Investigating the Economies of Scope and Cost Effectiveness in Manufacturing Companies with Interval Data

The success requirement of managers’ progress, development and performance improvement lie in their attention to product variety and company effectiveness. Economies of scope (ES) examine the advantages of production or the services diversification of a company based on cost versus production by companies that produce the same products or services separately. Data Envelopment Analysis (DEA) is known as a suitable method for evaluating ES and cost effectiveness. DEA models are introduced with certain input and output costs, while many companies and manufacturing industries in different sectors of production and service provision may not have accurate information on available costs and outputs because of calculation errors, old information, and multiple repeated measurements. The estimation DEA for ES and cost effectiveness are sensitive to changes, also some parameters, such as cost and price, are fluctuated. Therefore, it is a requirement to focus on the interval DEA. Our most important goals in this article are: (1) we develop new DEA models to measure the ES and cost effectiveness of decision-making units (DMUs) under data uncertainty. These models will become non-linear and non-convex models; hence, (2) we identify an appropriate range for ES and cost effectiveness of DMUs from the optimistic and pessimistic viewpoints, allowing decision-makers can use the upper and lower limits or their combination depending on the optimistic and pessimistic viewpoints, (3) we apply our developed models to assess the ES and cost-effectiveness performance of 24 institutions, considering data uncertainties that may affect the quality and reliability of the results. (4) The proposed models’ features have been analyzed, and the impact of interval data on cost effectiveness and ES has been evaluated. The application description of the proposed models for determining ES and cost effectiveness shows that a company can exhibit economies of scope without necessarily being Cost Effectiveness.


Introduction
From the beginning, humans have sought greater profitability in their endeavors, leading them to strive for reduced operational inputs.In a competitive environment, organizations aim to minimize costs and increase outputs to generate more profit, create societal welfare, and maintain their competitiveness.
The main idea of most economic systems is centered around mass production with minimal inputs.Consequently, managers seek effective ways to achieve maximum output with minimum costs, ensuring economic viability.Today, one of the most significant and debated issues in organizations and companies is the reduction of costs and the crucial role of cost effectiveness in achieving the objectives of managers, company owners, and economic enterprises.
Economies of scope (ES) examines the advantages of diversifying production or services within a company on the basis cost versus production by companies that produce and provide the same products or services separately.The concept ES was originally introduced by Baumol et al.
[1] in the context of cost functions.Baumol et al. stated that if the cost of producing two products by one company is lower than the cost of producing the same products separately in specialized production companies, then ES will occur.Fare [2,3] extended Baumol et al. method on based DEA.Various methodologies have been proposed to measure ES, Hajargasht et al. [4] introduced a dual measure for evaluating economies of scope; estimating the cost function is not required in their method.Sahoo and Tone [5] introduced two DEA model to estimate a cost frontier exhibiting ES in production one based on the factor-based technology set and the other based on the cost-based technology set.De Witte et al. [6] presented a non-parametric methodology to investigate the existence of ES between teaching and research.Carvalho and Marques [7] proposed a nonparametric methodology based on more robust partial frontier nonparametric methods to study the presence of scope and scale economies.Ahranjani et al. [8] introduced a two-stage network DEA model to ES between two products.
Ferreira et al. [9] proposed a generalized algorithm to obtain locally convex frontiers using the directional order-α frontier method and introduced a generalized economies-of-scope-based ratio that allows the introduction of any inefficiency source.Zaker Harofte and Hosseinzadeh Saljooghi [10] presented models for evaluating ES in two-stage supply chain systems.In their model, undesirable products will be given to the second stage for processing and modification and will be returned to the first stage after re-modification.Kao et al. [11] studied optimal expansion paths for hospitals of different types from an ES perspective they studied how ES can be exploited by Chinese hospitals.The degree of economies of scope is measured as efficiency gains through DEA.Extensive studies have been done on the ES with different applications using DEA.ES has been potentially used in various industries.Nayak [12] analyzed the relevance and significance of ES in the context of agriculture and smallholder farmers from an efficiency and sustainability perspective.Li and Marinc [13] examined the existence of Economies of scale and scope in financial market infrastructures.Villafuerte et al. [14] evaluated Economies of scale and scope in publicly funded biomedical and health research.In this context, we can mention other people such as Ferrier et al. [15] in the banking industry, Berger et al. [16] in the insurance industry, Cummins, Weiss and Zi [17] in the US insurance industry, Cherchye et al. [18], Lee et al. [19] in the hospital specialization, De Witte and Marques [20] in the water sector, Ferreira et al. [9] in the hospitals.

