A buyer seller game model for selection and negotiation of purchasing bids: Extensions and new models

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1 Eropean Jornal of Operational Research 154 (2004) A byer seller game model for selection and negotiation of prchasing bids: Extensions and new models Joe Zh * Department of Management, Worcester Polytechnic Institte, Worcester, MA 01609, USA Received 1 March 2001; accepted 30 Agst 2002 Abstract A nmber of efficiency-based vendor selection and negotiation models have been developed to deal with mltiple attribtes inclding price, qality and delivery performance. The efficiency is defined as the ratio of weighted otpts to weighted inpts. By minimizing the efficiency, Tallri [Er. J. Operat. Res. 143(1) (2002) 171] proposes a byer seller game model that evalates the efficiency of alternative bids with respect to the ideal target set by the byer. The crrent paper shows that this byer seller game model is closely related to data envelopment analysis (DEA) and can be simplified. The crrent paper also shows that setting the (ideal) target actally incorporates implicit tradeoff information on the mltiple attribtes into efficiency evalation. We develop a new byer seller game model where the efficiency is maximized with respect to mltiple targets set by the byer. The new model allows the byer to evalate and select the vendors in the context of best-practice. By both minimizing and maximizing efficiency, the byer can obtain an efficiency range within which the tre efficiency lies given the implicit tradeoff information characterized by the targets. The crrent stdy establishes the linage between byer seller game models and DEA. Sch a linage can provide the byer with correct evalation methods based pon existing DEA models regarding the natre of bidding. Ó 2002 Elsevier B.V. All rights reserved. Keywords: Game models; Linear programming; Efficiency; Data envelopment analysis 1. Introdction As pointed ot by Wise and Morrison (2000), one of the major flaws in the crrent bsinessto-bsiness (B2B) model is that it focses on pricedriven transactions between byers and sellers, and fails to recognize other important vendor attribtes sch as response time, qality and cstomization. In fact, a nmber of efficiency-based * Tel.: ; fax: address: jzh@wpi.ed (J. Zh). negotiation models have been developed to deal with mltiple attribtes inpts and otpts. For example, data envelopment analysis (DEA) is sed by Weber and Desai (1996) and Weber et al. (1998) to develop models for vendor evalation and negotiation. The efficiency is defined as the ratio of weighted otpts to weighted inpts. The weights reflect the tradeoffs among mltiple otpts and mltiple inpts. However, sch tradeoffs are sally not completely nown to the byer. Using DEA, one can estimate a set of weights and obtains the relative efficiencies in the context of bestpractice /$ - see front matter Ó 2002 Elsevier B.V. All rights reserved. doi: /s (02)

