Automation in Construction 16 (2007) 569 – 575 www.elsevier.com/locate/autcon Supply chain optimization tool for purchasing decisions in B2B construction marketplaces D. Castro-Lacouture a,⁎, A.L. Medaglia b , M. Skibniewski c a c Building Construction Program, Georgia Institute of Technology, Atlanta, GA 30332-0680, USA b Departamento de Ingeniería Industrial, Universidad de Los Andes, Bogotá, Colombia Department of Civil and Environmental Engineering, University of Maryland, College Park, MD 20742-3021, USA Accepted 28 August 2006 Abstract Current initiatives for e-commerce deployment in the construction industry are related to the establishment of business-to-business (B2B) marketplaces, which allow large communities of buyers and suppliers to meet and trade with each other. Although these efforts have included aspects of interoperability, database management and security protocols, trading mechanisms in the supply chain have not been properly evaluated. Buyers of construction materials interact with material suppliers in a marketplace environment, thus striving for the best possible spot buy offer. This paper addresses the purchase of construction materials as the last component in the supply chain. A tool for optimizing purchasing decisions in B2B construction marketplaces is presented. © 2006 Elsevier B.V. All rights reserved. Keywords: B2B; Marketplace; Steel reinforcement; Optimization; Data envelopment analysis (DEA) 1. Introduction The development and implementation of business-to-business (B2B) marketplaces in the construction industry have opened a gateway for buyers and suppliers. Over the last few years, construction companies have been conducting their business using web-based trading mechanisms. E-commerce has provided an expanded marketplace within which buyers and suppliers can communicate directly, enabling them to buy and sell from each other at a dynamic price which is determined in accordance with the rules of the exchange. An electronic marketplace is defined as an interorganisational information system that allows the participating buyers and sellers in a particular market to exchange information about prices and product offerings, thus matching them, facilitating the exchange of information, goods, services and payments and providing an institutional infrastructure that enables the efficient ⁎ Corresponding author. Tel.: +1 404 385 6964; fax: +1 404 894 1641. E-mail address: [email protected] (D. Castro-Lacouture). 0926-5805/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.autcon.2006.08.005 functioning of the market [1,2]. Although the behaviour of a pure market is still maintained, in which a buyer can compare different possible suppliers and select the best combination of design and price, the use of information technology is likely to decrease the costs of tasks such as selecting suppliers, establishing contracts, scheduling activities and budgeting resources [3]. There are different types of marketplace for on-line trading: commodity and differentiated markets. In the former, electronic marketplaces can promote price competition among sellers by increasing the availability of price information. In the latter, characterized by heterogeneous consumer tastes and a variety of product offerings, the role of an electronic marketplace is to provide information about the existence and the price of a seller. In the construction industry, the domain of the electronic market is very distinctive. Since construction materials can be considered commodities (e.g., cement, rebar, aggregate, etc.), the variety of product offerings and the need for considering both the price of a particular seller and the characteristics of the corresponding product offering make it a differentiated marketplace. It is true that rebar is sold in tons, but the technical characteristics of each bar are important at the time of purchase, 570 D. Castro-Lacouture et al. / Automation in Construction 16 (2007) 569–575 as well as delivery times that will have implications in the different stages of the project supply chain. The most important benefits provided by the web-based marketplaces are related to the permanent availability of trading information and to the possibility of facilitating trading transactions. Users log onto the system as either buyers or sellers of construction materials. Sellers can upload their product information and find out summary information about their customers and their transactions. Buyers can search certain types of products, giving their requirements such as quality, price, type, strength, etc., or they can browse the products on display. With the use of mobile agents, the B2B e-trading marketplaces can be connected, thus keeping member information in a database, querying and ordering construction materials based on availability and location [4]. This paper addresses the purchase of the construction materials as the last component in the supply chain. Although there is considerable literature on electronic marketplaces, and the framework for establishing an open e-trading marketplace in construction has been described in previous research, a tool for optimizing B2B transactions