternatives are performed. Various FAHP methods are suggested by different authors. A few proposed methods are cited in the paper as examples in [22]. This paper used procedure given in [23]. In practice, FAHP method combines fuzzy logic with linguistic terms and original AHP method [24]. Both AHP and FAHP reduce complex problems into hierarchical structure of simple problems and pairwise comparisons becomes the main task [25].

Data were provided by Sustainable Agriculture Tanzania (SAT) a non-governmental organisation with its headquarter in Morogoro municipality, Tanzania. The organisation seeks to address the existing social and environmental problems by having a sustainable agriculture in Tanzania. The organisation also advocates organic farming that is a farming practice without use of chemical fertilizers, pesticides or genetically modified organisms [26]. Yadav et al. [27] defined organic farming as a system of agriculture that neither uses synthetic fertilizers nor pesticides nor growth regulators nor livestock feed additives.

In collaboration with other partners, SAT also disseminates proper knowledge to small scale farmers in order to transform farming practices in Tanzania through a training centre it owns [28]. Besides working with farmers, SAT does agro-processing for food products in some of the produce from their farmers. Food processing plays a big role in economic development especially in developing countries like Tanzania [29]. The organisation owns a shop for selling part/all of the processed products. The products were doing well in the market yet observation revealed that still there was a room to improve more market performance of the products. Massive production of agro-processed products requires agro-industry.

Broadly, agro-industry refers to the establishment of enterprises and supply chains for developing, transforming and distributing specific inputs and products in the agricultural sector. A narrower definition portrays agro-industry as the sum of post-harvest activities involved in the transformation, preservation and preparation of agricultural products for intermediate or final consumption [30]. Agro-industry provides means of converting row food and/or agricultural materials into value added products which in due process, generate employment and higher income and hence contribute to poverty alleviation vis-à-vis socio-economic development [31].

This study assessed performance of agro-processed products at SAT branding, delivery and protection of the processed product; payment and acquisition technology and uniqueness of the employed technology; target market and market competition; customer willingness to pay, heterogeneity and demand for the products as they were scrutinised in [32].

As it is seen in the case study section, four criteria; product, technology, marketing and customer were used. Each criterion carried with it sub-criteria to narrow it down. Brand (one of sub-criterion in product) is a logo that has unique design, name, term, sign, symbol, word or a combination of them [33]. It identifies goods or services of a seller or group of sellers from those of other sellers [33]. Thus, branding is the process of creating brand that identifies itself from other products or services. “Brands are clusters of functional and emotional values which promise stakeholders unique experiences” [33]. Thus brand creates relationship with customers and consumers. It builds a kind of image in the customers’ mind [34]. It is the centre of which business processes and procedures function [35].

Appropriate technology is a very important factor/criterion in the growth of agro-processing industry [36]. In a review of the challenges affecting the agro-processing sector in Tanzania, It is shown that technology is one of the key factors in the sector [37]. But also it is a challenging task to the management and since it is not an easy task to get the right technology [38]. [29] pointed out that poor technology is among the challenges facing agro-processing industry in Tanzania. Inter alia, technology in agro-processing includes; processing itself, packaging, means of payment and delivery of products or services to customers.

In business, goods or services have to be delivered to the market or customers. Thus one should consider the issue of marketing in the business. In marketing goods and services, customers’ needs have to come first [39]. Marketing starts with identifying consumers’ or customers’ needs and then plan for production of goods and services or brands that will satisfy the customers’ needs [40]. [41] defined marketing as “an organisational function and a set of processes for creating, communicating, and delivering value to customers and for managing customer relationships in ways that benefit the organisation and its stakeholders”. Also marketing was defined as “performance of all business activities involved in the flow of goods and services from the point of initial production until they are in the hands of ultimate consumer” [42]. From these definitions, the key issue in marketing is the relationship between goods producers or service providers in one hand and consumers in another hand or more precisely sellers and buyers. In Tanzania, production and marketing of organic products is still at its beginning and there are factors that impede rapid growth of the sub-sector including consumption of the products [43].

The main goal of this paper is to assess performance of agro-processed products at SAT Morogoro, Tanzania in order to guide in the decision processes. Together with the introduction section, the rest of the paper is organised as follows; the methodology section that explains the fuzzy TOPSIS and fuzzy AHP methods, a case study section that comprises of data collection and presentation, data analysis and interpretation of results. A references section comes after the conclusion and recommendations section.

2 Methodology

2.1 The fuzzy TOPSIS method

For the sake of this study, the fuzzy TOPSIS method was divided into ten steps as follows;

Step 1: Formulation of a decision making committee

A committee D ={ D₁, D₂, ⋯ , D_K } of K decision makers who are experts in the area of interest was formed.

Step 2: Criteria and alternatives

A set C ={ C₁, C₂, ⋯ , C_n } of n evaluation criteria and a set A ={ A₁, A₂, ⋯ , A_n } of m alternatives under study were identified.

Step 3: Formulation of linguistic variables

Choose appropriate linguistic variables for assigning the importance weight of each criterion and for rating of each alternative.

Step 4: Aggregation of weights of criteria and alternative ratings

The importance weights assigned by the individual experts are aggregated for each criterion to get an aggregated fuzzy weight ${\tilde{w}}_{j}$ for each criterion C_j, j = 1, 2, ⋯ , n. This is done as follows:

Suppose ${\tilde{w}}_{jl} = (w_{jl 1}, w_{jl 2}, w_{jl 3})$ is the importance fuzzy weight assigned to criterion C_j by expert l, l = 1, 2, ⋯ , K.

Then, the aggregated fuzzy weight for criterion j is ${\tilde{w}}_{j} = (w_{j 1}, w_{j 2}, w_{j 3})$ , where,

$w_{j 1} = min_{i} {w_{jl 1}}$ , for all l in a particular C_j.

$w_{j 2} = \frac{1}{K} \sum_{l = 1}^{K} w_{jl 2}$ , for each criterion C_j.

$w_{j 3} = max_{i} {w_{jl 3}}$ , for all l in a particular C_j.

Thus, for all criteria, $\tilde{w} = {[{\tilde{w}}_{1}, {\tilde{w}}_{2}, \dots, {\tilde{w}}_{n}]}^{T}$ .

Similarly, the decision makers (experts) were pooled together to come up with the aggregated fuzzy rating ${\tilde{x}}_{ij}$ for each alternative A_i, i = 1, 2, ⋯ , m for each criterion C_j.

Let ${\tilde{x}}_{ijl} = (x_{ijl 1}, x_{ijl 2}, x_{ijl 3})$ be a fuzzy rating by decision maker l on criterion C_j for alternative A_i, i = 1, 2, ⋯ , m. Then, ${\tilde{x}}_{ij} = (x_{ij 1}, x_{ij 2}, x_{ij 3})$ is the aggregated fuzzy rating of alternative A_i on criterion C_j, where,

$x_{ij 1} = min_{i} {x_{ijl 1}}$ , for all l in a particular C_j on each A_i.

$x_{ij 2} = \frac{1}{K} \sum_{l = 1}^{K} x_{ijl 2}$ , for each criterion C_j on each A_i.

$x_{ij 3} = max_{i} {x_{ijl 3}}$ , for all l in a particular C_j on each A_i.

Step 5: Construction of fuzzy decision matrix and normalised fuzzy decision matrix

All ${\tilde{x}}_{ij}$ s in step 4 above are collected and systematically placed in matrix form such that $\begin{matrix} \tilde{X} = [\begin{matrix} {\tilde{x}}_{11} & {\tilde{x}}_{12} & \begin{matrix} \dots & {\tilde{x}}_{1 n} \end{matrix} \\ {\tilde{x}}_{21} & {\tilde{x}}_{22} & \begin{matrix} \dots & {\tilde{x}}_{2 n} \end{matrix} \\ \begin{matrix} ⋮ \\ {\tilde{x}}_{m 1} \end{matrix} & \begin{matrix} ⋮ \\ {\tilde{x}}_{m 2} \end{matrix} & \begin{matrix} \begin{matrix} ⋱ & ⋮ \end{matrix} \\ \begin{matrix} \dots & {\tilde{x}}_{mn} \end{matrix} \end{matrix} \end{matrix}] \end{matrix}$

The fuzzy decision matrix is normalised so that

•Each entry of the normalised triangular fuzzy numbers belongs to [0, 1],

•The linear scale transformation makes criteria scales into comparable numerical quantities for alternative ratings (Vafaei et al., 2018).