2 J. Zh / Eropean Jornal of Operational Research 154 (2004) Tallri (2002) develops an efficiency-based byer seller game model that evalates alternative bids based pon the ideal target set by the byer. In Tallri (2002), the efficiency of each vendor is minimized when the efficiency of the ideal target is set eqal to one. That is, min P a ry rp a ry r ideal a ry ri s¼1 b sx si 6 1 8i ð1þ where a r and b s represent nnown otpt and inpt weights, respectively. p represents a specific vendor bid nder evalation among n bids (i ¼ 1;...; n). Each bid i has v bid otpts y ri (r ¼ 1;...; v) and bid inpts x si (s ;...; ). y r ideal (r ;...; v) and x s ideal (s ;...; ) represent the rth bid otpt and ith bid inpt for the ideal target set by the byer. The ideal target represents the best vales for each inpts and otpts (i.e., minimm inpt vales and maximm otpt vales) across all n vendor bids, and therefore dominates all the vendors. Tallri (2002) states that althogh or models are gronded in the general efficiency theory, they are different from a traditional DEA sense. In fact, model (1) is closely related to the CCR ratio model (Charnes et al., 1978) where the efficiency of each individal vendor is maximized. The next section shows that becase the ideal target defined in Tallri (2002) dominates all vendors, the constraints on the vendors ðð P a ry ri = s¼1 b sx si Þ 6 1Þ are redndant and can be removed from both model (1) and the CCR ratio model. Note the fact that the efficiency of the ideal target mst be eqal to one in the CCR ratio model. Ths, the only difference between model (1) and the CCR ratio model is that the former minimizes the vendor efficiency and the latter maximizes the vendor efficiency. Throgh efficiency minimization in model (1), the byer identifies/selects the vendor who has the largest minimm efficiency with respect to the ideal target. Note that a vendor having the largest minimm efficiency does not necessary mean that the vendor is the best with respect to the ideal target. Becase nlie the CCR ratio model, model (1) does not evalate efficiency in the context bestpractice. Note that the constraint on the ideal target ðð a ry r ideal = Þ¼1Þ incorporates implicit tradeoff information into the efficiency evalation. i.e., this constraint can be viewed as incorporation of tradeoffs or weight restrictions in DEA (see, e.g., cone-ratio DEA model by Charnes et al., 1990). Note also that the ideal target defined in Tallri (2002) may not be achievable by all the vendors, and therefore may be nrealistic. As a reslt, the implicit tradeoff information characterized by ð a ry r ideal = Þ¼1 may not be valid, and the vendor efficiency in model (1) may not be correctly characterized with respect to the nrealistic target/invalid tradeoffs. Therefore, alternative targets which are realistic may be sed in evalating and selecting the vendors. In fact, sch targets represent the byerõs preferences over the weighting scheme. When mltiple bid otpts and inpts are present, mltiple targets may be needed in order to flly characterize the byerõs preferences over the bids. The crrent paper extends the model (1) into sitations where vendor efficiency is maximized so that the byer evalates and selects the vendors in the context of best-practice. The crrent paper also explicitly lins the Tallri (2002) model to the DEA. Conseqently, more byer seller game models can be obtained based pon DEA. For example, the Tallri (2002) model yields an invalid reslt in proposing negotiation strategies where the percentage of on-time delivery can exceed 100%. This invalid reslt can be circmvented by sing a DEA model where the best-practice exhibits variable-retrns-to scale. Althogh the ideal target implicitly incorporates some sort of tradeoff information, the tre efficiency of vendors is still not completely nown to the byer. With the crrent byer seller game model, the byer can obtain an efficiency range where the tre efficiency lies given the targets. This better helps the byer in selecting the competitive bids.

3 152 J. Zh / Eropean Jornal of Operational Research 154 (2004) The rest of the paper is organized as follows. The next section simplifies the models in Tallri (2002). We then develop new byer seller models. The new models are illstrated with nmerical examples inclding a pharmacetical company stdied in previos stdies. The final section concldes. Similarly, model (3) can also be simplified as X n min P a ry rp i¼1 a ry r ideal ð4þ 2. Simplified byer seller game model Becase the ideal target in Tallri (2002) is defined as dominating all vendors, constraints ð a ry ri = s¼1 b sx si Þ 6 1 are redndant in model (1). Conseqently, model (1) can be written as min P a ry rp a ry r ideal ð2þ Becase the ideal target dominates all the vendors, the DEA efficiency of the ideal target mst be eqal to one, and the DEA efficiency of vendors mst be less than one. Conseqently, if we change the objective from minimization to maximization in model (1), then we obtain the CCR ratio model. This finding lins model (2) (or model (1)) to the DEA methodology. As a reslt, more byer seller game can be obtained by changing the objective in varios DEA models. In Tallri (2002), another byer seller game model is developed when all vendorsõ efficiencies are minimized simltaneosly. That is The contribtion of the simplified models (2) and (4) lies in the fact that comptational brden can be sbstantially redced if the problem size is large. Also, the byer seller game model is lined to DEA. 3. New byer seller game models Note that the ideal target in Tallri (2002) may not be achievable by all vendors. As a reslt, the target may not be realistic and the reslting vendor efficiency may not be correctly evalated. In this section, we assme that the targets set by the byer do not have to dominate all the vendors. Sppose the byer selects G targets represented by y target r and x target s ( ;...; G). We modify model (1) into the following model where efficiency is maximized as in DEA 1 max a ry rp a ry target s¼1 b sx target a ry rp 6 1 ;...; G ð5þ 8i 6¼ p min X n i¼1 a ry rp a ry r ideal a ry ri s¼1 b sx si 6 1 8i ð3þ 1 Note that model (5) may be infeasible if the mltiple targets set by the byer cannot be fit into the same efficient facet. If that happens, the byer has to adjst the targets. For example, we can reqire that G 6 v þ 1. These G targets actally determines an efficient facet contains the origin. Therefore, we only need at most v þ 1 points to determine sch an efficient facet. Model (5) is also similar to the benchmaring model sed in Zh (2001) in measring the qality-of-life of cities.

