An Application to Quality Requirement Planning of a B ...

Chance Constrained Programming Approach to QFD: An Application to Quality Requirement Planning of a B-School V. Charles, Mukesh Kumar, CENTRUM Católica, Graduate School of Business, Pontificia Universidad Católica del Perú, Calle Daniel Alomía Robles 125 - 129, Los Álamos de Monterrico Santiago de Surco, Perú.

Email: [email protected], [email protected] Srinivas Suggu Infosys Limited, Bangalore - 560 076, India. E-mail: [email protected] ABSTRACT Quality function deployment (QFD) is becoming a widely used customer-oriented approach and tool in product design. Tang et al. (2002) proposed two types of fuzzy optimization models by introducing the concepts of planned degree, actual achieved degree, actual primary costs required and actual planned costs which aim not only to optimize the overall customer satisfaction but also the enterprise satisfaction with the costs committed to the product. In turn, this paper deals with a chance constrained model for QFD wherein, the proposed model maximizes the overall customer satisfaction and simultaneously minimizes the total cost subject to the preferred acceptable customer satisfaction along with the preferred requirements of the actual achieved degree of technical requirements for the given cost under the stochastic environment. The proposed chance constrained model for QFD has been illustrated with a hypothetical case for the requirement planning of a B-School to achieve the needs and expectations of students. Keywords and phrases: QFD, chance constrained, B-School.

1.1 Introduction Probabilistic or Stochastic programming is a framework for modeling optimization problems that involve uncertainty. The basic idea used in solving stochastic optimization problems has so far been to convert a stochastic model into an equivalent deterministic model and it is possible when the right hand side resource vector follows some specific distributions such as normal, lognormal and exponential distributions. One of the common problems in the practical application of mathematical programming is the difficulty for determining the proper values of model parameters. The values of these 39

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Journal of the Indian Society for Probability and Statistics

parameters are often influenced by random events that are impossible to predict i.e., some or all of the model parameters may be random variables. What is needed is a way to formulate the problem so that the optimization will directly consider the uncertainty. One such approach for mathematical programming under uncertainty is Stochastic Programming (SP). SP is an optimization technique in which the constraints and/or the objective function of an optimization problem contains certain random variables. In the stochastic linear programming literature, several researchers suggested various models (Kall and Wallace, 1994). Model coefficients of most of these models are assumed to follow independent normal distribution because deriving the deterministic equivalent of the objective function and/or constraints of the model is well known (Kall and Wallace, 1994) in this case. SP models were first formulated by Dantzig (1955) who suggested a two stage programming technique that involves conversion of SP models into their equivalent deterministic programming models. However, this technique suffers from the limitation that it does not allow any constraint to be violated even at specific probability level. This gave rise to the concept of Chance Constrained Programming (CCP), where constraints containing random variables are guaranteed to be satisfied with a certain probability. Charnes and Cooper (1959, 1963) developed the concept of CCP. Jagannathan (1974) discussed a single objective chance constraint programming with joint constraints. Biswal et al. (1998) obtained deterministic equivalent of the objective function and left hand side constraint coefficients following exponential distributions. Charles and Dutta (2001, 2003, 2006a,b) and Charles et al. (2010) also derived deterministic equivalent of the objective function(s) and constraint coefficients with normal random variables. Joint probabilistic constraints for independent random variables were used initially by Miller and Wagner (1965). The properties of stochastic programming problems and methods for obtaining optimal solution have been described in Rao (1989), Kall and Wallace (1994), Birge and Louveaux (1997) and Prekopa (1995). In various areas of real world, the problems are modeled as stochastic programming. For example, modeling of investment portfolio, modeling of strategic capacity investments, power systems, manpower planning in Jeeva et al. (2002, 2004), financial models can be seen in Charles (2005a,b,c), and reliability stochastic optimization in Yadavalli et al. (2007). Quality is a concept that lacks a clear and concise definition and is thus difficult to accurately measure, improve or even compare across different industries, products and services. The quality of a service refers to the extent to which the service fulfills the requirements and expectations of the customer. A basic element of quality is that it is not free: it always requires efforts typically reviews, testing, inspections etc. which cost but on the other hand it always adds some value to the customer. The cost and return of quality improvements suggest that there are diminishing returns to quality expenditures. Therefore the key management problem is how to make profitable decisions on quality expenditures. Typically quality characteristics may be required or not, or may be required to a greater or lesser degree, and trade-offs may be made among them. Given the trend of business globalization, companies face challenges from both national and international competitors. The survival of a company is heavily dependent on its capacity to identify new customer requirements and to develop a new product. (Shen et al., 2000). An effective and universally acceptable tool to translate customer needs and requirements into the quality characteristics to improve quality for an existing product to develop a new consumer product is Quality Function Deployment (QFD). QFD has been used as an important part of the product development process. In this approach, several steps are followed to