Motivation
The above literature shows the lack of DEA studies in exploring Economies of Scope and cost-effectiveness (CCE) models under interval and uncertainty sources and understanding the real impact of these models on ES and CCE measurement results.Although there are many studies focusing on ES evaluation, cost-effectiveness DEA models are mostly created with the assumption of deterministic inputs and outputs.However, in real-time situations and different departments of organizations or companies, we may encounter cases where we do not have accurate information about the costs (inputs) of the company/organization (such as the cost of sending goods) and outputs (such as the number of invoices), and it's impossible to determine the exact numerical value for these costs or outputs and we believe that many researches are best described by the intermediate case, where some imprecise input and output data are available.In this condition, the DEA model does not seem suitable for organizations/companies, and the used models should evaluate ES considering imprecise data (interval or fuzzy data).The purpose of this paper is to fill this gap by contributing to a better understanding and modelling of uncertainty in ES and CCE models in situations where a company is interested in viewing the best ES and CCE frontier this will lead to understanding the level of uncertainty to arrive at a more valid and accurate decision.
In addition to examining ES, we will examine the effectiveness of joint production and specialized production of companies using the CCE model of data envelopment analysis.Using the results of the model, it is possible to determine the effectiveness of the measures taken to achieve the predetermined goals of joint production companies compared to specialized production companies.
We believe that many research situations are best described by the intermediate case, where some uncertain input and output data are available.

Novelties
When the data are in the form of imprecise data, the ES and CCE scale calculated from the data must be uncertain as well.Thus, in this paper, considering imprecise data (interval or fuzzy data) for each organization, we have expanded the two concepts of Economies of Scope and cost effectiveness of production diversity to with interval sources, which have not been researched so far.
We extend the ES and CCE measurement theories from two distinctive perspectives-optimistic and pessimisticso that we can modify the incomplete price information by using the upper and lower bounds for the ES and CCE scales.

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We transform non-linear DEA models ES and CCE to linear programming equivalents based on the original dataset, applying transformations only to the variables.
We try to identify the right range for ES and CCE, similar to accurate data, to determine ES and CCE DEA units.
The presented models provide more useful information to organizations and industrial activities.

Structure of the Paper
The present study continues as follows: In Sects. 2 and 3, interval models and ES concepts are introduced, respectively.In Sect.4, we discuss about cost effectiveness.In Sects.5 and 6, the proposed model is presented to determine ES and CCE with interval data.The numerical example, discussion and conclusions are provided in Sects.7, 8 and 9, respectively.

