Robust Optimization of Stochastic Revenue Management in Hotel Industry

Mohammad Modarres* & Mehdi Najafi

Mohammad Modarres Sharif University of Technology,
Mehdi Najafi, Sharif University of Technology

Keywords ABSTRACT

Robust Optimization;
Revenue Management;
Robust Solution;
Robust Model,
Stochastic Problem,
Hotel Management The successful application of “revenue management” in airline industry motivated hotel industry management as well as some researchers to apply the concept and techniques of revenue management to enhance the income of hotels. In this paper, we develop a multi objective model for hotel management in stochastic environments. Since demand for rooms, and in some cases the supply, is random we introduce a robust model, in order to manage its uncertain parameters. Therefore, the objectives of the model are robustness as well as income enhancement. Furthermore, some other new assumptions such as overbooking, early (or late) departure of a passenger and different prices are also included in the model
æèííçåŶƬūéƵŹŚưƃ ŶǀƫƺţŢƿźƿŶƯƹƖƿŚƴƇƾſŶƴƸƯƾƬƬưƫřƲǀŝƶƿźƄƳ’

ƪŤƷŶƯōŹŵŹřŶƿŚěƽŻŚſƶƴǀƸŝƾƟŵŚƈţƽżƿŹƶƯŚƳźŝ
ƾƠŬƳƽŶƸƯƹŽŹŶƯŶưŰƯ
ƵŶǀĪģƽŶǀƬĩšŚưƬĩ
30467-5391