V. Charles, Mukesh Kumar and Srinivas Suggy

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expose customer expectations into the service process and ensure that at each level of expectation the highest possible quality is provided. This methodology is used for the development or deployment of features, attributes, or functions that give a product or service high quality. QFD is an investment in people and information. It uses a cross functional team to determine customer requirements. QFD is a systematic and analytical technique for meeting customer expectation. QFD is a planning process for translating customer requirements (voice of the customer) into the appropriate technical requirements for each stage of product development and production (i.e. marketing strategies, planning, product design and engineering, prototype evaluation, production process development, production, sales) (Sullivan, 1986) and (Revelle et al., 1998). A discussion of the methodology and principles of QFD may be found in Hauser and Clausing (1988); González et al. (2004); Akao and Mazur (2003); González et al. (2005). Quality Function Deployment was introduced by Akao (1972) in Japan as a method for developing a design quality aimed at satisfying the consumer by translating the consumer’s demands into product development. QFD has been applied successfully in many Japanese organizations to improve processes and to build competitive advantage. Today, companies are successfully using QFD as a powerful tool that addresses strategic and operational decisions in businesses. Akao and Mazur (2003) mentioned QFD as a method for defining design qualities that are in keeping with customer expectations and then translating those customer expectations into design targets and critical quality assurance points that can be used throughout the production/service development phase. In 1986, Kelsey Hayes used QFD for developing a coolant sensor that fulfilled critical customer needs such as “easy-to-add coolant”, “easy-toidentify unit”, and “provide cap removal instructions” (King, 1987; Prasad, 1998). QFD can be very helpful in answering the question “how to deliver quality products and services based on the needs of customers, or the voices of customers?” (Hwarng and Teo, 2001). QFD improves the communication of customer expectations and completeness of attributes that customer needs (González, 2001). QFD is performed by a multidisciplinary team representing marketing, design engineering, manufacturing engineering, and other functions considered critical by the company. In general, it provides a framework in which all participants can communicate their thoughts about a product. It has been used as a customer-oriented approach to facilitating product design by analyzing and projecting customer requirements into product attributes (Fung et al., 1996). Today almost all the industries across the globe have taken QFD technology into consideration to deliver products that are marketable and tasteful to their customers, and buildable by their production expert teams (Griffin and Hauser, 1992; Chan and Wu, 2002). There are many benefits in the implementation of QFD in both service and product development industries and can be detailed by the researchers Sullivan (1986), Hauser and Clausing (1988), Zairi and Youssef (1995), Chan and Wu (2002), and Terninko (1995). Griffin et al. (1995) have considered that QFD provides a means of communication among product life cycle stages. Benefits which arise from these and other reported QFD applications include lower design and service costs, fewer and earlier design changes, reduced product development time, fewer start-up problems, better company performance, more reliable input for marketing strategies, improved service quality and, above all, increased customer satisfaction (Jae et al., 1998). Though it was originally used in product development and design, several researchers have applied QFD to different service areas (Jeong and Oh, 1998; Trappey et al., 1996; Stuart and Tax, 1996; Cadogan et al., 1999; Pun et al., 2000; Peters, 1988; González et al., 2003, 2005).