The normalised fuzzy decision as given by Saghafian (2005) is a m × n matrix given by $\begin{matrix} \tilde{R} = [\begin{matrix} {\tilde{r}}_{11} & {\tilde{r}}_{12} & \begin{matrix} \dots & {\tilde{r}}_{1 n} \end{matrix} \\ {\tilde{r}}_{21} & {\tilde{r}}_{22} & \begin{matrix} \dots & {\tilde{r}}_{2 n} \end{matrix} \\ \begin{matrix} ⋮ \\ {\tilde{r}}_{m 1} \end{matrix} & \begin{matrix} ⋮ \\ {\tilde{r}}_{m 2} \end{matrix} & \begin{matrix} \begin{matrix} ⋱ & ⋮ \end{matrix} \\ \begin{matrix} \dots & {\tilde{r}}_{mn} \end{matrix} \end{matrix} \end{matrix}] \end{matrix}$

If C_j. is a benefit criterion then,

${\tilde{r}}_{ij} = (\frac{x_{ij 1}}{c_{j}^{*}}, \frac{x_{ij 2}}{c_{j}^{*}}, \frac{x_{ij 3}}{c_{j}^{*}})$ (1)

If C_j is a cost criterion then,

${\tilde{r}}_{ij} = (\frac{a_{j}^{-}}{x_{ij 3}}, \frac{a_{j}^{-}}{x_{ij 2}}, \frac{a_{j}^{-}}{x_{ij 1}})$ (2) where,

$c_{j}^{*} = max_{i} {x_{ij 3}}$ , j is in benefit criterion.

$a_{j}^{-} = min_{i} {x_{ij 1}}$ , j is in cost criterion.

Step 6: Construction of Weighted Normalised Fuzzy Decision Matrix

The importance weight of each criterion is now considered. The weighted normalized decision matrix is given in matrix form as $\begin{matrix} \tilde{V} = [\begin{matrix} {\tilde{v}}_{11} & {\tilde{v}}_{12} & \begin{matrix} \dots & {\tilde{v}}_{1 n} \end{matrix} \\ {\tilde{v}}_{21} & {\tilde{v}}_{22} & \begin{matrix} \dots & {\tilde{v}}_{2 n} \end{matrix} \\ \begin{matrix} ⋮ \\ {\tilde{v}}_{m 1} \end{matrix} & \begin{matrix} ⋮ \\ {\tilde{v}}_{m 2} \end{matrix} & \begin{matrix} \begin{matrix} ⋱ & ⋮ \end{matrix} \\ \begin{matrix} \dots & {\tilde{v}}_{mn} \end{matrix} \end{matrix} \end{matrix}] \end{matrix}$ for all criteria, where, ${\tilde{v}}_{ij} = {\tilde{r}}_{ij} (\cdot) {\tilde{w}}_{j} = (v_{ij 1}, v_{ij 2}, v_{ij 3})$ in components form.

Step 7: Determination of the Fuzzy Positive Ideal Solution (FPIS, A^*) and Fuzzy Negative Ideal Solution (FNIS, A^-)

By Soufi et al. (2015), FPIS is the alternative that one would want. It maximizes the benefit criteria while minimizing the cost criteria. The FNIS does the reverse.

Let $A^{*} = [{\tilde{v}}_{1}^{*}, {\tilde{v}}_{2}^{*}, \dots, {\tilde{v}}_{n}^{*}]$ and $A^{-} = [{\tilde{v}}_{1}^{-}, {\tilde{v}}_{2}^{-}, \dots, {\tilde{v}}_{n}^{-}]$ be the FPISs and FNISs respectively. Following the idea by Seyedmohammad (2018), the study used ${\tilde{v}}_{j}^{*} = (max_{j} {v_{ij 1}}, max_{j} {v_{ij 2}}, max_{j} {v_{ij 3}})$ and ${\tilde{v}}_{j}^{-} = (min_{j} {v_{ij 1}}, min_{j} {v_{ij 2}}, min_{j} {v_{ij 3}})$ .

Step 8: Calculation of the distances d (A_i, A^*) and d (A_i, A^-) for each criterion

A_i is in the weighted normalized fuzzy decision matrix. Thus, $d (A_{i}, A^{*}) \equiv d ({\tilde{v}}_{ij}, {\tilde{v}}_{j}^{*})$ and $d (A_{i}, A^{-}) \equiv d ({\tilde{v}}_{ij}, {\tilde{v}}_{j}^{-})$ for alternative A_i on criterion C_j. For any two triangular numbers $\tilde{A}$ and $\tilde{B}$ , the formula $d (\tilde{A}, \tilde{B}) = \sqrt{\frac{1}{3} \sum_{i = 1}^{3} {(a_{i} - b_{i})}^{2}}$ , where, $a_{i} \in \tilde{A}$ and $b_{i} \in \tilde{B}$ is used in the determination of both d (A_i, A^*) and d (A_i, A^-) (Saghafian, 2005).

Step 9: Calculation of the Closeness Coefficient CC_i of Each alternative

$\begin{matrix} {CC}_{i} = \frac{d_{j}^{-}}{d_{j}^{*} + d_{j}^{-}}, where, d_{j}^{*} = \sum_{j = 1}^{n} d ({\tilde{v}}_{ij}, {\tilde{v}}_{j}^{*}) \\ and d_{j}^{-} = \sum_{j = 1}^{n} d ({\tilde{v}}_{ij}, {\tilde{v}}_{j}^{-}) . \end{matrix}$

Step 10: Determine the Ranking Order of the Alternatives

The ranking is performed according to the closeness coefficients CC_i. The highest CC_i value gives the highest rank and corresponds to the best choice among/between alternatives.

2.2 The fuzzy AHP method

By Ayhan (2013) procedure, fuzzy AHP method falls under two main steps;

Determination of weights of criteria based on the goal,

Determination of weights of alternatives according to the given criteria.

2.2.1 Determination of Weights of Criteria w_j

Some simple steps are needed in the determination of criteria weights:

Step 1: Pair wise comparison of criteria with respect to the goal

Every expert (decision maker) was asked to compare one criterion to another, be it on the importance or any other preference. This was done via given linguistic terms. Each linguistic term was associated with a fuzzy number.

Suppose a positive fuzzy number ${\tilde{x}}_{ij}^{k}$ is the k^th decision maker’s preference of i^th criterion on j^th criterion. Then ${\tilde{x}}_{ji}^{k} = \frac{1}{{\tilde{x}}_{ij}^{k}}$ .