4 J. Zh / Eropean Jornal of Operational Research 154 (2004) Model (5) maximizes each vendorõs efficiency individally when the efficiencies of the targets are set eqal to one. Model (5) differs from model (1) in three aspects. First, model (5) provides the best efficiency scenario for the vendors whereas model (1) calclates the worst scenario. Second, the efficiency of vendor can exceed one if the targets do not dominate all the n vendors. Third, if new vendors enter the negotiation process and some of the new vendors dominate the targets, new targets need not to be established. If all the targets dominate the vendors, then ð a ry rp = Þ 6 1(8i 6¼ p) are redndant and can be removed from model (5). In fact, model (1) determines the minimm efficiency for vendor p, and model (5) determines the maximm efficiency when the efficiency of targets is assmed to be one. Let h p and h p be the optimal vales to models (1) and (5), respectively. Let h p be the tre efficiency of vendor p with respect to the fact that the efficiencies of the targets are set eqal to one. Then h p 6 h p 6 h p. When both models (1) and (5) are calclated, the byer can have an efficiency range within which the tre vendor efficiency lies. This better helps the vendor in evalating and selecting the vendors. 2 We can develop the following byer seller game model when all vendorsõ efficiencies are maximized simltaneosly. max X n i¼1 w a ry rp i a ry target s¼1 b sx target ;...; G ð6þ where w i (i ;...; n) are byer-specified weights reflecting the preference over vendorsõ efficiencies. By sing the Charnes Cooper transformation, all models (1) (6) can be transformed into linear programs. For example, model (5) is eqivalent to 2 We may se the average of h p and h p as the efficiency of vendor bid p in the Tallri (2002) bid selection model. max X v X v X v X s¼1 a r y rp a r y target a r y rp 6 X b s x sp 4. Illstration ¼ X s¼1 s¼1 b s x sp b s x target 8i 6¼ p ;...; G ð7þ We first se the nmerical example in Table 1 to illstrate the byer seller game models. We have 12 vendors (DMUs). Their performance is characterized by (1) distribtion costs, inclding transportation and handling costs, (2) cstomer response time (days), the amont of time between an order and its corresponding delivery, (3) percentage of shipping errors (nmber of incorrect shipments made), (4) manfactring cost, total cost of manfactring, inclding labor, maintenance, and re-wor costs, (5) nmber of items prodced (in each month), (6) percentage of ontime delivery, (7) fill rate, proportion of order filled immediately, (8) wealy profit, and (9) nmber of distribtion centers. Usally smaller vales on the first for measres and larger vales on the remaining five measre indicate better performance of these vendors. We therefore select the first for measres as inpts and the remaining as otpts. The last row of Table 1 provides the target based pon the best vales of the inpts and otpts across all 12 DMUs. The last colmn provides the efficiency scores obtained from the CCR ratio model. It can be seen that model (1) indicates that DMU3 is the best DMU, becase it has the largest minimm efficiency score. Model (5) indicates that DMU6 is the best DMU with respect to the target. This shows that the best DMU nder model (1) does not necessarily mean that the DMU is the

5 154 J. Zh / Eropean Jornal of Operational Research 154 (2004) Table 1 Nmerical example DMU Distribtion cost Response time Shipping errors (%) Manfactring cost Items prodced Ontime delivery (%) Fill rate (%) Profit Distribtion centers Model (1) Model (5) Target CCR ratio Table 2 Data for the vendors Variable Vendor 1 Vendor 2 Vendor 3 Vendor 4 Vendor 5 Vendor 6 Ideal target Price ($/nits) % Acceptance % OTD best in the context of best-practice in model (5). Therefore, it is critical that we also se model (5) when selecting the best vendor. Finally, the efficiency ranges presented by the 11th and 12th colmns yield more information regarding the performance of the DMUs. We next se the data in Weber and Desai (1996), Weber et al. (1998) and Tallri (2002) to demonstrate the models. Table 2 presents the data where the price is sed as the only inpt, and percentages of accepted items (measring qality) and on-time deliveries (OTD) (measring delivery performance) are sed as the two otpts. The company is a division of a Fortne 500 pharmacetical company. The last colmn reports the ideal target sed in Tallri (2002) which by definition dominates all the vendors. Table 3 presents the reslts obtained from model (2) with the last colmn reporting the efficiency scores (h p ) obtained from model (1). Note that h p ¼ h p in vendors 3 and 4. Althogh Table 3 reports a set of optimal weights, we shold point ot that cation shold be paid when sing these optimal weights becase of possible mltiple optimal soltions. The same comment can be applied to Tallri (2002). Now, we relax the assmption that the target mst be composed by the best vales of all attribtes of the vendors. Sppose a new vendor enters the bidding. The last colmn of Table 4 reports the new vendor. The second colmn of Table 4 reports the efficiency scores obtained from model (5). If we se model (1), then a new ideal target will be determined via changing the inpt for the old ideal target to The third colmn of Table 4 reports the efficiency scores obtained from model (1) with the new ideal target. Both models (1) and (5) indicate that the new vendor is the best one. We finally note that Tallri (2002) states vendors 1, 3, 4, 5 and 6 mst improve their percentage of OTD to 96.77%, %, %, %