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1.2 QFD Methodology QFD methodology translates customer needs and requirements into the quality characteristics to design any new product/service. QFD is also known as the house of quality (HoQ). QFD uses a matrix format to capture a number of issues pertinent and vital to the planning process. The QFD matrix consists of six parts. The first step starts with constructing a list of requirements as voiced by the customer. The second part of the house of the quality is the assessment of importance of customer requirements. The next step is to determine the technical attributes. These attributes, which are measurable and controllable that will impact on one or more customer requirements. The fourth phase is the correlation matrix to identify the interrelationship of each technical attributes. The fifth step is an evaluation of the strength of the relationship between the customer requirements and the technical attributes. The last step is the technical assessment. The HoQ does not detail about any product design but produces the requirements the customer wants (Vonderembse and Raghunathan, 1997).

Figure 1: House of Quality (HoQ)

Tang et al. (2002) proposed two fuzzy optimization models which independently aim to (i) maximize the overall customer satisfaction for at least a given pre-specified level of attainment of TR and (ii) achieve a preferred acceptable customer satisfaction at the smallest design cost. In this paper, we propose a chance constrained model for QFD that aims at maximizing the overall customer satisfaction and simultaneously minimizing the total cost subject to the preferred acceptable customer satisfaction along with the preferred requirements of the actual achieved degree of technical requirements for the given cost under the stochastic environment.


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The paper is organized as follows. Section 2 provides the formulation of chance constrained model for QFD in line with the two types of fuzzy optimization models proposed by Tang et al. (2002). A hypothetical case for the requirement planning of a B-School to achieve the needs and expectations of students has been discussed in Section 3. The last section concludes the study with direction for future research.

1.3 QFD Planning Models The House of Quality (HoQ) shown in Figure 1 helps translating the desires of Customer Requirements (CRs) into determination of Technical Requirements (TRs) in the new product design or enhancement of the existing product (Akao 1990, Hauser and Clausing 1988, Bode and Fung 1998). Assume that CR1, CR2, …,CRm are the m customer requirements and TR1, TR2, …, TRn are the n technical requirements in a product design. Let d1 , d2 ,..., dm be the degree of importance of m CRs representing the relative importance of individual CRs to overall customer satisfaction while w1 , w2 ,..., wn are the relative weights attached to n TRs and determined from the relationship between CRs and TRs; and importance of CRs. The relationship between CRs and TRs is represented by the matrix R  ( Rij ), where the element Rij denotes the strength of impact of the jth TR on the satisfaction of ith CR. These Rij values are quantified by using a scale of 1-3-9 to express weak, medium and strong relationships respectively. The normalized matrix R*  ( Rij* ) represents the contributed quota of the jth TR towards complete fulfillment of the ith CR when the target of the jth TR is met. The inter-relationships between the TRs can be represented by the matrix T  (Tkj ), j=k=1,2,…,n. These values will be shown in the roof of the HoQ and are quantified by using a scale of 1-3-9-18 for weak, medium, strong and perfect direct relationships respectively while negative numbers represent for indirect relationships in case of conflict between any two TRs. Post normalized Tkj matrix elements ranges between -1 and +1. Traditional methods for QFD planning are not considered with the financial factor which is the most important and sensitive aspect in current high competitive market. Tang et al. (2002) introduced two fuzzy optimization models which consider the enterprise satisfaction with committed costs of product along with overall customer satisfaction. He introduced four new concepts known as planned degree, actual achieved degree, actual primary costs required and actual planned costs in his model. Maximization Model

This model aims at maximizing the overall customer satisfaction by considering the planned degree of attainment of the TRs under limited budget and specific level of TR constraints (at least preferred level of attainment of TRs). Hence, the model maximizes the overall customer satisfaction and guarantees at least a pre-specified level of attainment of the TR.

Max

n

n

j 1

k 1

n

n

n

 w j Tkj yk   w jTkj yk   wk* yk k 1 j 1

subject to

k 1

(1) n

T k 1

kj

yk  0 , j  1, 2,..., n,

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  n c 1  T y Tkj yk  B,   kj k   j j 1  k j  k 1 n

n

T

kj

k 1

yk  1, j  1, 2,..., n,

0  yk  1, j  1, 2,..., n,

where n and k are the number of technical requirements (TRs), m: the number of customer requirements (CRs), yk : the planned degree of attainment of the kth TR, w j : weight of the jth TR m