The pairwise comparison matrix contributed by the k^th decision maker is thus given by: $\begin{matrix} {\tilde{X}}^{k} = [\begin{matrix} {\tilde{x}}_{11}^{k} & {\tilde{x}}_{12}^{k} & \begin{matrix} \dots & {\tilde{x}}_{1 n}^{k} \end{matrix} \\ {\tilde{x}}_{21}^{k} & {\tilde{x}}_{22}^{k} & \begin{matrix} \dots & {\tilde{x}}_{2 n}^{k} \end{matrix} \\ \begin{matrix} ⋮ \\ {\tilde{x}}_{m 1}^{k} \end{matrix} & \begin{matrix} ⋮ \\ {\tilde{x}}_{m 2}^{k} \end{matrix} & \begin{matrix} \begin{matrix} ⋱ \\ \dots \end{matrix} & \begin{matrix} ⋮ \\ {\tilde{x}}_{mn}^{k} \end{matrix} \end{matrix} \end{matrix}], k = 1, 2, \dots, K \end{matrix}$

The individual pairwise comparison matrices contributed by K decision makers are aggregated to come up with one matrix. In this study, the overall matrix $\tilde{X}$ was obtained by averaging the individual contribution matrices such that $\begin{matrix} \tilde{X} = \frac{1}{K} \sum_{k = 1}^{K} {\tilde{X}}^{k} = [\begin{matrix} {\tilde{x}}_{11} & {\tilde{x}}_{12} & \begin{matrix} \dots & {\tilde{x}}_{1 n} \end{matrix} \\ {\tilde{x}}_{21} & {\tilde{x}}_{22} & \begin{matrix} \dots & {\tilde{x}}_{2 n} \end{matrix} \\ \begin{matrix} ⋮ \\ {\tilde{x}}_{m 1} \end{matrix} & \begin{matrix} ⋮ \\ {\tilde{x}}_{m 2} \end{matrix} & \begin{matrix} \begin{matrix} ⋱ \\ \dots \end{matrix} & \begin{matrix} ⋮ \\ {\tilde{x}}_{mn} \end{matrix} \end{matrix} \end{matrix}] \end{matrix}$ where,

${\tilde{x}}_{ij} = \frac{1}{K} \sum_{k = 1}^{K} {\tilde{x}}_{ij}^{k}$ (3)

In such a case (for more than one decision maker), ${\tilde{x}}_{ij}$ is not necessary equal to ${\tilde{x}}_{ji}^{- 1}$ as for one decision maker.

Step 2: Determination of the geometric mean from the fuzzy comparison matrix

By Ayhan (2013), the geometric mean of fuzzy comparisons is obtained by $\begin{matrix} {\tilde{r}}_{i} = {(\prod_{j = 1}^{n} {\tilde{x}}_{ij})}^{\frac{1}{n}}, i = 1, 2, \dots, n \end{matrix}$

Step 3: Determination of the fuzzy weight of each criterion

Sub-Step 3a: Summation of ${\tilde{r}}_{i}$

The ${\tilde{r}}_{i}$ s are summed component-wise that is the vector summation of ${\tilde{r}}_{i}$

$s = \sum_{i = 1}^{n} {\tilde{r}}_{i}$ (5)

Sub-step 3b: Determine s^-1

Suppose s = (a, b, c), then $s^{- 1} = (\frac{1}{c}, \frac{1}{b}, \frac{1}{a})$ for non-zero a, b, c. When s^-1 is determined, the resulting components (for the case of triangular fuzzy numbers) are written in increasing order; the lower l followed by the middle m and ends by the upper value u. Hence, $\begin{matrix} s^{- 1} = (l, m, u) \end{matrix}$

Sub-step 3c: Determine the fuzzy weight ${\tilde{w}}_{j}$ of criterion j;

Each geometric mean ${\tilde{r}}_{i}$ is multiplied by (l, m, u) component-wise. ${\tilde{w}}_{j} = {\tilde{r}}_{i} \otimes (l, m, u) = ({lw}_{j}, {mw}_{j}, {uw}_{j})$ . In this case, i is replaced by j.

Step 4: Defuzzification of ${\tilde{w}}_{j}$ ;

Since ${\tilde{w}}_{j}$ was still fuzzy numbers, it was defuzzified to help in the ranking procedure. By Ayhan (2013), a centre of area method was used in the study. Thus in the case of triangular fuzzy numbers,

$M_{j} = \frac{{lw}_{j} + {mw}_{j} + {uw}_{j}}{3}$ (6)

Step 5: Normalisation of M_j

M_js were normalised such that w_j ∈ [0, 1] and $\sum_{j = 1}^{n} w_{j} = 1$ . Normalisation was by the formula

$w_{j} = \frac{M_{j}}{\sum_{j = 1}^{n} M_{j}}$ (7)

Same procedure could be applied if there could be sub-criteria.

2.2.2 Determination of the weights of alternatives

The procedure for determining weights of alternatives is the same as for criteria weights. The difference is that for each criterion or sub-criterion (if any), a pair wise comparison matrix of alternatives is formed. Decision makers’ preferences are done based on that criterion or sub-criterion. For n given criteria or sub-criteria, there are n pair wise comparison matrices. Each matrix takes same steps as for criteria weights.

Then, sums of products for weight of alternative and the respective criterion and sub-criterion are determined for each alternative. The alternative with highest score (sum) is the best preferred alternative.

Suppose there are K sub-criteria altogether, let $w_{k}^{A_{i}}$ be the weight of alternative A_i for sub-criterion c_jk of criterion C_j with weight w_j. If $w_{j}^{c_{jk}}$ is the weight of c_jk then, the score of alternative A_i is

${score}_{A_{i}} = \sum_{j = 1}^{n} (\prod_{k = 1}^{K} (w_{j} \times w_{j}^{c_{jk}} \times w_{k}^{A_{i}}))$ (8)

for each fixed i, i = 1, 2, ⋯ , m.

3 The case study of Sustainable Agriculture Tanzania (SAT) in Morogoro

In the exercise to assess the performance of processed crops at SAT, a set of four main criteria C ={ C₁, C₂, C₃, C₄ } was identified. These are; product itself, technology, marketing and customer in the order of set C. Under the product criterion; three sub-criteria were formulated as branding c₁₁, delivery c₁₂, and patent protection c₁₃. Product payment and acquisition c₂₁ and technology uniqueness c₂₂ are sub-criteria of technology. The sub-criteria of marketing are target markets c₃₁ and market competition c₃₂. Under the criterion customer, three sub-criteria were formulated as customer willingness to pay c₄₁, customer heterogeneity c₄₂ and customer demand c₄₃.

The alternatives for this study were the set of processed crops A ={ A₁, A₂, A₃, A₄, A₅ }. These are the processed beans, maize, sunflower seeds, Rosella, and rice at SAT in the order of set A. A committee of three decision makers who are experts at SAT and in the area of interest DM ={ D₁, D₂, D₃ } was formed.

Table 1 gives description of the intended meaning of each sub-criterion under this exercise.

Table 1
Description of evaluation sub-criteria for each criterion

Criteria Sub-criteria Description

Product C₁ Branding c₁₁ A combination of name, symbol, term and or/ design that identifies specific product

Delivery c₁₂ How the product reaches customers

Patent protection c₁₃ Is the product protected by the patent (issue of intellectual property right)

Technology C₂ Product payment and acquisition c₂₁ The means by which the product reach customers, especially customers outside Morogoro township

Technology uniqueness c₂₂ The level at which the product is unique compared to other products of the same type

Marketing C₃ Target markets c₃₁ Specific group(s) of customers at whom the firm serves its products

Competition c₃₂ Level of competition where other firms with similar products in the market with more advantageous conditions for customers

Customer C₄ Willingness to pay c₄₁ This can be shown by quantity/frequency customer buy

Heterogeneity c₄₂ Different levels of customers can make room for price based segmentation

Demand c₄₃ Level of customer demand of the product in/outside the market

Criteria	Sub-criteria	Description
Product C₁	Branding c₁₁	A combination of name, symbol, term and or/ design that identifies specific product
	Delivery c₁₂	How the product reaches customers
	Patent protection c₁₃	Is the product protected by the patent (issue of intellectual property right)
Technology C₂	Product payment and acquisition c₂₁	The means by which the product reach customers, especially customers outside Morogoro township
Technology uniqueness c₂₂	The level at which the product is unique compared to other products of the same type
Marketing C₃	Target markets c₃₁	Specific group(s) of customers at whom the firm serves its products
Competition c₃₂	Level of competition where other firms with similar products in the market with more advantageous conditions for customers
Customer C₄	Willingness to pay c₄₁	This can be shown by quantity/frequency customer buy
Heterogeneity c₄₂	Different levels of customers can make room for price based segmentation
Demand c₄₃	Level of customer demand of the product in/outside the market

Figure 1 shows the MCDM problem structure on performance of processed products of crops at Sustainable Agriculture Tanzania (SAT) in Morogoro.

Fig. 1

MCDM problem structure.

3.1 Field data collection and presentation

3.1.1 For fuzzy TOPSIS

A set of three decision makers DM ={ D₁, D₂, D₃ } was used to assign the importance weight of sub-attributes with respect to each main attribute. The decision makers were experts in agriculture and specifically in agro-processing.