in the purchase of construction materials is needed and is presented in this paper. Buyers of construction materials can interact with material suppliers, or sellers, in a marketplace environment. Steel reinforcement for concrete (rebar) will be used as a typical construction material. The methodology consists of presenting and validating a supply chain optimization tool with hypothetical data, in the context of a rebar marketplace. The data required by the supply chain optimization tool is comprised of past completed transactions in the marketplace or previous matches between buyers and sellers. This data is originated from the marketplace or from the purchasing agent who keeps track of the activity on a given rebar exchange. The supply chain optimization tool acts as a decision support system for analyzing past transactions in the marketplace in search of efficient and inefficient transactions. 2. Benefits of electronic markets B2B electronic markets function as digital intermediaries that focus on specific business functions and set up virtual marketplaces where firms participate in buying and selling activities after they obtain membership [5]. Marketplaces create value by bringing buyers and sellers together to create transactional immediacy and supply liquidity, and by supporting the exchange of demand and supply information. E-procurement is defined as utilizing electronic media, including the Internet, to streamline as many steps in the procurement process as possible. The major benefits of adopting e-procurement systems are reduced operating costs and searching costs, which lead to high returns on investments. This paper discusses the application of a supply chain optimization tool to the purchase of rebar. The rebar market is very suitable for this study given its applicability for both commodity and differentiated marketplaces. Most of the times, rebar is bid and purchased based on price-per-ton competition among suppliers, typical of a commodity marketplace. On the other hand, based on the design demands or unique project characteristics, a differentiated marketplace appears with heterogeneous consumer needs and a variety of product offerings aimed to fulfil those needs (e.g., high tensile strength rebar). The preliminary stages of rebar procurement feature long periods of quantity takeoff revisions, cost estimation, quality assurance and procurement. These delays translate into expensive activities for construction companies and rebar suppliers, and into time overspent for activities that could be automated. E-procurement should reduce uncertainties by enabling the ordering, at a convenient price, of the precise types and quantities of materials needed to install on the subsequent workday, resulting in higher quality of the implementation of the just-intime concept. Moreover, contractors may find the way of tracking the status of critical orders, thus knowing instantly when a supplier has run out of an already ordered item. Contractors want the purchase-related information to be entered just once, flowing easily throughout the life cycle of their projects, from the estimate and bid to the purchase order, and then into home–office systems such as job costing and accounting [6]. The steel industry made an attempt to create an interactive global marketplace named e-Steel, which provided an interactive online marketplace enabling both buyers and sellers to initiate a transaction, specify product details, target offers or inquiries to specific members, and negotiate and close contracts [7]. 3. Marketplaces in the procurement of construction materials The application of e-commerce approaches to the procurement of construction materials has been subject of research in the past few years [4,8]. These research efforts have proposed initiatives for e-commerce deployment in the construction industry with the creation of B2B e-trading marketplaces, which allow large communities of buyers and suppliers to meet and trade with each other. One initiative is the e-Union, a prototype system that utilizes web services technology to provide sharing between construction material e-commerce systems, therefore bridging the lack of interoperability between the material databases. The unique feature of a B2B e-trading marketplace is that it brings multiple buyers and sellers together in one central market space and enables them to buy and sell from each other at a dynamic price which is determined in accordance with the rules of the exchange. The automation of rebar workflows have been addressed in previous research, in which a B2B e-work system was designed and validated [9,10]. The scope of the B2B e-work system is limited to streamlining the workflow during the take-off, estimating and bidding stages of the preconstruction interactions among supplier, contractor and designer. This paper proposes the optimization of the transaction process using a supply chain optimization tool, a decision support system for B2B purchases. 