(j = 1,2,...,n) =

 d R , d : weight of importance of the ith CR indicates the relative importance i 1

i

ij

i

of the ith CR towards the overall customer satisfaction, Rij : strength of impact of the jth TR on the satisfaction of the ith CR. This matrix is known as relationship matrix. Tkj : correlation factor between the kth and the jth TR, wk* : the contribution of the kth TR to overall customer satisfaction due to their correlation of TRs when one unit of planned degree of attainment of the TR is n

committed. It is to be noted that  wk*  1.  0 : preferred acceptable degree of attainment of TR. k 1

This value is decided by the decision maker’s preference and subjectivity. Additionally, different level of  0 may be applied to different TRs in practical scenarios. The value  0 ranges from 0 to 1. c j : cost associated to the jth TR and B : the total available/limited budget. Owing to the uncertainties in the design process, such as ill-defined or incomplete understanding of the relationship between TRs and CRs, as well as the human subjective judgment on the dependence between TRs, c j and B are considered to be fuzzy in nature. Minimization Model The model aims to achieve a preferred acceptable customer satisfaction at the smallest design cost. This financial model minimizes the total cost and guarantees a minimum pre-specified level of customer satisfaction. n   n Min  c j 1   Tkj yk   Tkj yk j 1  k j  k 1 subject to (2) n

w y k 1

n

T k 1 n

kj

T k 1

* k

k

 0

yk   j , j  1, 2,..., n,

kj

yk  1, j  1, 2,..., n,

0  yk  1, j  1, 2,..., n,


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where  0 : preferred acceptable customer satisfaction and ranges between 0 and 1,  j : preferred requirements for the actual achieved degree of attainment of TRs and ranges between 0 and 1. c j is considered to be fuzzy in nature. Chance constrained model for QFD

Chance constrained model for QFD aims at the overall customer satisfaction by considering the planned degree of attainment of TRs under limited budget and specific level of TR constraints (at least preferred level of attainment of TRs) as well as achieves a preferred acceptable customer satisfaction at the smallest design cost. Hence, this model maximizes the overall customer satisfaction and guarantees at least a pre-specified level of attainment of TR and minimizes the total cost (design cost for the product development/improvement) and guarantees a minimum pre-specified level of customer satisfaction. The chance constrained model for QFD planning problem can be formulated as follows: n n   n Max  wk* yk   c j 1   Tkj yk   Tkj yk k 1 j 1  k j  k 1 subject to (3) n

Pr{ wk* yk  0 }   01 , k 1

  n Pr{ c j 1   Tkj yk   Tkj yk  B}   02 , j 1  k j  k 1 n

n

Pr{ Tkj yk   j }   j , j  1, 2,..., n, k 1

n

T k 1

kj

yk  1, j  1, 2,..., n,

0  yk  1, k  1, 2,..., n,

where  01 ,  02 and  j are the given confidence levels. c j : normalized cost associated to the jth TR and B : the total available/limited budget (normalized).  0 : preferred acceptable customer satisfaction and ranges between 0 and 1,  j : preferred requirements for the actual achieved degree of attainment of TRs and ranges between 0 and 1. Pr is the probability operator with respect to some probability distribution of c j , B, 0 and  j . The precise probability distribution that is used for each constraint will be made explicit later. Theorem 1: Assume that the stochastic vector  degenerates to a random variable with probability distribution  , and the function f ( y, ) has the form f ( y, )  h( y)  . Then Pr{ f ( y, )  0}   iff h( y)  K , where K is the maximal number such that Pr{K   }   . Remark 1: The probability Pr{K   } will increase if K is replaced with a smaller number. Hence Pr{h( y)   }   iff h( y)  K .

46


Remark 2: For a continuous random variable  , the equation Pr{K   }  1  ( K ) always holds and we have by theorem 1, K  1 (1   ), where  1 is the inverse function of . n