Interval DEA
The conventional DEA-based efficiency measurement methods assume accurate and definite data, whereas in various economic sectors, decision-makers are faced with inaccurate inputs and outputs for which the use of conventional DEA models will not be appropriate.Much research has been performed to calculate relative efficiency with imprecise DEA data.Cooper et al. [21] first conducted studies on how imprecise data, especially bounded data, behave in the DEA.The nonlinear DEA model obtained was designated as an imprecise DEA model or IDEA [21].Despotis and Smirlis [22] obtained the best relative efficiency under the most favorable conditions and the worst relative efficiency under the most unfavorable conditions.Entani and Yanaka [23] used the pessimistic and pessimistic perspectives, to represent a model for evaluating interval efficiency.They first developed their model for constant data and then for interval and fuzzy data.Wang et al. [24] pointed out weaknesses in interval DEA models related to the use of different production boundaries.They addressed this by considering a constant production boundary and approaching the problem from optimistic and pessimistic viewpoints.Mo et al. [25] introduced the Interval Modified Slacks-Based Measure model considering the undesirable outputs.This model was then used to evaluate the efficiency of a set of homogeneous units with imprecise and negative data [25].Toloo et al. [26] presented a pair of IDEA models based on pessimistic and optimistic perspectives to identify the unique status of each of the dual interval factors.Esmaeili et al. [27] proposed the gridded two-stage DEA model under interval data.Hosseinzadeh Lotfi et al. [28] developed Network DEA models on interval data.Hosseinzadeh Lotfi et al. [29] proposed Malmquist productivity index (MPI) for decision-making unit with interval data has been evaluated.Chakraborty et.al [30] described an impression of different representations of nonlinear pentagonal intuitionistic fuzzy number (PIFN) and its classification under different scenarios and presented a new de-antiunification technique of non-linear PIFN with its various graphical representations.Banik et al. [31] presented an improved Multi-criteria group decision-making (MCGDM) strategy in a pentagonal neutrosophic environment incorporating grey relational analysis and method on the removal effects of criteria (MEREC) techniques to address the relative advantages and disadvantages of these aspects in MCGDM their model can capture the underlying uncertainties in a robust way and can produce consistent results in a more rigorous way.Jin et al. [32] stated that the recently proposed basic uncertain information can directly present numerical uncertainties for given real values, but it cannot handle given interval values which themselves also have uncertainties Hence they analyzed some basic operations, weighted arithmetic mean and preference transformation for interval basic uncertain information.Jin et al. [33] presented some new definitions of relative basic uncertain information, relative certainty/uncertainty degree and comprehensive certainty/uncertainty with some related measurements and analysis.They defined two corresponding aggregation operators: the relative basic uncertain information valued weighted arithmetic mean operator and the interval induced relative basic uncertain information valued ordered weight averaging operator.Jin et al. [34] introduced two objective methods to derive incomplete fuzzy relation from a set of vectors and basic uncertain information (BUI) granules and also, they suggested two scaling methods to transform contradictive fuzzy relation into incomplete fuzzy relation in the research done in this field, we can also mention the work of people like Kim et al. [35], Lee et al. [36], Kao [37] etc. Haque et al. [38] defined and discussed various algebraic properties of logarithmic operational law for GSFN where the logarithmic base is a positive real number.They suggested a novel scheme to detect the best cloud service provider using logarithmic operational law in a generalized spherical fuzzy environment.

DEA Models with Interval Data
Consider n units under evaluation (DMU) .Each (j = 1, 2, … , n)DMU j u s e s m n e g a t i v e i n p u t (i = 1, 2, … , m)x ij to generate s negative output (r = 1, 2, … , s)y rj .These inputs and outputs are in the interval forms, x ij ∈ x l ij , x u ij and y rj ∈ y l rj , y u rj .x l ij and x u ij are the lower and upper bounds of the input x ij , and y l rj and y u rj are the lower and upper bounds of the output (y rj ) , respectively.Also x l ij > 0 and y l rj > 0.
Considering a fixed production frontier, the worst (lower bound) and best (upper bound) efficiency of the unit under evaluation are calculated from models (1) and (2), respectively.

Lower bound of efficiency
Upper bound of efficiency Theorem 1: If H l j and H u j are, respectively, the optimal value of the objective functions of models (1) and (2), then H l j ≤ H u j [24].

Economies of Scope (ES)
ES exist when the cost of producing several types of activities, products or services together by a company is lower than the cost of producing the same products separately by specialized companies.
To determine ES, consider the types of companies as follows: 1. Specialized companies (group s k ): These are companies that produce a specific product or good.2. Diversified companies (Group D): These are companies that produce several products at the same time.

Virtual (artificial) diversified companies (group V):
These companies are hypothetical ones, which are all created from the merger of specialized companies.

Suppose diversified companies (group D ) produce two products y D
1 ، y D 2 , respectively, with prices w D 1 and shows the minimum cost of production of all the products made by this company.Also, consider p specialized companies, group S 1 , producing only product 1 y s 1 1 with price w s 1 1 and q other specialized companies, group S 2 , producing only product 2 y s 2 2 with pricew 2 , 0 are, respectively, the minimum cost of production each of products 1 and 2 by specialized companies.Definition 2: ES exists when [5]: Also, Degrees of Economies of Scope (DES) is another criterion to determine ES.Definition 3: DES for company j is obtained by following the relationship [5].Now, we propose the following steps for the determination of ES in the framework of DEA: • The creation of virtual companies (group V) by combining two by two groups s 1 and s 2 .This Group consists of all combinations of firms in s 1 and • Calculate the cost efficiency of each DMU in the D group using model (3) [5].• Comparing the minimum costs of diversified companies in group D with virtual diversified companies in V using model ( 4) [5].