3.1. Online purchasing of construction materials in the US It is widely believed that e-commerce can provide advantageous situations for both suppliers and buyers in the supply chain. In spite of the massive use of the Internet for information D. Castro-Lacouture et al. / Automation in Construction 16 (2007) 569–575 571 more timely responses to information requests, cost reductions efforts, and enhanced employee productivity. In spite of the featured capabilities for data control, Home Depot and Lowe's still lack an integrated system that would allow customers to adequately perform transactions by fully satisfying the product requirements and delivery demands. The efficiency of every individual e-trading site has proven successful for the internal data flow management and inventory control, but not for transaction efficiency. 3.2. Estimation and procurement of rebar exchange, the construction industry has not found yet an adequate platform for material purchase and procurement. Construction material retailers are using portals to sell their products, thus lacking dynamic pricing or trading strategies that could benefit both suppliers and buyers. In the US construction industry, Home Depot and Lowe's are presently utilizing e-commerce approaches for their retailing practices [9]. These retail companies are also implementing B2B solutions for the sharing of information among customers, stores, suppliers and employees. Home Depot has point-of-sale (POS) terminals at stores or available in the website. When customers want to buy an appliance, they enter the order into the POS system, which is connected to the supplier's inventory and distribution systems. It allows consumers to buy appliances directly from the supplier through Home Depot and schedule an appointment to have the refrigerator delivered and installed at the customers' convenience. On a different e-procurement approach, Lowe's employees are able to understand what is going on at any store location. Regional vice presidents and district managers can quickly view sales and inventory trends at any store and make decisions that add value to the bottom line. As a result of using the MicroStrategy platform, Lowe's is realizing significant benefits through improved merchandising decisions, Quantities of reinforcing steel are calculated by adding up bar lengths according to size and converting the total length into weight. The size number of the bar indicates the diameter in eighths of an inch or millimetres for metric sizes. This weight can be expressed in pounds (lb), hundred weight (CWT) or tons. It is customary to include bar laps as additional quantities in the total weight. These additional quantities may represent around 10% of the steel reinforcing bars used in the structure. Lap lengths are calculated using the provisions of a building code. The factors that affect the cost of reinforcing steel can be classified as delivery factors and specification factors. Delivery factors are those that influence the shipment and procurement conditions of the purchase. For instance, some examples of delivery factors are supplier proximity, expected delivery time, potential delays, construction schedule, quantities, presence of alternative suppliers, etc. On the other hand, specification factors are inherent to the manufactured product, for example [11]: bar grade, tensile strength, whether bars are deformed, have ridges of indentations on their surface, or if they can be smooth, extent to which bars must be fabricated, cut and bent, and the complexity therein, and whether bars can be plain or must be galvanized or epoxy-coated. Reinforcing steel can be epoxy-coated either before or after it is fabricated, and epoxy coating will typically add about 25% to the installed price of reinforcement [12]. Reinforcing bars are usually deformed, featuring ridges that provide bonding with the adjacent concrete. Fig. 1 shows a typical steel reinforcing bar featuring various marks according to several specification factors such as manufacturer, size, type and grade. The grade corresponds to the steel yield point in thousands of pounds per square inch. In the United States, the size designations of rebar are specified by the American Society for Testing and Materials (ASTM). Most bars are deformed; they have a pattern rolled Fig. 2. Optimization tool owned by buyer. Fig. 3. Optimization tool owned by seller. Fig. 1. Steel reinforcing bar description (adapted from [13]). 572 D. Castro-Lacouture et al. / Automation in Construction 16 (2007) 569–575 ratio between outputs and inputs. In other words, a purchase transaction is efficient if it gets more outputs (e.g., quantity and number of features) for fewer inputs (e.g., lead time and price). 