Remark 3: The chance constraint Pr{ wk* yk  0 }   01 coincides with the form k 1

n

Pr{ f ( y, )  0}   by defining f ( y, )  0   wk* yk . k 1

n

Remark 4: The chance constraints Pr{ Tkj yk   j }   j , j  1, 2,..., n, coincide with the form k 1

n

Pr{ f ( y, )  0}   by defining f ( y, )   j   Tkj yk . k 1

Theorem 2: Assume that the stochastic vector   ( 0 ) and the function f ( y, ) has the form n

f ( y, )  0   wk* yk . If  0 is assumed to be independently normally distributed variable, k 1

then Pr{ f ( y, )  0}  01 iff

n

w y * k

k 1

k

 E[ 0 ]   1 ( 01 ) E[ 0 ], where  is the standardized

normal distribution. Theorem 3: Assume that the stochastic vector   (c1 , c2 ,..., cn , B) and the function f ( y, ) has n   n the form f ( y, )   c j 1   Tkj yk   Tkj yk  B. If c j and B are assumed to be independently j 1  k j  k 1 normally distributed variables, then Pr{ f ( y, )  0}  02 iff n   n   n 1 E [ c ] 1  T y T y   (  ) V [ c 1  T y Tkj yk ]  V [ B]  E[ B], where    kj k     kj k   j  kj k 02 j j 1 j 1  k j  k 1  k j  k 1 is the standardized normal distribution. n

Theorem 4: Assume that the stochastic vector   (  j ) and the function f ( y, ) has the form n

f ( y, )   j   Tkj yk . If  j is assumed to be independently normally distributed variable, k 1

then Pr{ f ( y, )  0}   j iff

n

T k 1

kj

yk  E[  j ]   1 ( j ) E[  j ], where  is the standardized

normal distribution. The deterministic equivalent of the chance constrained model for QFD in line with above mentioned theorems can be derived as: n n   n Max  wk* yk   E[c j ] 1   Tkj yk   Tkj yk k 1 j 1  k j  k 1

subject to

(4) n

w y k 1

* k

k

 E[ 0 ]   1 ( 01 ) E[ 0 ],




n

47

n  n   n 1 y T y   (  ) V [ c 1  T y      kj k kj k 02 j kj k   Tkj yk ]  V [ B ]  E[ B ], j 1  k 1  k j  k 1

 E[c ] 1  T j 1

j



k j

n

T k 1

kj

yk  E[  j ]   1 ( j ) E[  j ], j  1, 2,..., n, n

T k 1

kj

yk  1, j  1, 2,..., n,

0  yk  1, k  1, 2,..., n.

1.4 Hypothetical Case of Requirement Planning of a BSchool Service industries are playing an increasingly important role in the economy of developing and emerging nations. In today’s world of global competition, rendering quality service is a key for success, and many experts are of the opinion that the most powerful competitive trend currently shaping business and marketing strategy is the service quality. The same need is identified in academic institutions in order to have the competitive edge in the education sector. The academic institutions are required to provide quality education that fulfills the needs and expectations of students by enabling them to serve the nation and strengthen the economy. The B-Schools, in particular provide the learning environment in which students excel their professional skills by cooperating closely with corporate partners and by exposing them to a dynamic and intercultural business environment. Here we provide a hypothetical case for requirement planning of a BSchool, which aims at unveiling the importance of QFD to the design and improvement of quality of a B-School by identifying the vital factors that need to be implemented and enhanced to match the requirement of the students. Let there be six major criteria requirements of a B-School from students perspective, namely Qualified Faculty (CR1), Comfortable Environment (CR2), Research Opportunity (CR3), State of Art Technology (CR4), Reasonable Fees (CR5) and Solid Academic Program (CR6). There are five technical requirements (TRs) from the perspective of the B-school’s management, i.e., No of PhD Faculty (TR1), Class Room Facilities (TR2), Reasonable Publications (TR3), No of Discussion Rooms (TR4) and Professional Accreditations (TR5). The Figure 2 provides an overview of quality requirements from the student perspective as well as the technical requirements from the management perspective. As we mentioned earlier that the relationship between CRs and TRs is measured by the scale of 1-3-9 and the correlation between TRs by the scale of 1-3-9-18, where 1, 3, 9 represents the weak, medium and strong relationship between CRs and TRs, as well as among TRs, respectively, whereas 18 denotes the strongest dependency between TR and itself. For example, the relationship matrix given in Figure 3 shows that the student requirement as Quality Faculty

48

Journal of the Indian Society for Probability and Statistics Quality of a B-School

Research Opportunity State of Art Technology Reasonable Fees Solid Academic Program

C O R R E L A T I O N

Class Room Facilities

Reasonable Publications

No. of Discussion Room

REQUIREMENTS

STUDENTS

Comfortable Environment

No. of PhD Faculty

TECHNICAL

REQUIREMENTS

Qualified Faculty

Professional Accreditation

Figure 2: Model for quality requirement planning of a B-School

is strongly associated with the number of PhD faculty in the school as well as research publications by the faculty. Similarly, the inter-correlation of the technical requirements shows that the number of PhD faculty is strongly associated with research publications by the faculty as well as professional accreditation.