ŢƘƴƇƲƿřŵŚƿŻŢƷŚŞƃƹƾƿŚưǀěřƺƷƖƿŚƴƇŹŵŶƯōŹŵŢƿźƿŶƯżǀƯōŢǀƤƟƺƯƽŚƷŵźŝŹŚĩƹƱƹżƟřŻƹŹƁźŤƀĭÎ
ƹ ŚƸƫŶƯ ŶƯōŹŵ ƂƿřżƟř ƹ źŤƸŝ ŢƿźƿŶƯ ŢƸū Śţ ŢŴǀĮƳřźŝ řŹ ƱřźƿŶƯ ƹ ƱřźĮƄƷƹĦě ƽŹřŵƪŤƷ ŢƘƴƇ Śŝ Ʋƿř ƽŚƸǀĭĦƿƹ ƶŝ ƶūƺţ Śŝ ƹ ŶƴƷŵ Źřźƣ ƵŵŚƠŤſř ŵŹƺƯ żǀƳ ƽŹřŵƪŤƷ ŢƘƴƇ ƽřźŝ řŹŶƯōŹŵ ŢƿźƿŶƯ ƮǀƷŚƠƯ ƽŚƸƫŶƯƶŗřŹřƶŝƶƫŚƤƯƲƿřŹŵŶƴƷŵƶƘſƺţżǀƳřŹŢƘƴƇƲƿřŚŝŜſŚƴŤƯŶƯōŹŵŢƿźƿŶƯƾƋŚƿŹƽŚƸƫŶƯŢƘƴƇ ƶŝ ƶūƺţŚŝ ƾƟźƏŻř ŵƺƃ ƾƯƶŤųřŵźě ƾƟŵŚƈţ ƽŚƸƐǀŰƯ Źŵ ƪŤƷ ŶƯōŹŵ ŢƿźƿŶƯƾſŹźŝ ŢƸū ƶƟŶƷ Ŷƴģ ƶŝŶŝŚƿƾƯƶƘſƺţŚƸƫŶƯƲƿřŹřŶƿŚěƽŻŚſƶƴǀƸŝƶƫŚƤƯƲƿřŹŵƶƋźƗƽŵŹřƺƯŹŵƹŚƋŚƤţƱřżǀƯƱŵƺŝƾƟŵŚƈţ ƪƇŚůŶƯōŹŵƱŵźĩźŨĩřŶůƵřźưƷƶŝƩŶƯƽŹřŶƿŚěƹśřƺūƽŹřŶƿŚěŻřŶƴţŹŚŞƗŚƷƩŶƯƲƿřƝřŶƷřŜǀţźţƲƿř
šƹŚƠŤƯƲǀƴĤưƷƹƞƗŚƌƯƹŹŻŹžƭŚĮƴƷźƿŵƭŚĮƴƷŵƹŻũƹźųźǀƔƳŶƿŶūšŚǀƋźƟƲŤƟźĭźƔƳŹŵŚŝŚƴưƋ
ŶŝŚƿƾƯƶƘſƺţŶƯōŹŵŢƿźƿŶƯśƺģŹŚģŹŵƽźĮƿŵƾƋŚƿŹƽŚƸƫŶƯƞƬŤŴƯƎƿřźƃŢŰţŚƷŢưǀƣƱŵƺŝ ŶƯōŹŵŢƿźƿŶƯ
ŹřŶƿŚěƽŻŚſƶƴǀƸŝ
ƾƟŵŚƈţƽżƿŹƶƯŚƳźŝ
śřƺūƽŹřŶƿŚě ƩŶƯƽŹřŶƿŚě ƪŤƷŢƿźƿŶƯ

ííëíƩƺƇƹŲƿŹŚţ ííææëŜƿƺƈţŲƿŹŚţ
Modarres@sharif.eduƞƿźƃƾŤƘƴƇƵŚĮƄƳřŵƖƿŚƴƇƾſŶƴƸƯƵŶĪƄƳřŵŽŹŶƯŶưŰƯźŤƧŵ M_Najafi@ie.sharif.eduƞƿźƃƾŤƘƴƇƵŚĮƄƳřŵƖƿŚƴƇƾſŶƴƸƯƵŶĪƄƳřŵƾƠŬƳƽŶƸƯ
ƾƠŬƳƽŶƸƯƹŽŹŶƯŶưŰƯƪŤƷŶƯōŹŵŹřŶƿŚěƽŻŚſƶƴǀƸŝƾƟŵŚƈţƽżƿŹƶƯŚƳźŝæç
ƵżǀƳŚĪƯƽŚƷƮŤƀƀǀſŚƷơŚţřƽŵƺūƺƯƩźŤƴĩƽŹřŸĭŢſŚǀſřŹ ŶƯōŹŵƮƸſƪǀƬŰţƹƶƿżŬţšŚưǀưƈţƽŻŚſƶƴǀƸŝŶƯōŹŵŢƿźƿŶƯ šŚƤǀƤŰţ ŢƟźƄǀ쎌ſřźŝ ŵźĪƬưƗ ƾŝŚƿŻŹř ƹŹřŻŚŝ ƽŚƷŢưƀƣ ƽŚƸƃƹŹƾſŹźŝƹŹƹźƯƶŝ>æç@ĬƳřĥŶƴĩƾƯƾƟźƘƯŶƳƹŹƹŵƺŞưĩ ŚƿƹŹƺŰƯŹřŻŚŝŹƺŰƯƶƴƿżƷƪǀŞƣŻřƪŤƷƽŹřŸĭŢưǀƣƞƬŤŴƯ ĨƿƶƘſƺţƶŝ>æ@ĬƳƺƫƱƹƹĬƳƺĩĬƴǀĩŢſřƶŤųřŵźěƾŞǀĩźţ