5. Optimization of business practices Fig. 4. Optimization tool owned by marketplace. onto them that helps the concrete get a grip on the bar. Plain bars are also manufactured, but are used only in particular situations in which the bars are expected to slide (e.g., crossing expansion joints in a bridge). The size designations are based on the diameter in eighths of an inch of a plain round bar having the same weight per foot as the deformed bar. 4. Business problem In an e-business marketplace environment, a purchasing agent is faced with the problem of defining an advantageous offer. For instance, suppose that in a rebar net market, an agent needs to bid on rebar to be used in a construction project. Before submitting the bid to the exchange, the agent needs to fully understand the possible tradeoffs between quantity, delivery time, price to be paid, and features (i.e., grade, surface, shape, etc), among others. Through the use of the supply chain optimization tool, the agent is expected to solve the following situations: ▪ ▪ ▪ ▪ ▪ Initial price to be offered in the bid Quantity of material for an efficient offer Tradeoffs between bid price, lead time and material quantity Competitiveness of bid price Sensitivity of bid price to product requirements By offering an initial price for a certain weight of rebar, efficiency in the transaction can be understood as a generalized In order to obtain better results in terms of cash flow, working capital turnover cost reduction or quality, businesses are continuously searching for new tools. In recent years, online reverse auctions have emerged as popular means for reducing the price of purchased materials used in the production of durable goods. This dynamic bidding process typically results in significantly lower prices than the buyer has historically paid [14]. However, net savings may also be lower due to various factors such as buyer not selecting the lowest bid, changes in price through post-online auction negotiation or buyer purchasing neither all nor any of the line items [15]. This situation, coupled with the lack of knowledge of market prices and their behaviour and especially the local optimization of product design at the expense of optimizing the performance of the entire enterprise, makes businesses prone to unfavourable outcomes [16]. With the use of the proposed supply chain optimization tool, the analytical capabilities of three different entities in the transaction environment (i.e., sellers, buyers and marketplace) can be enhanced. The ownership of the supply chain optimization tool determines the application and treatment of the information to be entered in the transaction. Fig. 2 shows a case in which the supply chain optimization tools is owned by the buyer. This information can be used for purchasing analysis. Fig. 3 illustrates the situation when the supply chain optimization tool is owned by the seller. A third case, shown in Fig. 4, occurs when the marketplace owns the supply chain optimization tool. Information about past transactions is collected and owned by the marketplace. This arrangement allows marketplace parties involved (i.e., sellers and buyers) to acquire the value-added analytical service and information through the payment of an Table 1 Sample data for a rebar marketplace Date Buyer Transaction Paid Lead time Weight Bar size Grade Surface Shape ID (×1000 USD) (days) (tons) (×1000 lb/in.2) (EC = epoxy-coated, Reg = Regular) 10/03/2004 10/04/2004 11/14/2004 11/15/2004 11/15/2004 11/18/2004 11/22/2004 11/26/2004 12/03/2004 12/04/2004 12/05/2004 12/06/2004 12/07/2004 12/07/2004 Amb. Steel Steelia Ltd ZBW Steelers STB Co. ARC Inc. Van Steel Consteel Inc. Imatek STB Co. Van Steel Consteel Inc. Steelers Associates Amb. Steel STB Co. CRS-001 CRS-002 CRS-003 CRS-004 CRS-005 CRS-006 CRS-007 CRS-008 CRS-009 CRS-010 CRS-011 CRS-012 CRS-013 CRS-014 5.18 6.63 3.98 6.90 10.35 1.74 7.74 8.70 8.18 9.81 7.28 17.02 5.26 2.63 15.00 18.00 18.00 15.00 15.00 18.00 24.00 18.00 18.00 18.00 10.00 12.00 12.00 20.00 3.00 5.00 3.00 4.00 6.00 1.00 4.00 5.00 5.00 6.00 7.00 14.00 4.00 2.00 4.00 4.00 5.00 6.00 6.00 4.00 5.00 6.00 7.00 8.00 8.00 6.00 7.00 8.00 50 50 50 50 50 60 60 60 60 60 60 70 70 70 EC Reg Reg EC EC EC EC EC EC EC Reg Reg Reg Reg Straight Single bend Single bend Straight Straight Straight Single bend Straight Straight Straight Straight Straight Single bend Single bend D. Castro-Lacouture et al. / Automation in Construction 16 (2007) 569–575 Fig. 5. Efficient front generated by the optimization tool with one input and one output. access fee. The supply chain optimization tool uses quantitative data from previous transactions. This information is valuable for sellers and buyers and helps them make decisions regarding purchases or sales over the marketplace. 6. Computational experiments To illustrate the reports available with the proposed supply chain optimization tool, hypothetical data was used, arising in the context of a rebar marketplace (see Table 1). The data required by the supply chain optimization tool is comprised of past completed transactions in the marketplace or previous matches between buyers and sellers. This data is originated from the marketplace or from the purchasing agent who keeps track of the activity on a given rebar exchange. Table 1 contains sample data from a deformed rebar exchange. Each row in the table contains information on a successful purchase transaction for a given type of rebar. First, a simple, yet illustrative case is considered in which the amount paid (in thousands of dollars) is considered an input and the weight (in tons) is considered an output. The rationale behind the model is that if a company pays more, it should get more tons of steel. Current efforts are well underway to incorporate categorical outputs