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The corresponding normalized relationship between CRs and TRs and correlation between TRs are shown in Figure 4 along with the respective derived weights. Let  01 =0.95,  02 =0.95,

j

= 0.90, j  1, 2,...,5; the total available normalized budget ( B ) assumed to follow normal distribution with the mean 0.95 and variance 0.003. Preferred acceptable customer satisfaction (  0 ) also assumed to follow normal distribution with the mean 0.8 and variance 0.0004. Table 1 provides the mean and variance of the cost associated to the respective TRs and preferred requirements for the actual achieved degree of attainment of TRs.

wk*

0.500 0.500 0.500 0.167 1.000

Professional Accreditation

wj

0.056 0.167 0.167 1.000 0.167

# Discussion Rooms

Qualified Faculty Comfortable Environment Research Opportunities State of Art Technology Reasonable Fee Solid Academic Program

0.500 0.056 1.000 0.167 0.500

Reasonable Publications

Student/Customer Requirements (↓)

0.056 1.000 0.056 0.167 0.500

Class Room Facilities

0.243 0.135 0.189 0.108 0.135 0.189

Technical Requirements (→)

1.000 0.056 0.500 0.056 0.500

# PhD Faculty

Importance for Students

# PhD Faculty Class Room Facilities Reasonable Publications # Discussion Rooms Professional Accreditation

0.310 0.043 0.333 0.333 0.273 0.120 0.240

0.034 0.391 0.111 0.111 0.273 0.360 0.199

0.310 0.130 0.333 0.111 0.091 0.040 0.188

0.034 0.391 0.111 0.333 0.091 0.120 0.153

0.310 0.043 0.111 0.111 0.273 0.360 0.219

0.463

0.358

0.454

0.268

0.559

Figure 4: House of Quality (HoQ) with standardized values

Table 1: Mean and variance of random variables Random Variables

cj

j

Statistics Mean SD Mean SD

1 0.150 0.003 0.800 0.010

2 0.350 0.004 0.800 0.015

3 0.100 0.002 0.800 0.030

4 0.300 0.001 0.800 0.010

5 0.100 0.003 0.800 0.020

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Using the above given statistics one can use the system (4) to derive the table 2. Table 2 provides the planned verses actual achieved degree of TRs as well as the actual planned standardized cost and total actual cost in standardized form. One can see the best balance between the planned and actual degree of TRs. The optimal objective value of system (4) for the given n n   n statistics is  wk* yk   E[c j ] 1   Tkj yk   Tkj yk = 0.9171 – 0.5737 = 0.3434. k 1 j 1  k j  k 1 Table 2: Planned verses Actual achieved degree of TRs with cost details Criteria

Planned degree of TRs

TR1 0.5625 TR2 0.6597 TR3 0.5864 TR4 0.5739 TR5 0.0000 Total actual standardized cost

Actual achieved degree of TRs 0.9243 0.8192 1.0000 0.8128 1.0000

Actual planned standardized cost 0.0885 0.2410 0.0586 0.1856 0.0000 0.5737

From the results, it can be observed that system (4) could achieve 0.9171 level of customer satisfaction when compared to preferred acceptable customer satisfaction level as

E[0 ]  1 (01 ) E[ 0 ] = 0.8329 with 0.7114 units of the budget which is relatively less than the given total available normalized budget.

1.5 Conclusion In this paper, we have demonstrated a single model under stochastic environment, known as chance constraint programming model for QFD as a replacement technique to the two fuzzy models proposed by Tang et al. (2002), which has been illustrated with a hypothetical case for the requirement planning of a B-School. Unlike their model, our model does not require human interaction to solve the problem. However, the parameters under consideration are expected to follow some known distributions which could be relaxed in the future research.

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