ŹŵƪŤƷŶƯōŹŵŢƿźƿŶƯŢƸūƶƟŶƷĨţƾƟŵŚƈţƽżƿŹƶƯŚƳźŝƩŶƯ
ŶƳřƶŤųřŵźěƾƟŵŚƈţƎǀŰƯ
ƽŶƿŶūŵŚƘŝřƲƨǀƫŢſř>æ@ƶƫŚƤƯƽŚŤſřŹŹŵƢǀƤŰţƲƿřƶģźĭř
ŶƃŚŝƶŤƃřŵƽźŤƄǀŝƢŝŚƐţšŚǀƘƣřƹŚŝŚţŢſřƵŶƃƶŤƟźĭźƔƳŹŵ ƲƿřżƿŚưţƵƺūƹŢſřšƹŚƠŤƯżǀƳƽŵŚƸƴƄǀěƽŚƸƫŶƯƹŹƲǀưƷŻř śžƵŶƃƶŗřŹřƽŚƷƩŶƯƱŵƺŝƶƟŶƷŶƴģƞƫřŻřŶƴţŹŚŞƗƢǀƤŰţ ƪŤƷŢǀƟźƓƱřżǀƯ>æ@ŚŝƶŝŚƄƯŚƷŚƋŚƤţƱŵƺŝƾƟŵŚƈţźŝƵƹLjƗ ƶŤƟźĭźƔƳŹŵƞƬŤŴƯƽŚƷƺƿŹŚƴſŢŰţƹƲŘưƐƯŚƳšŹƺƈŝżǀƳ ƲǀƴĤưƷƹƭŚĮƴƷźƿŵƭŚĮƴƷŵƹŻũƹźųƲŤƟźĭźƔƳŹŵũžŵƺƃƾƯ
ƽŚƷƺƿŹŚƴſŹŵŚƷŢưǀƣƱŵƺŝšƹŚƠŤƯƱŚĪƯřŵžƞƗŚƌƯƹŹŻŹ
ŵŹƺƯƢǀƤŰţƲƿřŹŵźǀƳŢưǀƣźƷƽřźŝƁƹźƟƱřżǀƯƹƞƬŤŴƯ çƂŴŝŢſřźƿŻŭźƃƶŝƶƫŚƤƯƲƿřŹŚŤųŚſŵźǀĭƾƯŹřźƣƾſŹźŝ
ƩŶƯèƂŴŝŹŵŵŻřŵźěƾƯŚƸƫŶƯƽŚƷŵřŵŹřźƣƹšŚǀƋźƟƱŚǀŝƶŝ ŵŵźĭƾƯƾƟźƘƯƪŤƷŶƯōŹŵŢƿźƿŶƯƾƟŵŚƈţƽżƿŹƶƯŚƳźŝƶƿŚě ƽŹřŶƿŚěƝŶƷƹŵŚŝƾƟŵŚƈţƽżƿŹƶƯŚƳźŝƩŶƯŹřŶƿŚěƽŻŚſƶƴǀƸŝ ƶĪǀƿŚŬƳōŻřŶŝŚƿƾƯƶƘſƺţéƂŴŝŹŵƩŶƯƽŹřŶƿŚěƹśřƺū ŚƿźţŵƹŻƹƵŵƺưƴƳƪưƗŵƺųƶǀƫƹřƶƯŚƳźŝƢŝŚƐƯŚƷźƟŚƀƯŻřƾųźŝ ŦůŚŞƯŶƴƿŚưƳƾƯŢƯŚƣřƪŤƷŹŵƵŶƃƾƴǀŝƂǀěƱŚƯŻŻřźţźƿŵ ƞƗŚƌƯƹŹŻŹŚŞƣŚƘŤƯƶĩŵƺƃƾƯŭźƐƯƭŚĮƴƷźƿŵƭŚĮƴƷŵƹŻũƹźų ƹƶƫƺƤƯƲƿřƾſŹźŝƶŝêƂŴŝŹŵƹźƴƿřŻřŢƃřŵŶƷřƺųƾěŹŵřŹ ŹŵơƺƟŦůŚŞƯźŝƵƹLjƗŵƺƃƾƯƶŤųřŵźěƩŶƯŹŵƱōƽźǀĭŹŚĪŝ
ƞƬŤŴƯƽŚƷŢưǀƣƽřŹřŵŚƋŚƤţƞƬŤŴƯƽŚƷƺƿŹŚƴſŵŹřƺƯŻřƽŹŚǀƀŝ ƶŤƟźĭźƔƳŹŵƺţŹŵƺţźǀƛŚƿƹƺţŹŵƺţšŹƺƈŝŶƴƳřƺţƾƯƶĩŶƴŤƀƷ ŚŝƾƟŵŚƈţƽżƿŹƶƯŚƳźŝƶƟŶƷŶƴģƩŶƯƶƫŚƤƯƲƿřëƂŴŝŹŵŶƳƺƃ ƶĩ ƾŤƫŚůŹŵ ƞƗŚƌƯ ƹŹŻŹ ƱŚĪƯř ƹ ƭŚĮƴƷźƿŵƭŚĮƴƷŵƹŻ ũƹźų ƶƘſƺţŶƴŤƀƷƺţŹŵƺţšŹƺƈŝƹšƹŚƠŤƯƽŚƷŢưǀƣƽřŹřŵŚƷƺƿŹŚƴſ