such as bar size, grade, surface coating, and shape, often present in the rebar Net market. Underlying optimization models can be incorporated to deal with this type of outputs [17–20]. After running the model behind the supply chain optimization tool (see the Appendix), the efficient front shown in Fig. 5 is obtained. The transactions with Transaction ID of CRS-006, CRS-011, CRS-012, and CRS-014 are efficient and are labeled with stars. They define an envelope in which all the transactions below it are called inefficient. In other words, the efficient transactions serve as a reference set of best buy offers. Suppose that the buyer is thinking of making a bid similar to the transaction CRS-010, this is, paying $9810 for 6 tons of rebar. Fig. 6 shows the recommendation made by the optimization tool. After analyzing the buyer's bid, the optimization tool will recommend the bidder to lower its offer by $2530 and to 573 Fig. 6. Projecting an inefficient transaction onto the efficient front. aggressively negotiate an additional ton of steel. After this simple analysis of the previous transactions made in the marketplace, the buyer may want to rethink the offer. To unveil the full potential of the supply chain optimization tool, a model is analyzed with two inputs: amount paid (thousands of dollars) and lead time (days); and one output: weight (in tons). The rationale behind this model is that if more is to be paid or delivery time is to be longer, then more tons of rebar are to be received. The supply chain optimization tool can help analyze the past transactions in the marketplace in search of efficient and inefficient transactions. In Fig. 7 efficient transactions are labeled with a star and inefficient transactions are labeled with a triangular cylinder. A quick exploration of the output of the optimization procedure suggests that under this new model, transactions CRS-006, CRS-011, CRS-012, CRS-013, and CRS-014 are considered efficient. On the other hand, Fig. 8 shows the projection onto the efficient front of the inefficient transactions. For instance, if an analyst is planning to make a spot buy offer in the marketplace similar to transaction CRS-005, Fig. 8 (e) tells Fig. 7. Efficient (stars) and inefficient (triangle) transactions generated by the optimization tool with two inputs and one output. 574 D. Castro-Lacouture et al. / Automation in Construction 16 (2007) 569–575 Fig. 8. Projection of inefficient transactions onto the efficient front. the analyst that there is significant room to improve the bid. The optimization tool suggests, in this case, that in order to make an efficient bid, the analyst could ask the supplier to shorten the delivery time in 5 days, obtain 1 more ton of steel, and lower its price by $3070. Although this might not be possible at the current market conditions, the analyst could use this information to fine tune his offer or negotiate a better informed deal. In a similar fashion, results from the optimization procedure suggest that transactions CRS-007, CRS-008, CRS-009 and CRS-010 also have room for improvement by shortening the delivery time, obtaining more tons of steel, and lowering the price. The outcome of the optimization tool will lead to an optimized spot buy offer. In the case of transaction CRS-007, in order to make an efficient bid, the analyst could ask the supplier to shorten the delivery time in 14 days, obtain 3 more tons of steel, and lower its price by $460. For the purposes of negotiating a better deal with current inefficient spot offers, a similar approach can be taken for transactions CRS-008, CRS-009 and CRS-010. The purchasing analyst may use the information provided by the optimization output in order to project other inefficient transactions onto the efficient front. For instance, for that pur- pose the analyst could ask the supplier to reduce the delivery time of transaction CRS-003 in 3 days, or while shortening the delivery time of transaction CRS-004 in 4 days, the buyer could obtain 2 more tons of steel. 7. Conclusions and further research Current developments on the design of B2B e-trading marketplaces are concentrated on building robust architectures and frameworks, while the evaluation by practitioners and researchers of mechanisms for optimizing purchasing decisions has not been properly addressed. This paper addresses the purchase of construction materials as the last component in the supply chain, presenting a tool for optimizing purchasing decisions in B2B construction marketplaces. With the use of a supply chain optimization tool, the analytical capabilities of different entities in the B2B construction marketplace are improved, focusing on rebar as a typical construction material. Implications for current practices of rebar procurement are related to satisfaction of contractors and rebar suppliers in terms of product price and specifications. Both D. Castro-Lacouture et al. / Automation in Construction 16 (2007) 569–575 buyers and sellers will have a tool that will allow them to successfully complete transactions, thus getting the best arrangement in relation to price, delivery time and quantity of rebar material. Several optimization