6 J. Zh / Eropean Jornal of Operational Research 154 (2004) Table 3 Vendor efficiency Efficiency ( h p ) Weights Efficiency (h p ) Price % Acceptance % OTD Vendor Vendor Vendor Vendor Vendor Vendor Table 4 Vendor efficiency with a new vendor Efficiency Efficiency ( h p ) (h p ) New vendor Vendor Price: Vendor % Acceptance: 100 Vendor % OTD: 98 Vendor Vendor Vendor New vendor and %, respectively. Obviosly, one cannot have OTD exceeding 100%. As a reslt, the negotiation strategies proposed in Tallri (2002) are npractical and invalid. Tallri (2002) does not demonstrate how these vales are obtained. However, we shold note the fact that we cannot se model (1) to generate these vales, becase model (1) does not evalate efficiency in the context of best-practice. Becase of the linage between DEA and the byer seller model established by the crrent stdy, one can se a variable-retrns-to-scale DEA model to overcome this problem. i.e., the a ry ri shold be replaced by a ry ri þ o, where o is nconstrained in sign. This o corresponds to the convexity constraint in the dal form to model (7), and this convexity constraint garantees that the OTD will not exceed 100% (Baner et al., 1984). Note that Weber and Desai (1996) employ the correct DEA model with variable-retrns-to-scale. 5. Conclsions Tallri (2002) proposes minimizing efficiency to develop a byer seller game model for selection and negotiation of prchasing bids. The crrent paper proposes maximizing efficiency in developing the byer seller game model so that the performance is measred against best-practice. The crrent paper shows that the models developed in Tallri (2002) and in the crrent paper are closely related to DEA. As a reslt, new models based pon varios DEA models can be established. For example, if we want to consider the retrns to scale or treat ncontrollable bid otpt/inpt vales, related DEA models can be sed with respect to efficiency minimization and maximization. Finally, Zh (2002) provides an Excel Add-In for calclating the models discssed in the crrent paper. Acnowledgements The athor is gratefl to the helpfl comments and sggestions made by two anonymos referees. References Baner, R.D., Charnes, A., Cooper, W.W., Some models for estimating technical and scale efficiencies in data envelopment analysis. Management Science 30, Charnes, A., Cooper, W.W., Rhodes, E., Measring the efficiency of decision maing nits. Eropean Jornal of Operational Research 2, Charnes, A., Cooper, W.W., Hang, Z.M., Sn, D.B., Polyhedral cone-ratio DEA models with an illstrative application to larger commercial bans. Jornal of Econometrics 46, Tallri, S., A byer seller game model for selection and negotiation of prchasing bids. Eropean Jornal of Operational Research 143 (1),

7 156 J. Zh / Eropean Jornal of Operational Research 154 (2004) Weber, C.A., Desai, A., Determination of paths to vendor maret efficiency sing parallel coordinates representation: A negotiation tool for byers. Eropean Jornal of Operational Research 90 (1), Weber, C.A., Crrent, J.R., Desai, A., Non-cooperative negotiation strategies for vendor selection. Eropean Jornal of Operational Research 108 (1), Wise, R., Morrison, D., Beyond the exchange: The ftre of B2B. Harvard Bsiness Review 78 (6), Zh, J., A mltidimensional qality-of-life measre with an application to FortneÕs best cities. Socio-Economic Planning Sciences 35 (4), Zh, J., Qantitative Models for Performance Evalation and Benchmaring: Data Envelopment Analysis with Spreadsheets. Klwer Academic Pblishers, Boston.

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