íƂŴŝŹŵŚŤƿŚƸƳƹŵƺƃƾƯƶŗřŹřƶƳƺưƳƾƫŚŨƯìƂŴŝŹŵŶŝŚƿƾƯ
ŵŵźĭƾƯŭźƐƯƾţōšŚƤǀƤŰţŵŚƸƴƄǀěƹƽŶƴŝƖưū
ƪŤƷŶƯōŹŵŢƿźƿŶƯƩŶƯç
ƂŴŝ Ʋƿř Źŵ ƾƟŵŚƈţ ƽżƿŹƶƯŚƳźŝ ƽŚƸƫŶƯ ƶŗřŹř Żř ƪŞƣ
ŵŵźĭƾƯƾƟźƘƯƩŶƯƽŚƷźǀƜŤƯƹŚƷźŤƯřŹŚěŚƷźĭŚſŚƴƃšŚǀƋźƟ
ƩŶƯšŚǀƋźƟæç
ƹŵƶŝƱřƺţƾƯřŹƢǀƤŰţƲƿřŹŵƵŶƃƭŚŬƳřšŚǀƋźƟƾƬĩŹƺƐŝ
ƶĩŶƴŤƀƷƾƯƺưƗšŚǀƋźƟƶƗƺưŬƯƩƹřƶŤſŵŵƺưƳƮǀƀƤţƶŤſŵ
ƶƯŶƤƯæ
ƶŝŜſŚƴƯšŚƯŶųŚƿƩƺƈŰƯƁƹźƟŻřšŹŚŞƗŶƯōŹŵŢƿźƿŶƯ
ŢưǀƣŚŝƹŜſŚƴƯƩŚƳŚĩƢƿźƏŻřŜſŚƴƯƱŚƯŻŹŵŜſŚƴƯƽźŤƄƯ
ŹŵŶƯōŹŵƱŵźĩźŨĩřŶůŢƸūƶƫƺƤƯƲƿřŵźŝŹŚƧŢſřŜſŚƴƯ ƹƵŵƺŝƾƴŤƟŹƲǀŝŻřƵŹƹŵźƷŹŵƱōšŚƯŶųŚƿLJŚĩƶĩŢſřƾƘƿŚƴƇ ƽřźŝŶƯōŹŵŢƿźƿŶƯŶƳŹřŶƳřŹŶƘŝƵŹƹŵƽřźŝƱŶƃƵźǀųŷŢǀƬŝŚƣ ŭźƐƯƾƿŚưǀěřƺƷŢƘƴƇŹŵŚƷŢưǀƣƽŻŚſŵřŻōŻřžěŹŚŝƲǀƫƹř ŹřźƣƵŵŚƠŤſřŵŹƺƯŢƘƴƇƲƿřŹŵƩŚſéåŻřƂǀŝƱƺƴĩŚţƹƵŶƃ
ƽŹLJŵƱƺǀƬǀƯŵƺŞƸŝŜūƺƯƱōżǀƯōŢǀƤƟƺƯƽŚƷŵźŝŹŚĩƹ>é@ƶŤƟźĭ
>ëê@ŢſřƵŶƃŢƘƴƇƲƿřŶƯōŹŵŹŵ ƾƫƹŶƃŭźƐƯƾƿŚưǀěřƺƷŢƘƴƇƽřźŝřŶŤŝřƮǀƷŚƠƯƲƿřƶģźĭř ƶŝŚƄƯƖƿŚƴƇŶƿŵźĮƳŵƹŶŰƯŢƘƴƇƲƿřƶŝŚƸƴţƱōƽŚƷŵźŝŹŚĩ ŶƳřƵŵƺưƳƵŵŚƠŤſřŵƺųŶƯōŹŵŵƺŞƸŝŢƸūƮǀƷŚƠƯƲƿřŻřżǀƳźĮƿŵ ŦůŚŞƯƽźǀĭŹŚĪŝŢƸūŵŚƿŻŹŚǀƀŝƪǀƀƳŚŤěŚŝƖƿŚƴƇŻřƾĪƿ ŢƷŚŞƃŹŵźƯřƲƿřƪǀƫŵŢſřƽŹřŵƪŤƷŢƘƴƇŶƯōŹŵŢƿźƿŶƯ ƱřƺƴƗƶŝŢſřƶŤƠƸƳƾƿŚưǀěřƺƷŢƘƴƇŚŝŢƘƴƇƲƿřŵŚƿŻŹŚǀƀŝ
ŚưǀěřƺƷƽŚƷƾƫŶƴƇŶƴƳŚƯƪŤƷƽŚƸƣŚţřŻřƽŹřŵźŝƵźƸŝƞƫřƶƳƺưƳ
ŢƀǀƳŶƘŝƵŹƹŵƽřźŝƵźǀųŷƪŝŚƣƹƵŵƺŝƾƴŤƟŹƲǀŝŻřƽŚƷLJŚĩƕƺƳŻř
ƵŵƺŝŢŝŚŧšŶƯƵŚţƺƧŹŵŚưǀěřƺƷŢǀƟźƓŶƴƳŚƯżǀƳƪŤƷŢǀƟźƓś