Tables 2 3 give some linguistic variables and the corresponding triangular fuzzy numbers to be used in fuzzy TOPSIS model.

Table 2
Linguistic variables for importance weight of each criterion/sub-criteria

LV NI SI MI I VI

FN (0.1, 0.1, 0.3) (0.1, 0.3, 0.5) (0.3, 0.5, 0.7) (0.5, 0.7, 0.9) (0.7, 0.9, 0.9)

LV	NI	SI	MI	I	VI
FN	(0.1, 0.1, 0.3)	(0.1, 0.3, 0.5)	(0.3, 0.5, 0.7)	(0.5, 0.7, 0.9)	(0.7, 0.9, 0.9)

LV- Linguistic Variable; FN- Fuzzy Number; NI- Not Important; SI- Slightly Important; MI- Moderately Important; I- Important; VI- Very Important.

Table 3

Linguistic variables for rating the alternatives against criteria

LV	VP	P	MG	G	VG
FN	(1, 1, 3)	(1, 3, 5)	(3, 5, 7)	(5, 7, 9)	(7, 9, 9)

VP- Very Poor; P- Poor; MG- Medium Good; G- Good; VG- Very Good.

MP = (2,4,6) and F = (4,6,8) were for intermediate values Medium Poor and Fair respectively.

Table 4 shows the importance weight assigned by each expert on each sub-criterion by using the linguistic variable shown in Table 2 for the given criteria. Table 5 shows ratings of alternative by the experts obtained by using the linguistic variables given in Table 3.

Table 4

Decision makers’ assignment of the importance weight of sub-criteria

Criteria	Sub-criteria	Importance Weight
		D ₁	D ₂	D ₃
Product C₁	Branding	VI	I	VI
	Delivery	I	VI	VI
	Patent protection	VI	I	I
Technology C₂	Product payment and acquisition	I	VI	I
	Technology uniqueness	VI	I	I
Marketing C₃	Target markets	VI	I	I
	Competition	VI	VI	I
Customer C₄	Willingness to pay	MI	I	I
	Heterogeneity	MI	I	I
	Demand	I	VI	I

Table 5

Rating of alternatives (processed crops)

Alternatives	DM	C ₁	C ₂	C ₃	C ₄
Beans A₁	D ₁	P	P	P	P
	D ₂	G	F	VG	VG
	D ₃	VP	MG	F	F
Maize A₂	D ₁	VG	VG	G	G
	D ₂	G	F	VG	VG
	D ₃	MP	G	P	F
Sunflower A₃	D ₁	VG	G	VG	G
	D ₂	G	VG	VG	VG
	D ₃	MG	G	VG	F
Rosella A₄	D ₁	MP	MP	P	P
	D ₂	G	F	G	G
	D ₃	P	F	VP	G
Rice A₅	D ₁	MP	P	MP	P
	D ₂	G	G	G	F
	D ₃	MG	F	G	G

3.1.2 For fuzzy AHP

When at the field, data were collected for comparison purpose. The criteria/sub-criteria were pairwise compared to each other via linguistic scale shown in Table 6.

Table 6
Linguistic variables for weight comparison of pairs of criteria/sub-criteria

LV EI NI SI MI I VI

FN (1, 1, 1) (1, 1, 3) (1, 3, 5) (3, 5, 7) (5, 7, 9) (7, 9, 9)

LV	EI	NI	SI	MI	I	VI
FN	(1, 1, 1)	(1, 1, 3)	(1, 3, 5)	(3, 5, 7)	(5, 7, 9)	(7, 9, 9)

EI- Equally Important.

From Table 7 that shows experts comparisons between criteria, a linguistic pairwise comparison matrix for the criteria was formed as shown in Table 8.

Table 7

Pairwise importance weight comparison of criteria on goal

Importance			Criteria	Equally			Criteria	Importance
weight				Important				weight
D ₁	D ₂	D ₃		D ₁	D ₂	D ₃		D ₁	D ₂	D ₃
I	MI		C₁				C₂			MI
	MI		C₁				C₃	VI		MI
			C₁	EI			C₄		MI	MI
I	MI		C₂				C₃			MI
MI	SI		C₂				C₄			MI
MI	MI	MI	C₃				C₄

Table 8

Linguistic pairwise comparison matrix of criteria on goal

		C₁	C₂	C₃	C₄
C₁	D₁	EI	I	VI^- 1	EI
	D₂	EI	MI	MI	MI^- 1
	D₃	EI	MI^- 1	MI^- 1	MI^- 1
C₂	D₁	I^- 1	EI	I	MI
	D₂	MI^- 1	EI	MI	SI
	D₃	MI	EI	MI^- 1	MI^- 1
C₃	D₁	VI	I^- 1	EI	MI
	D₂	MI^- 1	MI^- 1	EI	MI
	D₃	MI	MI	EI	MI
C₄	D₁	EI	MI^- 1	MI^- 1	EI
	D₂	MI	SI^- 1	MI^- 1	EI
	D₃	MI	MI	MI^- 1	EI

Also sub-criteria for each criterion were pairwise compared to each other.

Table 9 is the comparison of sub-criteria for criterion ’product’. Likewise, the linguistic comparison matrix for these sub-criteria was done as for the criteria.

Table 9

Pairwise importance weight comparison of sub-criteria on criterion ’product’

Importance			Sub-	Equally			Sub-	Importance
weight			criteria	important			criteria	weight
D₁	D₂	D₃		D₁	D₂	D₃		D₁	D₂	D₃
I	MI		c₁₁				c₁₂			I
	MI		c₁₁				c₁₃	MI		VI
			c₁₂	EI			c₁₃		MI	MI

Similarly, pairwise comparisons of sub-criteria for other criteria; technology, marketing and customer were made.

Also the alternatives were pairwise compared on each sub-criterion.

Table 10 is the pairwise comparison of alternatives with regard to sub-criterion ’branding’. The same means of data collection was applied for other sub-criteria.

Table 10

Pairwise comparison of alternatives based on sub-criterion c₁₁

Ratings			Alternatives	Equal			Alternatives	Ratings
D ₁	D ₂	D ₃		D ₁	D ₂	D ₃		D ₁	D ₂	D ₃
G	G	VG	Beans A₁				Maize A₂
G			Beans A₁				Sunflower seedsA₃		VG	VG
			Beans A₁				Rosella A₄	G	G	G
			Beans A₁				Rice A₅	G	G	VG
			Maize A₂				Sunflower seedsA₃	MG	MG	VG
	VG	VG	Maize A₂	E			Rosella A₄
G		VG	Maize A₂		E		Rice A₅
MG	MG	MG	Sunflower seeds A₃				Rosella A₄
VG	VG	VG	Sunflower seeds A₃				Rice A₅
			Rosella A₄		E		Rice A₅	P		G

c₁₁- Branding.

3.2 Data analysis by fuzzy TOPSIS model

3.2.1 Aggregation of experts’ assessment of the importance weights of sub- criteria

By Table 2 and Table 4 and the sub-criteria branding; VI ≡ (0.7, 0.9, 0.9), I ≡ (0.5, 0.7, 0.9) and VI ≡ (0.7, 0.9, 0.9).

Thus, importance weight of sub-criterion ‘branding’ is given by ${\tilde{w}}_{1} = (w_{11}, w_{12}, w_{13})$ , where, w₁₁ = min { 0.7, 0.5, 0.7 } = 0.50, $w_{12} = \frac{1}{3} (0.9 + 0.7 + 0.9) = 0.83$ , w₁₃ = max { 0.9, 0.9, 0.9 } = 0.90, correct to two decimal places.

It implies that ${\tilde{w}}_{1} = (0.50, 0.83, 0.90)$ .

By similar procedure, fuzzy numbers ${\tilde{w}}_{2}, {\tilde{w}}_{3}$ and ${\tilde{w}}_{4}$ for importance weights of other criteria were obtained. Table 11 shows importance weight of each sub-criterion and corresponding aggregated importance weight of the criterion.