Cost efficiency
Comparison of the minimum cost of group D compared to virtual companies The companies of group D with respect to sets of the virtual diversified companies in V with the abbreviation D-V in model ( 4) is indicative.
• Calculate the DES using the following equation [5]: the local economies of scope at DMU o using the above relationship and following conditions: • DES j > 0 Economies of scope exist • DES j = 0 Indifference (implies that costs are additive in nature) 4) is infeasible, the efficiency value is considered to be 1 and hence the DES j = 0.

DEA-Based Convex Cost-Effectiveness Measures
The cost-effectiveness measures the DMU's ability to get the current output levels at minimum cost and sometimes it is addressed as the standard technique for evaluating efficiency, and is measured in terms of the ratio of the costs and outputs of a production or project, respectively.Suppose c ij and p rj denote, respectively, the prices of ith input and rth output for DMU j so that all prices are known exactly Hence CCE meas- ure is obtained based on the DEA model following [39,40].
where o = ∑ m i=1 c io x io and o = ∑ s r=1 p ro y ro are the total cost and the total price, respectively.

Determination of ES in the Presence of Interval Costs and Outputs
As mentioned above, conventional DEA models are ES with the assumption of exact input (costs) and outputs.However, in many cases, companies may have imprecise and uncertain inputs and outputs.Hence, in the present study, we will evaluate the ES due to production diversity when the inputs, costs, and outputs of the companies are imprecise, and we will propose ES interval DEA models.
For evaluating ES with interval data, we consider three production groups (groups D, S k (k = 1, 2) and V) and assume that the input and output of companies are in a bounded interval.So, the inputs, prices, and outputs are The superscript t indicates the type of group (V, D, S k ).Also, we assume that the upper and lower bounds of this input and outputs are positive.We now perform the following steps to evaluate the interval ES.
First, we use models ( 5) and ( 6) to obtain the minimum cost interval of the D group companies.xD l * i in model ( 6) is the optimal solution of model (5).Models ( 5) and ( 6) have the smallest and largest possible solutions, respectively.C D l and C D u are the optimal solutions of models ( 6) and ( 5).Note that we have considered the highest price with the lowest output for the undesirable state and the lowest price with the highest output for the desirable state.However, other units are in the best state of the production process (lowest input with highest output).Therefore, the optimal solution of model ( 5) is the upper limit and the optimal solution of model ( 6) is the lower limit in the cost efficiency range.So, we have Upper bound of the minimum cost Lower bound of the minimum cost CE D l and CE D u are the worst and the best cost efficiencies, respectively.

Definition 5:
The company under evaluation is cost efficiency if the upper bound of cost efficiency is equal to one, in other words CE D u = 1 Theorem 3: The cost efficiency in the worst conditions is less than or equal to the cost efficiency in the best conditions The virtual joint production companies (group V) are based on the combination of companies of group S k (k = 1, 2).Consider the number of these companies as b.All inputs, outputs and input prices of virtual companies are generated by combining of inputs, outputs and prices of specialized production companies S 1 and S 2 , which are as follows: Inputs interval: P r i c e s i n t e r v a l : i n t e r v a l : Theorem 4: The set of production possibilities of virtual companies in the presence of interval data can be created from the combination of efficient interval companies of groups S 1 and S 2 .Now, to determine the minimum interval cost, we compare companies in group D with companies in group V using models (7) and (8).

Pessimistic state
Optimistic state Although models (7) and ( 8) are pessimistic and optimistic, respectively, since model (8) uses the lowest price to calculate the highest input, it forms the lower bound of the minimum cost interval.Also, since model (7) consumes the highest price for calculating a smaller number of inputs, the optimal value of its objective function is considered as the upper bound of the minimum cost interval.

Theorem 5:
The optimal value of the minimum cost in the optimistic state is less than or equal to the optimal value of the minimum cost in the pessimistic state, in other words, if C D−V l and C D−V u are, respectively, the optimal val- ues of the objective functions of models (8) and (7), then

Definition 6:
The cost efficiency interval is given by the following relationship: It is noteworthy that here always CE D−V l ≤ CE D−V u .