ŢſřƵřźưƷƽŵŚƿŻŢŝŚŧƶƴƿżƷŚŝƹšŶƯŻřŹŵŹŵŚƸƴţƱōƂƿřżƟřƹ
ƹŵźƷŹŵŚƋŚƤţŵŵŹřŵŵƺūƹƹŵźƷƽřźŝƂǀěŻřƹŹŻŹƱŚĪƯřũ
ŢſřƾƟŵŚƈţŵŹƺƯ ŢƿźƿŶƯƢƟƺƯƽŚƷŹřżŝřƽźǀĭŹŚĪŝƶĩŢƀǀƳŹŚƔŤƳřŻřŹƹŵƲƿřźŝŚƴŝ ƽŚƷƩŶƯƽřźŝƽŵƺūƺƯƆǀƈŴţƹ>ë@ÏƞƗŚƌƯƹŹŻŹźǀƔƳŶƯōŹŵ
ŶƃŚŝźƿŸěƱŚĪƯřżǀƳŢƘƴƇƲƿřƽřźŝ>ê@ÑƺţŹŵƺţźǀƛŚƿƹÐƺţŹŵƺţ ƖƿŚƴƇ ŶƯōŹŵ ŢƿźƿŶƯ ƵŻƺů Źŵ ƵŶƃ ƭŚŬƳř šŚƤǀƤŰţ ƶģźĭř ŢƘƴƇƽŚƸǀĭĦƿƹƶŝƶūƺţŚŝƲƨǀƫŢſřƵŵźŤƀĭŹŚǀƀŝƾƿŚưǀěřƺƷ
ŢƸūƽźŤƄǀŝšŚƤǀƤŰţƾŗŚưǀěřƺƷŢƘƴƇŚŝƱōżƿŚưţƹƽŹřŵƪŤƷ ŵƺƃƭŚŬƳřŶƿŚŝƽŹřŵƪŤƷŢƘƴƇŹŵŶƯōŹŵŢƿźƿŶƯŚƸƫŶƯƶƘſƺţ ŹƺƏƶŝƱřƺţƾƯřŹƵŻƺůƲƿřŹŵƵŶƃƭŚŬƳřšŚƤǀƤŰţŻřƾƌƘŝ>æ@ ƹ ƱřźŤǀŝ ƹ >ì@ŵŹƺƟŹŶƿƹ ŵźƧ ƱŚǀŝ źƿŻ ŭźƃ ƶŝ ƶƇLjų ƂƿŚƯŻōŢƸūƽŻŚſƶǀŞƃƽŚƷƩŶƯŻřƵŵŚƠŤſřƶŝ>í@ƲǀĤſŵƺƯ ŵŹƺƯŹŵƮǀưƈţƽřźŝƱŚƃŵƺųƎſƺţƵŶƃƶŗřŹřƽŹŚƨŤŝřƽŚƷƁƹŹ ƭŚŬƳřšŚƘƫŚƐƯƶƘſƺţƶŝ>î@źĪǀŝŶƳřƶŤųřŵźěŚƷŚƋŚƤţŵŹŚƿƁźƿŸě ŵƺųƽŹŚƨŤŝřƽŚƷƩŶƯŚŝŚƷƩŶƯƲƿřƶƀƿŚƤƯƶŝƹƶŤųřŵźěơƺƟƵŶƃ ƶƀƿŚƤƯƶŝ>æå@ƺǀĭƹŶǀŤſřŢſřƶŤųřŵźěƞƗŚƌƯƹŹŻŹŵŹƺƯŹŵ ƶĩŶƳŵřŵƱŚƄƳƹƶŤųřŵźěƪŤƷŹŵƽŹřŸĭŢưǀƣƞƬŤŴƯƽŚƷƁƹŹ ŹřŻŚŝƎƿřźƃŹŵŶƿŚŝŶƯōŹŵŢƿźƿŶƯƮǀƷŚƠƯŽŚſřźŝƽŹřŸĭŢưǀƣ ƾſŹźŝƶŝŵƺųƶƘƫŚƐƯŹŵżǀƳ>ææ@ĬƳřƺƷŵźƿŸěšŹƺƇƾŤŝŚƣŹ
ƱōƾƬƇřŌżūƂƃƹƶŤųřŵźěƪŤƷŹŵŶƯōŹŵŢƿźƿŶƯƾƬƇřƽřżūř

Overbooking
Nested
Non-Nested
æèƾƠŬƳƽŶƸƯƹŽŹŶƯŶưŰƯƪŤƷŶƯōŹŵŹřŶƿŚěƽŻŚſƶƴǀƸŝƾƟŵŚƈţƽżƿŹƶƯŚƳźŝ

ŹŵƽŵƹŹƹƱřźƟŚƀƯƶŝŢŞƀƳƽŵƺƘƇźǀƛƖŝŚţƶƏƺŝźƯƖŝŚţƶĩŢſř
ŢſřŻƹŹƱō ƶŝƞƗŚƌƯƹŹŻŹƮŤƀǀſŹŵŢǀƟźƓźŝŵřŻŚƯƽŵƹŹƹƱřźƟŚƀƯ 
ŢųřŵźěŚƷƱōƶŝƽřƶưƿźūĢǀƷƹŶƳƺƃƾƯƵŵŚŤſźƟźĮƿŵƽŚƷƪŤƷ