Table 11
Aggregated importance weights of expert’s assessments on criteria/sub-criteria

Criteria Sub-criteria Importance weight

Sub-criteria criteria

Product Branding (0.5, 0.83, 0.9) ((0.5, 0.81, 0.9))

Delivery ((0.5, 0.83, 0.9))

Patent protection ((0.5, 0.77, 0.9))

Technology Product payment and acquisition ((0.5, 0.77, 0.9)) ((0.5, 0.77, 0.9))

Technology uniqueness ((0.5, 0.77, 0.9))

Marketing Target market ((0.5, 0.77, 0.9)) ((0.5, 0.8, 0.9))

Competition ((0.5, 0.83, 0.9))

Customer Willingness to pay ((0.3, 0.63, 0.9)) ((0.3, 0.67, 0.9))

Heterogeneity ((0.3, 0.63, 0.9))

Demand ((0.5, 0.77, 0.9))

Criteria	Sub-criteria	Importance weight
Product	Branding	(0.5, 0.83, 0.9)	((0.5, 0.81, 0.9))
	Delivery	((0.5, 0.83, 0.9))
	Patent protection	((0.5, 0.77, 0.9))
Technology	Product payment and acquisition	((0.5, 0.77, 0.9))	((0.5, 0.77, 0.9))
	Technology uniqueness	((0.5, 0.77, 0.9))
Marketing	Target market	((0.5, 0.77, 0.9))	((0.5, 0.8, 0.9))
	Competition	((0.5, 0.83, 0.9))
Customer	Willingness to pay	((0.3, 0.63, 0.9))	((0.3, 0.67, 0.9))
	Heterogeneity	((0.3, 0.63, 0.9))
	Demand	((0.5, 0.77, 0.9))

3.2.2 Construction of the decision matrix

As an example, entry ${\tilde{x}}_{11} = (x_{111}, x_{112}, x_{113}) = (1, 3.7, 9)$ of Table 12 is calculated by using entries from Tables 3 5 as follows; three decision makers in Table 5 rated A₁ for C₁ as P, G and VP. This is equivalent to (1, 3, 5) , (5, 7, 9) and (1, 1, 3) respectively in Table 3.

Table 12
The decision matrix

Product C₁ Technology C₂ Markets C₃ Customer C₄

Beans A₁ (1, 3.7, 9) (1, 4.7, 8) (1, 6, 9) (1, 6, 9)

Maize A₂ (2, 6.7, 9) (4, 7.3, 9) (1, 6.3, 9) (4, 7.3, 9)

Sunflower seeds A₃ (3, 7, 9) (5, 7.7, 9) (7, 9, 9) (4, 7.3, 9)

Rosella A₄ (1, 4.7, 9) (2, 5.3, 8) (1, 3.7, 9) (1, 5.7, 9)

Rice A₅ (2, 5.3, 9) (1, 5.3, 9) (2, 6, 9) (1, 5.3, 9)

	Product C₁	Technology C₂	Markets C₃	Customer C₄
Beans A₁	(1, 3.7, 9)	(1, 4.7, 8)	(1, 6, 9)	(1, 6, 9)
Maize A₂	(2, 6.7, 9)	(4, 7.3, 9)	(1, 6.3, 9)	(4, 7.3, 9)
Sunflower seeds A₃	(3, 7, 9)	(5, 7.7, 9)	(7, 9, 9)	(4, 7.3, 9)
Rosella A₄	(1, 4.7, 9)	(2, 5.3, 8)	(1, 3.7, 9)	(1, 5.7, 9)
Rice A₅	(2, 5.3, 9)	(1, 5.3, 9)	(2, 6, 9)	(1, 5.3, 9)

Hence, $x_{111} = \min {1, 5, 1} = 1, x_{112} = \frac{1}{3} (3 + 7 + 1) = 3.7$ and x₁₁₃ = max { 5, 9, 3 } = 9. Similar procedure is applies for other entries ${\tilde{x}}_{ij}$ of the decision matrix.

3.3.2.1 3.2.3 The normalised decision matrix

Since none of the criteria is a cost criterion, Equation (1) was applied.

Consider the fuzzy numbers along C₁ in Table 12. The set of third entries along C₁ is {9, 9, 9, 9, 9} and $c_{1}^{*} = \max {9, 9, 9, 9, 9} = 9$ . So each entry along C₁ was divided by 9. In the same manner, other entries ${\tilde{x}}_{ij}$ were obtained and their results are shown in Table 13 of the normalised decision matrix.

Table 13
The normalised decision matrix

C ₁ C ₂ C ₃ C ₄

A₁ (0.11, 0.41, 1) (0.11, 0.52, 0.89) (0.11, 0.67, 1) (0.11, 0.67, 1)

A₂ (0.22, 0.74, 1) (0.44, 0.81, 1) (0.11, 0.70.1) (0.44, 0.81, 1)

A₃ (0.33, 0.78, 1) (0.56, 0.86, 1) (0.78, 1, 1) (0.44, 0.81, 1)

A₄ (0.11, 0.52, 1) (0.22, 0.59, 0.89) (0.11, 0.41, 1) (0.11, 0.63, 1)

A₅ (0.22, 0.59, 1) (0.11, 0.59, 1) (0.22, 0.67, 1) (0.11, 0.59, 1)

	C ₁	C ₂	C ₃	C ₄
A₁	(0.11, 0.41, 1)	(0.11, 0.52, 0.89)	(0.11, 0.67, 1)	(0.11, 0.67, 1)
A₂	(0.22, 0.74, 1)	(0.44, 0.81, 1)	(0.11, 0.70.1)	(0.44, 0.81, 1)
A₃	(0.33, 0.78, 1)	(0.56, 0.86, 1)	(0.78, 1, 1)	(0.44, 0.81, 1)
A₄	(0.11, 0.52, 1)	(0.22, 0.59, 0.89)	(0.11, 0.41, 1)	(0.11, 0.63, 1)
A₅	(0.22, 0.59, 1)	(0.11, 0.59, 1)	(0.22, 0.67, 1)	(0.11, 0.59, 1)

3.2.4 The weighted normalised decision matrix

Consider $\begin{matrix} \tilde{w} = [\begin{matrix} {\tilde{w}}_{1} \\ {\tilde{w}}_{2} \\ \begin{matrix} {\tilde{w}}_{3} \\ {\tilde{w}}_{4} \end{matrix} \end{matrix}] = [\begin{matrix} (0.50, 0.81, 0.90) \\ (0.50, 0.77, 0.90) \\ \begin{matrix} (0.50, 0.80, 0.90) \\ (0.30, 0.67, 0.90) \end{matrix} \end{matrix}] \end{matrix}$

from Table 11 and the matrix in Table 13. The weighted normalised matrix $\tilde{v}$ was obtained. For example, the entry

$\begin{matrix} {\tilde{v}}_{11} = {\tilde{r}}_{11} (\cdot) {\tilde{w}}_{1} = [\begin{matrix} 0.11 \\ 0.41 \\ 1.00 \end{matrix}] (\cdot) [\begin{matrix} 0.50 \\ 0.81 \\ 0.90 \end{matrix}] = [\begin{matrix} 0.0550 \\ 0.3321 \\ 0.9000 \end{matrix}] \end{matrix}$

and

$\begin{matrix} {\tilde{v}}_{21} = {\tilde{r}}_{21} (\cdot) {\tilde{w}}_{1} = [\begin{matrix} 0.22 \\ 0.74 \\ 1.00 \end{matrix}] (\cdot) [\begin{matrix} 0.50 \\ 0.81 \\ 0.90 \end{matrix}] = [\begin{matrix} 0.1100 \\ 0.5994 \\ 0.9000 \end{matrix}] \end{matrix}$

Other ${\tilde{v}}_{ij}$ entries were obtained in the same manner. Results are in Table 14.