models are analyzed with two inputs: amount paid (thousands of dollars) and lead time (days); and one output: weight (in tons). The supply chain optimization tool acts as a decision support system for analyzing past transactions in the marketplace in search of efficient and inefficient transactions. It was demonstrated for the case of various transactions that, in order to make an efficient bid, the purchasing analyst could ask the supplier to shorten the delivery time in, obtain more tons of steel, and lower its price for the purposes of negotiating a better deal with current inefficient spot offers. The research efforts in this area should focus on characterizing the optimization model to represent rebar in the context of a differentiated marketplace, not only limiting the envelope analysis to quantities of steel, delivery times and price. Current efforts from the writers are well underway to incorporate categorical outputs such as bar size, grade, surface coating, and shape, often present in the rebar marketplace. Appendix A. The supply chain optimization tool The mathematical core behind the supply chain optimization tool is based on the theory of data envelopment analysis (DEA) [17]. In DEA, it is necessary to solve a sequence of linear programming models, one for each transaction. We have implemented the DEA additive model [17] using the SAS language. Let J be the set of past transactions in the rebar marketplace. In the jargon of DEA, these transactions are also called the decision making units (DMU). Let n be the number of past transactions in the marketplace. Let I be the set of factors in a transaction considered to be inputs (i.e., price). Let m be the total number of inputs. Let R be the set of factors in a transaction considered to be outputs (i.e., tons of steel). Let s be the total number of outputs. Let xij be the amount of input i in transaction j. Let yrj be the amount of output r in transaction j. Let λj be the efficient front projection multiplier for transaction j. Let sr+ and si−, be the efficient front slack for output r and input i, respectively. In the DEA additive model [17], for each transaction j′ = 1, …, n, the following linear optimization model is solved: min k̄;sþ ;s− zj V ¼ − subject to: P yrj kj −sþ r jaJ P − xij xij kj þ si jaJ P kj jaJ kj z0; sþ r z0; s−i z0; X þ X − raR iaI sr − si ¼ yr; j V; 8raR ¼ xi; j V; 8iaI ¼ 1 8jaJ 8raR 8iaI 575 References [1] J.Y. Bakos, Reducing buyers search costs: implications for electronic marketplaces, Management Science 43 (12) (1997) 1–27. [2] Y. Bakos, The emerging role of electronic marketplaces on the internet, Communications of the ACM 41 (8) (1998) 35–42. [3] T.W. Malone, J. Yates, R. Benjamin, Electronic markets and electronic hierarchies, Communications of the ACM 30 (6) (1987) 484–497. [4] H. Li, J. Cao, D. Castro-Lacouture, M. Skibniewski, A framework for developing a unified B2B e-trading construction marketplace, Automation in Construction 12 (2002) 201–211. [5] Q. Dai, R. Kauffman, Business models for internet-based e-procurement systems and B2B electronic markets: an exploratory assessment, 34th Hawaii International Conference on Systems Science, Maui, HI, January, 2001. [6] J. Kraker, Buyers expect systems soon will deliver for them, Engineering News Record, 11 Dec 2000, 2 May 2001 (2000) (bhttp://www.enr.com/ new/C1211a.aspN). [7] J. King, B2B exchanges tighten buyer/seller data links, Computerworld 34 (10) (March 2000) 42. [8] S. Kong, H. Li, T. Hung, J. Shi, D. Castro-Lacouture, M. Skibniewski, Enabling information sharing between E-commerce systems for construction material procurement, Automation in Construction 13 (2004) 261–276. [9] D. Castro-Lacouture, “B2B e-work intranet solution design for rebar supply interactions”. Doctoral Dissertation. Purdue University, West Lafayette, IN, 2003. [10] D. Castro-Lacouture, M. Skibniewski, Applicability of e-work models for the automation of construction materials management systems, Production Planning and Control 14 (8) (2003) 789–797. [11] C. Schexnayder, R. Mayo, Construction Management Fundamentals, McGraw-Hill, New York, 2004. [12] Concrete Reinforcing Steel Institute (CRSI), Epoxy-coated Reinforcement bhttp://www.crsi.org/ECR/faq.htmlN, 12-02-2004. [13] S. Nunnally, Construction Methods and Management, 6th ed., Prentice Hall, New Jersey, 2004. [14] M. Emiliani, D.J. Stec, Realizing savings from online reverse auctions, Supply Chain Management 7 (1) (2002) 12–23. [15] FreeMarkets, Annual Report, SEC Form 10-K, 23 February, 2001. [16] M. Emiliani, Cracking the code of business, Management Decision 38 (2) (2000) 60–79. [17] A. Charnes, W.W. Cooper, A.Y. Lewin, L.M. Seiford, Data Envelopment Analysis: Theory, Methodology, and Application, Kluwer Academic Publishers, 1994. [18] W.A. Kamakura, A note on “The use of categorical variables in data envelopment analysis”, Management Science 34 (1) (1988) 1273–1276. [19] F. Førsund, Categorical variables in DEA, International Journal of Business and Economics 1 (1) (2002) 31–41. [20] J.J. Rousseau, J.H. Semple, Categorical outputs in data envelopment analysis, Management Science 39 (3) (1993) 384–386.
© Copyright 2026 Paperzz