【正文】 （17） 目標(biāo)函數(shù)（16）在時(shí)達(dá)到最大值。約束條件（17）服從可行性條件，表明。那么，相應(yīng)的可以被導(dǎo)出。直接分化給比較靜力學(xué)最優(yōu)的解決方案。通過(guò)比較在命題8中的，和定理1中的，我們可以得到.命題9的證明：該公司的收益函數(shù)在這個(gè)案例中為。程序在第四節(jié)收益率公司的優(yōu)化問(wèn)題將作為決策變量，(18) , (19) , (20). (21)通過(guò)對(duì)定理2的證明,優(yōu)化問(wèn)題(18)(21)可以概括為單個(gè)決策變量。此外，的唯一最優(yōu)解由下式?jīng)Q定。因?yàn)?，所以跨越間隔，并且在E上減少。最佳的質(zhì)量水平和經(jīng)濟(jì)生產(chǎn)規(guī)模可以根據(jù)它們可以表示為純函數(shù)而被導(dǎo)出。根據(jù)定理2的證明，我們可以把最優(yōu)解的專用的情況用代替。對(duì)于任意的，我們可以得到，以及。參考文獻(xiàn)[1] Belloni,A.,Freund,R.,Selove,M.,Simester,D.,2008. Optimizing product line designs:efficient methods and Science 54(9),1544–1552.[2]Benjaafar,S.,Gupta,D., versus focus:issues of flexibility,capacity,and number of production Transactions 30(5), 413–425.[3]Chambers,C.,Kouvelis,P.,Semple,J., ased petition, profitability, and variable Science 52(12), 1884–(2012) 314–328 327[4]Choudhary,V.,Ghose,A.,Mukhopadhyay,T.,Uday,R. , pricing and quality differentiation. Management Science 51(7), 1120– Groote, X., 1994. The flexibility of production processes: a general Science 40(7), 933–945.[5] Desai, P., Kekre, S., Radhakrishnan, S., Srinivasan, K., 2001. Product differentiation and monality in design: balancing revenue and cost drivers. Management Science 47 (1), 37–51.[6] Dobson, G., Yano, ., 2002. Product offering, pricing,and maketostock/maketoorder decisions with shared capacity. Production and Operations Management 11 (3), 293–312.[7] Gabszewicz, ., Thisse, ., 1979. Price petition, quality and ine disparities. Journal of Economic Theory 20, 340–359.[8] Gaur, V., Honhon, D., 2006. Assortment planning and inventory decisions under a locational choice model. Management Science 52 (10), 1528–1543.[9] Hotelling, H., 1929. Stability in petition. Economic Journal 39, 41–57.[10] Jiang, K., Lee, ., Seifert, ., 2006. Satisfying customer preferences via mass customization and mass production. IIE Transactions 38, 25–38.[11] Jones, R., Mendelson, H., 2011. Information goods vs. industrial goods: cost structure and petition. Management Science 57 (1), 164–176.[12] Karmarkar, ., 1996. Integrative research in marketing and operations management. Journal of Marketing Research 33 (2), 125–133.[13] Kim, K., Chhajed, D., 2002. Product design with multiple qualitytype Science 48 (11), 1502–1511.[14] Kim, K., Chhajed, D., 2008. Commonality in product design: cost saving and cannibalization. European Journal of Operational Research 125 (3), 602–621.[15] Krishnan, V., Zhu, W., 2006. Designing a family of developmentintensive products. Management Science 52 (6), 813–825.[16] Lancaster, K., 1990. The economics of product variety: A survey. Marketing Science 9 (3), 189–206.[17] Lancaster, K., 1975. Socially optimal product differentiation. American Economic Review 65 (4), 567–585.[18] Matsubayashi, N., Ishii, Y., Watanabe, K., Yamada, Y., 2009. Fullline or specialization strategy? The negative effect of product variety on product line Journal of Operational Research 196 (2), 795–807.[19] Moorthy, ., 1984. Market segmentation, selfselection, and product line design. Marketing Science 3 (4), 288–305.[20] Moorthy, ., 1988. Product and price petition in a duopoly. Marketing Science 7 (2), 141–168.[21] Mussa, M., Rosen, S., 1978. Monopoly and product quality. Journal of Economic Theory 18, 301–317.[22] Netessine, S., Taylor, ., 2007. Product line design and production Science 26 (1), 101–117.[23]Schmidt, ., Porteus, ., 2000. The impact of an integrated marketing and manufacturing innovation. Manufacturingamp。Service Operations Management 2 (4), 317–336.[24] Tirole, J., 1988. The Theory of Industrial Organization. MIT Press, Cambridge, MA.[25] VillasBoas, ., 1998. Product line design for a distribution channel. Marketing Science 17 (2), 156–169