ŵƺƃƾưƳ

ƩŶƯƽŚƷźĭŚſŚƴƃçç
ŻřŶƴţŹŚŞƗƩŶƯƽŚƷźĭŚſŚƴƃƵŶƃŭźƐƯšŚǀƋźƟƶŝƶūƺţŚŝ
ƾſŹźŝŵŹƺƯƵŹƹŵŹŵƽŵƹŹƹƽŚƷŻƹŹƶƗƺưŬƯI
ƾſŹźŝŵŹƺƯƵŹƹŵŹŵƾūƹźųƽŚƷŻƹŹƶƗƺưŬƯJ
ƾſŹźŝŵŹƺƯƵŹƹŵŹŵŚƷŻƹŹƶƗƺưŬƯK
ƾſŹźŝŵŹƺƯƵŹƹŵƩƺƏT
ŚƋŚƤţƞƬŤŴƯƽŚƷƺƿŹŚƴſƶƗƺưŬƯS ƪŤƷŢǀƟźƓƞƬŤŴƯƽŚƷƺƿŹŚƴſƶƗƺưŬƯM ƾſŹźŝŵŹƺƯƵŹƹŵŹŵƱřźƟŚƀƯŵƹŹƹŻƹŹƶŝƍƺŝźƯźĭŚſŚƴƃi ƾſŹźŝŵŹƺƯƵŹƹŵŹŵƱřźƟŚƀƯũƹźųŻƹŹƶŝƍƺŝźƯźĭŚſŚƴƃj ƾſŹźŝŵŹƺƯŻƹŹƶŝƍƺŝźƯźĭŚſŚƴƃk
ŚƋŚƤţƽƺƿŹŚƴſƶŝƍƺŝźƯźĭŚſŚƴƃs
ŢǀƟźƓƽƺƿŹŚƴſƶŝƍƺŝźƯźĭŚſŚƴƃm
ŚƷźŤƯřŹŚěèç
ŻřŶƴţŹŚŞƗƩŶƯƽŚƷźŤƯřŹŚě
ŹŵƭřjŻƹŹŹŵũƹźųƹƭřiŻƹŹŹŵźƟŚƀƯŵƹŹƹŻřƾƃŚƳŶƯōŹŵRij
ƱŚƀĪƿŢưǀƣŢƫŚů
ƭřjŻƹŹŹŵũƹźųƹƭřiŻƹŹŹŵźƟŚƀƯŵƹŹƹŻřƾƃŚƳŶƯōŹŵRijs
sƺƿŹŚƴſŢŰţ
ƪŤƷƾƬƇřŢǀƟźƓC
ƭřmƽƺƿŹŚƴſŢŰţƪŤƷŢǀƟźƓC m
ŢƫŚůƭřjŻƹŹŹŵũƹźųƹƭřiŻƹŹŹŵŵƹŹƹƽřźŝŚƋŚƤţƱřżǀƯUij
ƶƿŚě
ŢŰţƭřjŻƹŹŹŵũƹźųƹƭřiŻƹŹŹŵŵƹŹƹƽřźŝŚƋŚƤţƱřżǀƯUijs ƭřsƺƿŹŚƴſ
ƮǀưƈţƽŚƷźǀƜŤƯéç
šƹŚƠŤƯŚƿƹƱŚƀĪƿŢưǀƣƽřŹřŵŚƷƺƿŹŚƴſƶĩƲƿřƶŝƶūƺţŚŝ
ŻřŶƴţŹŚŞƗƮǀưƈţƽŚƷźǀƜŤƯŶƴƃŚŝ
ũƹźųƹƭřiŻƹŹŹŵŵƹŹƹŢƸūƵŶƃƶŤƟźƿŸěƱřźƟŚƀƯŵřŶƘţxij
ƱŚƀĪƿŢưǀƣŢƫŚůŹŵƭřjŻƹŹŹŵ
ũƹźųƹƭřiŻƹŹŹŵŵƹŹƹŢƸūƵŶƃƶŤƟźƿŸěƱřźƟŚƀƯŵřŶƘţ xijs
šƹŚƠŤƯŢưǀƣŢƫŚůŹŵƺƿŹŚƴſŢŰţƭřjŻƹŹŹŵ
ŵŹřƹƭřkŻƹŹŹŵƶĩƾƳřźƟŚƀƯƪĩŵřŶƘţŢƠĭƱřƺţƾƯƹźƴƿřŻř ŚŝŢſřźŝřźŝŶƳƺƃƾƯ ƍƺŝźƯƶĩŶƴŤƀƷƾţŚǀƋźƟƭƹŵƶŤſŵŶƳƺƃƾƯƪƯŚƃřŹŚƸƫŶƯƶǀƬĩ
ŶƴŤƀƷƢǀƤŰţƲƿřŹŵƵŶƃƶŗřŹřƽŚƸƫŶƯŻřƾųźŝƶŝ
ŻřŶƴţŹŚŞƗƩƹřƶŤſŵšŚǀƋźƟ
ƲǀƴĤưƷƹƶŤƃřŶƳŹƺƌůƪŤƷŹŵźƟŚƀƯĢǀƷźƠƇƱŚƯŻŻřƪŞƣ ƶŝŶƳŵźĭũŹŚųƪŤƷŻřƵŹƹŵƽŚƸŤƳřŚţŶƿŚŝƽŵƹŹƹƱřźƟŚƀƯƶǀƬĩ