Table 14
The weighted normalised decision matrix

C ₁ C ₂ C ₃ C ₄

A₁ (0.055, 0.3321, 0.9) (0.055, 0.4009, 0.801) (0.055, 0.536, 0.9) (0.033, 0.4489, 0.9)

A₂ (0.11, 0.5994, 0.9) (0.22, 0.6251, 0.9) (0.055, 0.56.0.9) (0.132, 0.5427, 0.9)

A₃ (0.055, 0.4212, 0.9) (0.28, 0.6622, 0.9) (0.39, 0.8, 0.9) (0.132, 0.5427, 0.9)

A₄ (0.055, 0.4212, 0.9) (0.11, 0.4543, 0.801) (0.055, 0.328, 0.9) (0.033, 0.4221, 0.9)

A₅ (0.11, 0.4779, 0.9) (0.055, 0.4543, 0.9) (0.11, 0.536, 0.9) (0.033, 0.3953, 0.9)

	C ₁	C ₂	C ₃	C ₄
A₁	(0.055, 0.3321, 0.9)	(0.055, 0.4009, 0.801)	(0.055, 0.536, 0.9)	(0.033, 0.4489, 0.9)
A₂	(0.11, 0.5994, 0.9)	(0.22, 0.6251, 0.9)	(0.055, 0.56.0.9)	(0.132, 0.5427, 0.9)
A₃	(0.055, 0.4212, 0.9)	(0.28, 0.6622, 0.9)	(0.39, 0.8, 0.9)	(0.132, 0.5427, 0.9)
A₄	(0.055, 0.4212, 0.9)	(0.11, 0.4543, 0.801)	(0.055, 0.328, 0.9)	(0.033, 0.4221, 0.9)
A₅	(0.11, 0.4779, 0.9)	(0.055, 0.4543, 0.9)	(0.11, 0.536, 0.9)	(0.033, 0.3953, 0.9)

3.2.5 Determination of FPIS and FNIS

For FPIS, ${\tilde{v}}_{j}^{*} = (max_{j} {v_{ij 1}}, max_{j} {v_{ij 2}}, max_{j} {v_{ij 3}}), j = 1, 2, 3, 4 \forall i$ .

Hence, for j = 1; $\begin{matrix} {\tilde{v}}_{1}^{*} = (max_{1} {v_{i 11}}, max_{1} {v_{i 12}}, max_{1} {v_{i 13}}) \\ = (max_{1} {0.055, 0.11, 0.055, 0.055, 0.11}, \\ max_{1} {0.3321, 0.5994, 0.4212, 0.4212, 0.4779}, \\ max_{1} {0.9, 0.9, 0.9, 0.9}) = (0.11, 0.5994, 0.9) \end{matrix}$

By similar procedure, ${\tilde{v}}_{2}^{*}, {\tilde{v}}_{3}^{*}$ and ${\tilde{v}}_{4}^{*}$ were obtained.

Hence, $\begin{matrix} A^{*} = [\begin{matrix} \begin{matrix} {\tilde{v}}_{1}^{*} \\ {\tilde{v}}_{2}^{*} \end{matrix} \\ {\tilde{v}}_{3}^{*} \\ {\tilde{v}}_{4}^{*} \end{matrix}] = [\begin{matrix} \begin{matrix} (0.1100, 0.5994, 0.9000) \\ (0.2800, 0.6622, 0.9000) \end{matrix} \\ (0.3900, 0.8000, 0.9000) \\ (0.1320, 0.5427, 0.9000) \end{matrix}] \end{matrix}$

For FNIS, $\begin{matrix} {\tilde{v}}_{j}^{-} = & (min_{j} {v_{ij 1}}, min_{j} {v_{ij 2}}, min_{j} {v_{ij 3}}), \\ j = 1, 2, 3, 4 \forall I \end{matrix}$

Thus, $\begin{matrix} {\tilde{v}}_{1}^{-} = (min_{1} {v_{i 11}}, min_{1} {v_{i 12}}, min_{1} {v_{i 13}}) \\ = (min_{1} {0.055, 0.11, 0.055, 0.055, 0.11}, \\ min_{1} {0.3321, 0.5994, 0.4212, 0.4212, 0.4779}, \\ min_{1} {0.9, 0.9, 0.9, 0.9}) = (0.055, 0.3321, 0.9) \end{matrix}$

${\tilde{v}}_{2}^{-}, {\tilde{v}}_{3}^{-},$ and ${\tilde{v}}_{4}^{-}$ were obtained in the same manner.

Therefore, $\begin{matrix} A^{-} = [\begin{matrix} \begin{matrix} {\tilde{v}}_{1}^{-} \\ {\tilde{v}}_{2}^{-} \end{matrix} \\ {\tilde{v}}_{3}^{-} \\ {\tilde{v}}_{4}^{-} \end{matrix}] = [\begin{matrix} \begin{matrix} (0.0550, 0.3321, 0.9000) \\ (0.0550, 0.4009, 0.8010) \end{matrix} \\ (0.0550, 0.3280, 0.9000) \\ (0.0330, 0.3953, 0.9000) \end{matrix}] \end{matrix}$

3.2.6 Calculations of distances d (A_i, A^*) and d (A_i, A^-)

For FPIS, $d (A_{i}, A^{*}) \equiv d ({\tilde{v}}_{ij}, {\tilde{v}}_{j}^{*})$ for example,

$d ({\tilde{v}}_{11}, {\tilde{v}}_{1}^{*}) = d ((0.055, 0.3321, 0.9), (0.11, 0.5994, 0.9)) = \sqrt{\frac{1}{3} ({(0.11 - 0.055)}^{2} + {(0.5994 - 0.3321)}^{2} + {(0.9 - 0.9)}^{2})} = 0.1576$ correct to 4 decimal places.

Similar procedure was used to obtain other entries in Table 15.

Table 15
Distances between alternatives and FPIS

C ₁ C ₂ C ₃ C ₄ $d_{j}^{}$

$d ({\tilde{v}}_{1 j}, {\tilde{v}}_{j}^{})$ 0.1576 0.2071 0.2463 0.0787 0.6897

$d ({\tilde{v}}_{2 j}, {\tilde{v}}_{j}^{})$ 0 0.0407 0.2379 0 0.2786

$d ({\tilde{v}}_{3 j}, {\tilde{v}}_{j}^{})$ 0.1077 0 0 0 0.1077

$d ({\tilde{v}}_{4 j}, {\tilde{v}}_{j}^{})$ 0.1077 0.1653 0.3342 0.0901 0.6973

$d ({\tilde{v}}_{5 j}, {\tilde{v}}_{j}^{})$ 0.0701 0.1769 0.2222 0.1025 0.5717

	C ₁	C ₂	C ₃	C ₄	$d_{j}^{*}$
$d ({\tilde{v}}_{1 j}, {\tilde{v}}_{j}^{*})$	0.1576	0.2071	0.2463	0.0787	0.6897
$d ({\tilde{v}}_{2 j}, {\tilde{v}}_{j}^{*})$	0	0.0407	0.2379	0	0.2786
$d ({\tilde{v}}_{3 j}, {\tilde{v}}_{j}^{*})$	0.1077	0	0	0	0.1077
$d ({\tilde{v}}_{4 j}, {\tilde{v}}_{j}^{*})$	0.1077	0.1653	0.3342	0.0901	0.6973
$d ({\tilde{v}}_{5 j}, {\tilde{v}}_{j}^{*})$	0.0701	0.1769	0.2222	0.1025	0.5717

$d_{j}^{*} = \sum_{j = 1}^{4} d ({\tilde{v}}_{ij}, {\tilde{v}}_{j}^{*})$ for each i, i = 1, 2, 3, 4, 5.

For FNIS, $d (A_{i}, A^{-}) \equiv d ({\tilde{v}}_{ij}, {\tilde{v}}_{j}^{-})$ and same procedure was used to obtain $d ({\tilde{v}}_{ij}, {\tilde{v}}_{j}^{-})$ as for $d ({\tilde{v}}_{ij}, {\tilde{v}}_{j}^{*})$ . The results are shown in Table 16.