Definition 7:
The DES interval for company j is given by the following relationship: , then DES l j > 0 and DES u j > 0 In this case, the company under evaluation has ES in any situation.We define this mode as the set As a result, the company under evaluation has ES on the one hand and lacks ES on the other hand.We define this group of companies with a set < 1 and CE D−V u < 1 , then DES l j < 0 and DES u j < 0 and therefore the evaluated company in every situation shows the absence of ES (diseconomies of scope).We define this group of companies with a set E − = j|DES l j < 0, DES u j < 0 .IV.If CE D−V l = 1 and CE D−V u = 1 , then DES l j = 0 and DES u j = 0 , which indicates indifference of ES (nei- ther presence nor of absence ES) for the company.We define this group of companies with the set E = j|DES l j = 0, DES u j = 0 .V. When models ( 15) and ( 16) become infeasible, we set It should be noted that other situations may also occur.We also note that in general, when the DES becomes less than zero, the place of the upper and lower limits will change.
These models can be extended to several companies.The flowchart of the proposed method is shown in Fig. 1.

Determining Cost Effectiveness in Production Firms with the Presence of Interval Data
Cost effectiveness is a suitable method for measuring and evaluating various dimensions of organizations and enterprises, including costs and the amount of consumption.
Cost effectiveness is defined as the relationship between the level of performance and the costs associated with reaching this level of operations.In this section, we will examine and determine the cost effectiveness of joint production companies compared to specialized production companies.
Consider n decision-making units (DMU).DMU j (i = 1, 2, …, m) uses the non-negative input x ij (i = 1, 2, …, m) for creating s non-negative output as y rj (r = 1, 2, …, s).Input and output costs are w ij (i = 1, 2, …, m) and p rj (r = 1, 2, …, s), respectively, Also, assume that input and output, and their costs are intervals, i.e. x ij ∈ x l ij , x u ij , y rj ∈ y l rj , y u rj , w ij ∈ w l ij , w u ij , and p rj ∈ p l rj , p u rj .In addition, x l ij > 0 and y l rj > 0 .By considering a fixed production frontier, we determine the effectiveness of companies.
All cost-effectiveness steps are the same as in the previous section, with the difference that to determine the minimum cost of group D companies and compare this minimum cost with the cost of virtual companies from models ( 9)-( 10) and ( 11)- (12), respectively.

Determining the cost efficiency of the D group companies Upper bound of cost efficiency
Lower bound of cost efficiency min  CCE D−V l and CCE D−V u are the cost effectiveness of group D companies compared to virtual companies in optimistic and pessimistic mode, respectively.

Definition 9:
The Degrees cost effectiveness (DCCE) interval for the company j is obtained using the following relationship: We can check the cost effectiveness as follows: > 0 , As a result, the company under evaluation shows that it has CCE on the one hand and lacks CCE on the other hand.We d e f i n e t h i s s t a t e a s t h e s e t > 0 , in this case, the company under evaluation has CCE in any condition.We d e f i n e t h i s s t a t e a s t h e s e t < 0 and DCCE D−V u j < 0 and therefore the company under evaluation shows a lack of CCE in any situation.

We d e f i n e t h i s s t a t e a s t h e s e t E
and DCCE D−V u j = 0 , which it shows the indifference of CCE (neither presence nor absence of CCE) for the company.We define this group of companies with V. When models (11) and ( 12) become feasible, we set CCE D−V l = 1 and CCE D−V u = 1 , which expresses this previous state.VI.It should be noted that other situations may also occur.We also note that in general, when the degree of cost effectiveness becomes less than zero, the place of the upper and lower limits will change.
Theorem 6: Cost effectiveness in an optimistic state is less than or equal to cost effectiveness in a pessimistic state.CCE D l ≤ CCE D u .

Proof:
The proof is similar to Theorem 3.
The cost effectiveness of group D companies in the optimistic (pessimistic) state is less than or equal to the cost effectiveness of those companies in the same state.In other words, Proof: The proof process is similar to Theorem 7.