ƶŤƃřŵŵƺūƹƪŤƷŹŵźƟŚƀƯĢǀƷŶƿŚŞƳƵŹƹŵƽŚƸŤƳřŹŵźĮƿŵšŹŚŞƗ
ŶƃŚŝ
ĢǀƷƹŢſřźƿŸěƱŚĪƯřƱřźƟŚƀƯŵƹŹƹŚƸƴţźƠƇƱŚƯŻŻƹŹŹŵ 
ŵŵźĭƾưƳũŹŚųƪŤƷŻřźƟŚƀƯ ƶƯŚƳźŝƵŹƹŵƽŚƷŻƹŹŹŵũƹźųƹŵƹŹƹƽřźŝŚƋŚƤţƖƿŻƺţƖŝŚţ ŹŵũƹźųƹƮŬƴěŻƹŹŹŵŵƹŹƹƽřźŝŚƋŚƤţŵřŶƘţƩŚŨƯƱřƺƴƗƶŝƽżƿŹ
ŢſřƾƴǀŝƂǀěƪŝŚƣƶŤƃŸĭƽŚƷƵŵřŵƶŝƶūƺţŚŝƆŴƄƯƮƸƳŻƹŹ
ŚƿƱŚƀĪƿƽŚƷŢưǀƣƽřŹřŵŶƴƳřƺţƾƯŚƋŚƤţƞƬŤŴƯƽŚƷƺƿŹŚƴſ 
ŶƴƃŚŝšƹŚƠŤƯ
ŻřŶƴţŹŚŞƗëŚţéƽŚƸƄŴŝƶŝƍƺŝźƯƭƹŵƶŤſŵšŚǀƋźƟ
ŻřŻƹŹźƷŹŵƶĩŵŹřŵŵƺūƹŚƋŚƤţƽřźŝƾƠƬŤŴƯƽŚƷƺƿŹŚƴſ  ƭŻLJƶŤŞƫřŵŹřŵƕƺƣƹƱŚĪƯřƾƇŚųƩŚưŤůřŚŝŚƷƱōŻřĨƿźƷƵŹƹŵ ƶŝƵŹƹŵŻřƆŴƄƯŻƹŹĨƿŹŵƺƿŹŚƴſĨƿƕƺƣƹƶĩŢſřźĩŷƶŝ
ƲĪưƯźĮƿŵšŹŚŞƗƶŝŢƀǀƳŶƘŝƽŚƷŻƹŹŹŵƺƿŹŚƴſƱōŹřźĪţƾƴƘƯ
sƽƺƿŹŚƴſtæŻƹŹŹŵƾƫƹŶƷŵűŹƭřsƺƿŹŚƴſtŻƹŹŹŵƶĩŢſř
ŵŶƳƺǀĜŝƕƺƣƹƶŝ
ƶƯŚƳźŝƱŚƯŻŹŵƶĩƵŶƄƳƾƴǀŝƂâěƵĦƿƹƽŚƷƶƯŚƳźŝƶŝƶūƺţŚŝ  ƅŚƈŤųřƪǀƫŵƶŝŚƿƹƽŹřźƐƋřźǀưƘţźǀƔƳŢƀǀƳƆŴƄƯƽżƿŹ
ƽŚƷơŚţřŵřŶƘţƾŤƫƹŵƽŚƷƱŚĭŹřƶŝƪŤƷƽŚƷơŚţřŻřƾųźŝƽŹŚŞūř ƽŚƷƺƿŹŚƴſƲƨǀƫŢƀǀƳƆŴƄƯLjƯŚĩƱřźƟŚƀƯƽřźŝƵŵŚƠŤſřƪŝŚƣ ĨƿƕƺƣƹŵƺưƳŹƺƔƴƯƱřƺţƾƯřŹƆŴƄƯšLJŚưŤůřŚŝƾƠƬŤŴƯ
ƽŚƷŻƹŹŹŵƱōŹřźĪţƾƴƘƯƶŝƵŹƹŵŻřƆŴƄƯŻƹŹĨƿŹŵƺƿŹŚƴſ
ŢƀǀƳźĮƿŵ
ƽŚƷŢưƀƣŹŵƶĩŶƴŤƀƷźƿŻšŚǀƋźƟƭƹŵƶŤſŵšŚǀƋźƟźĮƿŵŻř
ŶƳƺƃƾƯƶŤƟźĭŹŚƧƶŝƶƫŚƤƯƮƄƃƹƮŬƴě šŶƯƶŝźŨĩřŶůŢƯŚƣřšŶƯƱŵźĩƾƳLJƺƏŚƿźţƵŚţƺĩƱŚĪƯř ƹƭŚĮƴƷŵƹŻũƹźųŜǀţźţƶŝƶĩŵŹřŵŵƺūƹƲƿźƟřŚƀƯƽřźŝŻƹŹƹŵ ƽŵƹŹƹƽŚƷźƟŚƀƯƩŚŨƯƱřƺƴƗƶŝŶƳƺƃƾƯƵŶǀƯŚƳƭŚĮƴƷźƿŵũƹźų řŹƮƸƳŻƹŹŹŵũƹźųƾƴƘƿƮƸƳŻƹŹŚţŢƯŚƣřŶƈƣƶĩƮŬƴěŻƹŹŹŵ