Table 16

Distances between alternatives and FNIS

	C ₁	C ₂	C ₃	C ₄	$d_{j}^{-}$
$d ({\tilde{v}}_{1 j}, {\tilde{v}}_{j}^{-})$	0	0	0.1201	0.0309	0.151
$d ({\tilde{v}}_{2 j}, {\tilde{v}}_{j}^{-})$	0.1576	0.1706	0.1339	0.1025	0.5646
$d ({\tilde{v}}_{3 j}, {\tilde{v}}_{j}^{-})$	0.0514	0.2071	0.3342	0.1025	0.6952
$d ({\tilde{v}}_{4 j}, {\tilde{v}}_{j}^{-})$	0.0514	0.0443	0	0.0443	0.1112
$d ({\tilde{v}}_{5 j}, {\tilde{v}}_{j}^{-})$	0.09	0.0649	0.1242	0	0.2791

$d_{j}^{-} = \sum_{j = 1}^{4} d ({\tilde{v}}_{ij}, {\tilde{v}}_{j}^{-})$ for each i, i = 1, 2, 3, 4, 5.

3.2.7 Determination of closeness coefficient CC_i

3.3 Data analysis by fuzzy AHP model

Using the linguistic scale of Table 6 and Equation (3), the comparisons of criteria by experts in Table 8 were aggregated and the results are shown in Table 18.

Table 18
The aggregated pairwise comparison matrix of criteria on goal

C ₁ C ₂ C ₃ C ₄

C ₁ (1, 1, 1) (2.7143, 4.0667, 5.4444) (1.08471.7704, 2.4906) (0.4286, 0.4667, 0.5556)

C ₂ (0.1837, 0.2459, 0.3684) (1, 1, 1) (2.7143, 4.0667, 5.4444) (1.3810, 0.7333, 4.1111)

C ₃ (0.4013, 0.5649, 0.9220) (0.1837, 0.2459, 0.3684) (1, 1, 1) (3, 5, 7)

C ₄ (1.8000, 2.1429, 2.3333) (0.2432, 0.3659, 0.7241) (0.1429, 0.2000, 0.3333) (1, 1, 1)

	C ₁	C ₂	C ₃	C ₄
C ₁	(1, 1, 1)	(2.7143, 4.0667, 5.4444)	(1.08471.7704, 2.4906)	(0.4286, 0.4667, 0.5556)
C ₂	(0.1837, 0.2459, 0.3684)	(1, 1, 1)	(2.7143, 4.0667, 5.4444)	(1.3810, 0.7333, 4.1111)
C ₃	(0.4013, 0.5649, 0.9220)	(0.1837, 0.2459, 0.3684)	(1, 1, 1)	(3, 5, 7)
C ₄	(1.8000, 2.1429, 2.3333)	(0.2432, 0.3659, 0.7241)	(0.1429, 0.2000, 0.3333)	(1, 1, 1)

Then the fuzzy geometric means ${\tilde{r}}_{i}$ , i = 1, 2, 3, 4 of the criteria comparison values were determined by Equation (4). For example, $\begin{matrix} {\tilde{r}}_{2} = ({(0.1837 \times 1 \times 2.7143 \times 1.381)}^{\frac{1}{4}}, \\ {(0.2459 \times 1 \times 4.0667 \times 2.7333)}^{\frac{1}{4}}, \\ {(0.3684 \times 1 \times 5.4444 \times 4.1111)}^{\frac{1}{4}}) \\ = (1.4514, 1.2858, 1.6946) \end{matrix}$

In the same manner, other ${\tilde{r}}_{i}$ were found and the results are in Table 19.

Table 19

Criteria geometric means and others

i	Criteria	${\tilde{r}}_{i}$			Relative fuzzy weight ${\tilde{w}}_{i}$			M _i	N _i
1	Product C₁	1.0599	1.3539	1.6567	0.1942	0.3237	0.4481	0.3220	0.3057
2	Technology C₂	1.4514	1.2858	1.6946	0.2659	0.3074	0.4584	0.3439	0.3265
3	Marketing C₃	0.6858	0.9129	1.2418	0.1256	0.2183	0.3359	0.2266	0.2151
4	Customer C₄	0.5002	0.6293	0.8663	0.8663	0.1505	0.2343	0.1608	0.1527
$s = \sum_{i = 1}^{4} {\tilde{r}}_{i}$		3.6973	4.1819	5.4594
s ^-1		0.2705	0.2391	0.1832

The resulting s^-1 (the inverse of Equation (5)) was then written in increasing order (s^-1 = (l, m, u) = (0.1832, 0.2391, 0.2705)) to make sense for a triangular fuzzy number before multiplying it with each ${\tilde{r}}_{i}$ to obtain the relative fuzzy weights ${\tilde{w}}_{i}$ . For example, $\begin{matrix} {\tilde{w}}_{1} = (1.0599 \times 0.1832, 1.3539 \times 0.2391, \\ 1.6567 \times 0.2705) = (0.1942, 0.3237, 0.4481) \end{matrix}$

Then M_i and N_i (≡ w_j, i, j = 1, 2, 3, 4) were determined by applying Equations (7) respectively. All results are found in Table 19.

Thus the crisp weights of criteria are; $\begin{matrix} {[\begin{matrix} w_{1} & w_{2} & \begin{matrix} w_{3} & w_{4} \end{matrix} \end{matrix}]}^{T} \\ = {[\begin{matrix} 0.3057 & 0.3265 & \begin{matrix} 0.2151 & 0.1527 \end{matrix} \end{matrix}]}^{T} \end{matrix}$

Similar procedure was applied to determine the relative weights for sub-criteria and relative weights of alternatives with respect to each sub-criterion. The results of the calculated values are shown in Table 20.

Table 20

Computed relative weights of sub-criteria and alternatives

		Alternative weights on every sub-criterion
Sub-criteria	Crisp weights	A ₁	A ₂	A ₃	A ₄	A ₅
c ₁₁	0.3500	0.0976	0.1212	0.5463	0.1018	0.1331
c ₁₂	0.1902	0.4489	0.1409	0.2529	0.0493	0.1080
c ₁₃	0.4598	0.0979	0.2991	0.4493	0.0566	0.0971
c ₂₄	0.5621	0.1796	0.1704	0.3969	0.1307	0.1225
c ₂₅	0.4379	0.0947	0.0986	0.5933	0.1112	0.1021
c ₃₆	0.6588	0.2353	0.3310	0.1662	0.0971	0.1704
c ₃₇	0.3412	0.3378	0.2754	0.1740	0.0485	0.1642
c ₄₈	0.3160	0.3662	0.1826	0.2641	0.0497	0.1374
c ₄₉	0.2734	0.2174	0.0912	0.4172	0.0434	0.2308
c ₄₁₀	0.4106	0.3731	0.3626	0.0726	0.0274	0.1645

Then, using entries of Table 19 (rightmost column) and Table 20, the overall score of each alternative was computed by Equation (8) and the results are shown in Table 21.

Table 21

Computed scores of alternatives

C _j	w _j	c _jk	$w_{k}^{A_{i}}$	Alternative scores
				A ₁	A ₂	A ₃	A ₄	A ₅
C ₁	0.3057	c ₁₁	0.3500	0.0104	0.0130	0.0585	0.0109	0.0142
		c ₁₂	0.1902	0.0261	0.0082	0.0147	0.0029	0.0063
		c ₁₃	0.4598	0.0138	0.0420	0.0632	0.0080	0.0136
C ₂	0.3265	c ₂₄	0.5621	0.0330	0.0313	0.0728	0.0240	0.0225
		c ₂₅	0.4379	0.0135	0.0141	0.0848	0.0159	0.0146
C ₃	0.2151	c ₃₆	0.6588	0.0333	0.0469	0.0236	0.0138	0.0241
		c ₃₇	0.3412	0.0248	0.0202	0.0128	0.0036	0.0121
C ₄	0.1527	c ₄₈	0.3160	0.0177	0.0088	0.0127	0.0024	0.0066
		c ₄₉	0.2734	0.0091	0.0038	0.0174	0.0018	0.0096
		c ₄₁₀	0.4106	0.0234	0.0227	0.0046	0.0017	0.0103
		Total score		0.2051	0.2110	0.3651	0.0850	0.1339
		Rank		3	2	1	5	4

3.4 Interpretation of results

Upon looking at the closeness coefficients CC_i column of Table 17, it was found that CC₃ > CC₂ > CC₅ > CC₁ > CC₄. This guided ranking of alternatives (processed crops) in Table 17. The higher the value of CC_i, the better the choice or option.