Application Example
Data were collected from 24 institutions over a period of 4 years.Table 1 shows the information relating to 8 institutions that are specialised in the field of education only (y 1 ).The inputs of these companies are the number of professors (x 1 ) and the number of administrative personnel (x 2 ).Table 2 shows information about 8 research institutions that only conduct research (y 2 ).The inputs for these institutions are the number of professors (x 1 ) and the number of administrative personnel (x 2 ).Table 3 is the information related to 8 educational research institutions that, in addition to educating students (y 1 ), also conduct research activities (y 2 ).Inputs, prices and outputs are collected during a period of time and as a result are estimated as intervals (minimum and maximum).According to Sect. 5, we first calculate the minimum interval cost of group D institutions according to models ( 5) and (6).The results of the minimum cost and cost efficiency are listed in Table 4.As can be seen, out of eight institutions in the third group, three institutions are cost effective.
For evaluating the hypothesis of combining each of the educational institutions with the research institutions as virtual institutions, we will have 64 virtual institutions, but according to theorem 4, these institutions can only be obtained from the combination of efficient institutions of the two groups.Therefore, the final number of virtual institutions is reduced to 24 institutions (in the first group, four Minimum cost range and cost efficiency of three institutions DMU First institute (Group S 1 ) Second institute (group S 2 ) Third institute (Group D)  institutions and in the second group, six institutions are efficient).The next step is to measure the minimum cost range of group D institutions compared to virtual companies, based on models (7) and ( 8), and their cost-effectiveness is calculated.The results of models ( 7) and ( 8), the amount of DES interval, the presence and absence of ES are shown in Table 5.
Based on the above results, educational research institutions 1, 2, 4, and 5 have economic efficiency in both optimistic and pessimistic states.But, institutions 3 and 7 do not show economic efficiency in optimistic and pessimistic states, so these companies should not engage in teaching students and research at the same time unless they reduce their costs.Institution 8 does not state the presence or absence of ES.In fact, this institution has fully used its resources and has not spent more or less than the institutions of specialized groups 1 and 2. As a result, according to the average DES interval, group D, which consists of educational and research institutions, was able to reduce the cost to the lowest level compared to the specialized group of institutions.
Sensitivity analysis If we reduce the inputs by 20%, it can be seen that the cost efficiency of diversified companies increases compared to virtual companies, but the results are still the same as before, that is, companies 1, 2, 4, and 5 are in a group E ++ .Now, if we only increase the outputs by 40%, the level of DES will change, and on the other hand, institution 4, which was previously in the category E ++ , will be placed in the category E .Therefore, with the increase or decrease of the input and output of companies, the measure of cost efficiency, the DES and the type of the category of companies will change.The graphs of these changes are shown in Fig. 2. Now, we use models ( 9)-( 12) to determine the costeffectiveness of the institutions.Table 6 shows the results of the models.In group of eight educational research institutions, 1, 2, 4 and 5 institutions with ES were also Cost effectiveness.Institution 8 shows that cost effectiveness is not useful.The remaining institutions also show the absence of cost effectiveness.
It is worth noting that an institution may have ES, but does not show cost effectiveness.For example, Institution 6, which has ES in an optimistic state, but not Cost effectiveness.If we compare Tables 5 and 6, we can see that the results obtained in Table 6 are less than or equal to the results obtained in table (5).

Sensitivity analysis
As before, if we reduce the inputs by 20%, we see that the CCE measure of diversified institutions increases compared to virtual institutions, in this case, as before, units 1, 2, 4 and 5 are in a group E ++ .Now, if we only increase the outputs by 40%, the DCCE level will change and, on the other hand, unit 4, which was previously in the category E ++ , will be placed in the category E and unit 6 will also be placed in the category E + .Therefore, with the increase or decrease of the input and output of instithe measure of CCE, DCCE and the type of the category of units will change.The comparison of the above situation is shown in Fig. 3.

Discussion
According to the results, it can be concluded that the existence of Economies of Scope in the educational-research institution is indicative of this matter, institutions that specialize only in education or only in student research will have higher costs and hence, it is not recommended to provide services separately.Therefore, creating conditions that allow education and research institutions to operate simultaneously can play a significant role in reducing costs.Also, the results of the example show that educational research institutions may be Economies of Scope, but they will not necessarily be cost-effective.As a result, the best unit is an institution that has Economies of Scope and cost effectiveness.It should be mentioned that if managers of companies or organizations tend to use the production of several products to reduce their costs, first, it is necessary to examine their current situation and then proceed to produce several products at the same time.
The calculation results show that the presented models are more accurate than the usual model and the ability to effectively detect Economies of Scope and cost-effectiveness has been significantly improved.It is also possible to check Economies of Scope in the simultaneous production of products in the proposed models.
According to the data of virtual companies, we have min y = 1530 According to the above content of this example, it can be said that if one of the outputs of the diversified companies is greater than the maximum of the virtual company, the diversified companies are placed in a category E , and if the lower abound of both outputs of the diversified companies is less than the minimum of each of the outputs of the virtual companies, it will be placed in a category.,It will be placed in a category E − , and also if its outputs are in the maximum and minimum range of the virtual company, it will be placed in a category E ++ .

Conclusion
One of the most effective factors for managers in developing organizations is to pay attention to economic issues.Determining economic efficiency and cost effectiveness are two methods of economic evaluation.In conventional data envelopment analysis methods to evaluate the Economies of Scope and cost-effectiveness, the data are accurate and specific, whereas in reality we may encounter cases where we do not have accurate information about the costs (inputs) of the company/organization and outputs (such as the number of invoices), and it's impossible to determine the exact numerical value for these costs or outputs.Hence, in the present study, we propose models for determining Economies of Scope and cost-effectiveness in the presence of imprecise and uncertain (interval) inputs and outputs and a new approach for obtaining Economies of Scope and cost-effectiveness measurement with a data set of interval numbers are considered.Using the proposed models, we can determine whether joint production has the best efficiency compared to specialized production in the presence of interval data or not.The proposed models are useful in identifying the most cost-effective option to achieve predetermined goals.Because in many environmental we have encountered data that are imprecise and ambiguous, or the knowledge about their production process is imprecise, the presented models provide more useful information to organizations and industrial activities.
Finally, we implemented the proposed models on an applied example.The results show that by considering interval data in educational research institutions, it affects the Economies of Scope and cost effectiveness.Further, the findings conclude that the company/organization with diseconomies of Scope and absence of cost-effectiveness are recommended to more efforts for their continued existence in highly competitive markets through input reductions and output augmentations to become best practice branches.
It is expected to expand and generalize the above models to evaluate the Economies of Scope and cost-effectiveness to cases where fuzzy and stochastic data could be incorporated into the model.In addition, consider the uncertain input price in evaluating the Economies of Scope and cost-effectiveness of the closed loop supply chain and network systems and changing these models to two-stage system models.We know that in minimization, every optimal solution is less than the feasible solution So C D l ≤ C D u .

Theorem 2 :Definition 4 :
The minimum cost of model 6 is lower than the minimum cost of model 5, in other words C D l ≤ C D u We can get the cost-efficiency interval of the company under evaluation from the following relationship:

Fig. 2 a
Fig. 2 a Interval ES measure before and after input's changes.b Interval ES measures before and after the output's changes

V l 1 = 2 = 2 =>780 = max y D u 1 < max y V u 1 = 1520 and also for output 2 w e h a v e 130 = min y D 2 > min y V l 2 = 100 a n d 139 = max y D u 2 < max y V u 2 =< 2 < min y V l 2 = 2 >
95 and max y V u 1 = 1520 (the minimum and maximum output of the first output is 95 and 1520) and also min y V l 100 and max y V u 1530 .From the results of Table5and the above article, it can be seen that a company like Company 1, which was placed in a category E ++ t h e o u t p u t s o f t h i s c o m p a n y a r e a s 770 = min y D 1 1530 While a company like the third company, which is in a category E − , its output compared to the output of the virtual company is as 25 = min y D 1 100 or a company like the company 8, which is in a category E , its output compared to the output of the vir tual company is as 2000 = max y D u max y V u 2

Fig. 3 a
Fig. 3 a Interval CCE measure before and after changing the inputs.b Interval CCE measure before and after changing the outputs

Proof of Theorem 2 :
Suppose * j , x D l * i is the optimal solution of model(6).Put j = * j , x D u * i = x D l * i .It is clear that j , x D u * iis a feasible solution for model(5).

Theorem 3 :Proof of Theorem 4 :
Given that x D l * i ≤ x D * i ≤ x D u * i and x D l io ≤ x D io ≤ x D u io we have On the other by dividing the above two relations, we have With the addition of existing limits on i, we have As a result, CE D ∈ CE D l , CE D u and therefore CE D l ≤ CE D u .Consider the PPS groups S k (k = 1, 2) as follows:where input, input price and outputs are intervals.P S 1 is formed by the finite number of cost-efficient DMUs.Therefore, the cost-inefficient company x 1 can be written from the convex combination of cost-efficient companies with at least one non-zero slack in the input or output iE 1 is a set of efficient companies in S 1 group.Also

Table 1
Data of educational institutions (group S 1 )

Table 2
Data of research institutions (group S 2 )

Table 3
Data of educational-research institutions (group D)

Table 5
Cost efficiency interval, DES rate and economic efficiency in education-research institutions