Table 17
Calculated closeness coefficients CC_i

$d_{j}^{}$ $d_{j}^{-}$ $d_{j}^{} + d_{j}^{-}$ CC _i Rank

A ₁ 0.6897 0.1510 0.8407 0.1796 4

A ₂ 0.2786 0.5646 0.8432 0.6696 2

A ₃ 0.1077 0.6952 0.8029 0.8659 1

A ₄ 0.6973 0.1112 0.8085 0.1375 5

A ₅ 0.5717 0.2791 0.8508 0.3280 3

	$d_{j}^{*}$	$d_{j}^{-}$	$d_{j}^{*} + d_{j}^{-}$	CC _i	Rank
A ₁	0.6897	0.1510	0.8407	0.1796	4
A ₂	0.2786	0.5646	0.8432	0.6696	2
A ₃	0.1077	0.6952	0.8029	0.8659	1
A ₄	0.6973	0.1112	0.8085	0.1375	5
A ₅	0.5717	0.2791	0.8508	0.3280	3

${CC}_{i} = \frac{d_{j}^{-}}{;} d_{j}^{*} + d_{j}^{-}$ for each i = 1, 2, 3, 4, 5 and j = 1, 2, 3, 4.

One can defuzzify the aggregated importance weights of criteria in Table 11 (not a step of fuzzy TOPSIS method) in order to easily visualise the importance weights of criteria.

By centre of area method, each ${\tilde{w}}_{j}$ is defuzzified by using $\begin{matrix} M_{j} = \frac{w_{1}^{j} + w_{2}^{j} + w_{3}^{j}}{3} \end{matrix}$ where, $w_{i}^{j}, i = 1, 2, 3$ are the components of ${\tilde{w}}_{j}$ . The resulting M_js are normalised by the formula $\begin{matrix} w_{j} = \frac{M_{j}}{(\sum_{j = 1}^{4} M_{j})} \end{matrix}$

Results of all these are in Table 22.

Table 22

Defuzzified importance weights w_j of criteria

Main criteria C_j	Importance weight	Defuzzified	Normalised
	(fuzzy) ${\tilde{w}}_{j}$	weight (crisp) M_j	defuzzified weight w_j
Product	(0.50, 0.81, 0.90)	0.737	0.2617
Technology	(0.50, 0.77, 0.90)	0.723	0.2567
Marketing	(0.50, 0.80, 0.90)	0.733	0.2603
Customer	(0.30, 0.67, 0.90)	0.623	0.2212

According to aggregation procedure applied in the whole process, it was observed that crisp weights of criteria do not differ much from one another but more weight went to the product criterion (0.2617) and less weight to customer criterion (0.2212) as seen in Table 22.

A comparison of the results (Table 19) by fuzzy AHP showed that more weight went to technology criterion (0.3265) and less weight also to customer criterion (0.1527) (Fig. 2).

Fig. 2

Comparison of weights obtained by FTOPSIS method against FAHP method.

The ranking of alternatives was compared also by fuzzy AHP and it was guided by the total scores (Table 21) of the alternatives; TscoreA₃ > TscoreA₂ > TscoreA₁ > TscoreA₅ > TscoreA₄ (Fig. 3).

Fig. 3

Comparison of CC_i obtained by FTOPSIS method against total scores by FAHP method.

4 Conclusion and recommendations

The main goal of this study was to assess the market performance of processed products of crops at Sustainable Agriculture Tanzania (SAT) in Morogoro in order to assist in decision making process of the organisation.

According to the results by fuzzy TOPSIS model in Table 17, the CC_i of the alternatives were such that A₃ > A₂ > A₅ > A₁ > A₄. This means that according to the criteria product, technology, marketing and customer as they were defined by their sub-criteria for each criterion, the processed product of the crop sunflower seeds performed better than the rest of the other four processed products of the crops beans, maize, rosella and rice.

As a general remark, the sunflower seeds processed product performed best followed by processed maize, processed rice and then processed beans. The product of the processed rosella performed in the lowest position.

Results by fuzzy TOPSIS model were compared by results of fuzzy AHP model. It was found that, processed products from sunflower seeds, maize and rosella retained their ranks while beans and rice exchanged positions (Fig. 4).

Fig. 4

Comparison of ranks obtained by FTOPSIS model against FAHP model.

Four criteria with ten sub-criteria altogether were considered and applied in evaluating performance of the products of processed crops. SAT may work out on products that performed lower in the market by looking at their corresponding weights of criteria/sub-criteria.

Similar studies may be conducted by considering other different criteria deemed useful to improve performance of same products. The study too may be replicated for other agro-processed products at SAT or elsewhere.

Footnotes

Acknowledgments

Thanks to my supervisors; Prof. Valentina E. Balas, Department of Automation and Applied Informatics, Aurel Vlaicu University of Arad, Romania and Dr. David Koloseni, Department of Mathematics, University of Dar es Salaam, Tanzania for their assistance with particular techniques, methodologies, and comments that greatly improved the manuscript. Thanks to Madam Janet, the managing director and the management of SAT in general for the warm welcome to work with their organisation.

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Application of fuzzy TOPSIS in assessing performance of agro-processed organic crops: A case of Sustainable Agriculture Tanzania (SAT) in Morogoro

Abstract

Keywords

1 Introduction

2 Methodology

2.1 The fuzzy TOPSIS method

2.2.1 Determination of Weights of Criteria wj

3.1.1 For fuzzy TOPSIS

Table 2 Linguistic variables for importance weight of each criterion/sub-criteria LV NI SI MI I VI FN (0.1, 0.1, 0.3) (0.1, 0.3, 0.5) (0.3, 0.5, 0.7) (0.5, 0.7, 0.9) (0.7, 0.9, 0.9)

Table 6 Linguistic variables for weight comparison of pairs of criteria/sub-criteria LV EI NI SI MI I VI FN (1, 1, 1) (1, 1, 3) (1, 3, 5) (3, 5, 7) (5, 7, 9) (7, 9, 9)

3.2.1 Aggregation of experts’ assessment of the importance weights of sub- criteria

3.2.6 Calculations of distances d (A i , A*) and d (A i , A-)

3.3 Data analysis by fuzzy AHP model

Table 17 Calculated closeness coefficients CC i d j * d j - d j * + d j - CC i Rank A 1 0.6897 0.1510 0.8407 0.1796 4 A 2 0.2786 0.5646 0.8432 0.6696 2 A 3 0.1077 0.6952 0.8029 0.8659 1 A 4 0.6973 0.1112 0.8085 0.1375 5 A 5 0.5717 0.2791 0.8508 0.3280 3

Footnotes

Acknowledgments

References

2.2.1 Determination of Weights of Criteria w_j

Table 2
Linguistic variables for importance weight of each criterion/sub-criteria

LV NI SI MI I VI

FN (0.1, 0.1, 0.3) (0.1, 0.3, 0.5) (0.3, 0.5, 0.7) (0.5, 0.7, 0.9) (0.7, 0.9, 0.9)

Table 6
Linguistic variables for weight comparison of pairs of criteria/sub-criteria

LV EI NI SI MI I VI

FN (1, 1, 1) (1, 1, 3) (1, 3, 5) (3, 5, 7) (5, 7, 9) (7, 9, 9)

3.2.6 Calculations of distances d (A_i, A^*) and d (A_i, A^-)

Table 17
Calculated closeness coefficients CC_i

$d_{j}^{}$ $d_{j}^{-}$ $d_{j}^{} + d_{j}^{-}$ CC _i Rank

A ₁ 0.6897 0.1510 0.8407 0.1796 4

A ₂ 0.2786 0.5646 0.8432 0.6696 2

A ₃ 0.1077 0.6952 0.8029 0.8659 1

A ₄ 0.6973 0.1112 0.8085 0.1375 5

A ₅ 0.5717 0.2791 0.8508 0.3280 3