ũƹźųŶƴƃŚŝƶŤƃřŵŹƺƌůƮŤƄƷŚƿƮŤƠƷŻƹŹŚţŢſřƲĪưƯŶƳŹřŵ
ŚƿƮƷŵŻƹŹŚţƹƵŵźĩźţƾƳLJƺƏřŹŵƺųŢƯŚƣřšŶƯŚƿƹƭŚĮƴƷŵƹŻ
ƭŚĮƴƷźƿŵũƹźųŶƳƺƃũŹŚųƮƷŵŻŚƿ
ƶŝƹåæŚŝźŝřźŝŻƹŹĨƿšŶƯƶŝŢƯŚƣřƱŚƯŻŹŵźǀǀƜţƩŚưŤůř 
ŵƺƃƾƯƉźƟååêŚŝźŝřźŝŻƹŹƹŵšŶƯ ƹŹŻŹƱŚĪƯřŢƀǀƳƆŴƄƯƪƯŚĩŹƺƐŝŚƷŚƋŚƤţƶĪǀƿŚŬƳōŻř ŻƹŹƽřźŝƱōŹřŶƤƯƶĩŵŹřŵŵƺūƹƪŤƷŢƿźƿŶƯƽřźŝżǀƳƞƗŚƌƯ
ƾƸƿŶŝŢſřŻƹŹƱōŹŵƽŵƹŹƹƱřźƟŚƀƯƽŚƋŚƤţƶŝƶŤƀŝřƹƆŴƄƯ
xij Uij i  0,…,T 1 j 1,…,T í
xij  0 i  0,…,T 1 j 1,…,T î
ƾƠŬƳƽŶƸƯƹŽŹŶƯŶưŰƯƪŤƷŶƯōŹŵŹřŶƿŚěƽŻŚſƶƴǀƸŝƾƟŵŚƈţƽżƿŹƶƯŚƳźŝæé
ƱŵźĩƶƴǀƄǀŝƩŚŞƳŵƶŝêƝŶƷƖŝŚţŵƺƃƾƯƵŶƷŚƄƯƶĩŹƺƐƳŚưƷ ë ŢƿŵƹŶŰƯ Ţſř ƱřźƟŚƀƯ ƶŝ ƾƷŵžƿƹźſ Żř ƶƬƇŚů ŵƺſ ŢǀƟźƓźŨĩřŶůŻřƪŤƷŹŵźƋŚůƱřźƟŚƀƯŵřŶƘţƶĩŶƴĩƾƯƲǀưƌţ ŻřƂǀŝƱřźƟŚƀƯŵƹŹƹƭŶƗźĮƳŚǀŝìŢƿŵƹŶŰƯŶƿŚưƴƳŻƹŚŬţƪŤƷ ƲǀưƌţíŢƿŵƹŶŰƯŚŤƿŚƸƳƹƵŵƺŝźƠƇŻƹŹŹŵƪŤƷŢǀƟźƓ ƹƭřiŻƹŹŹŵŵƹŹƹŢƸūƵŶƃƶŤƟźƿŸěƱřźƟŚƀƯŵřŶƘţƶĩŶƴĩƾƯ
ŶƃŚŞƳ źŤƄǀŝ ƵŶƃ ƾƴǀŝƂǀě ƽŚƋŚƤţ Żř ƭř j ŻƹŹ Źŵ ũƹźų
ŢſřƶƏƺŝźƯƩŶƯźǀƜŤƯźĮƳŚǀŝżǀƳîŢƿŵƹŶŰƯ
ŹřŶƿŚěƽŻŚſƶƴǀƸŝƹƾƟŵŚƈţƽżƿŹƶƯŚƳźŝƩŶƯé
ƽżƿŹƶƯŚƳźŝêîƾƋŚƿŹƩŶƯŵƺƃƾƯƵŶƷŚƄƯƶĩŹƺƐƳŚưƷ Źŵ CźŤƯřŹŚěƶĩŢƃřŵƶūƺţŶƿŚŝŢſřƾƐųƹŮǀŰƇŵŶƗ íŢƿŵƹŶŰƯŹŵ Uij ƲǀƴĤưƷƹìƹëƽŚƷŢƿŵƹŶŰƯ
ŶƳŹřŵƞƬŤŴƯƽŚƷƺƿŹŚƴſƶŝƾĮŤƀŝƱōźƿŵŚƤƯƹŶƴŤƀƷƾƘƐƣźǀƛ ŢƟźĭŹŚĪŝƱřƺţƾƯřŹƾƠƬŤŴƯƽŚƷŵźĪƿƹŹƪĪƄƯƲƿřƪůŢƸū
ƾƴƘƿŚƷźŤƯřŹŚěƲƿřƾƋŚƿŹŶǀƯřŻřƵŵŚƠŤſřŚƷŵźĪƿƹŹƲƿřŻřƾĪƿ źƔƳƶŝ ƵŵŚſ ŹŚǀƀŝ ƶģźĭř ŵźĪƿƹŹ ƲƿřŢſř EC ƹ EUij  ƶūƺƯƽŚƷśřƺūŶǀƫƺţƲǀưƌţƭŶƗƱōƵŶưƗƞƘƋŚƯřŶſŹƾƯ żǀƫŚƳōŻřƽŵŹřƺƯŹŵƞƘƋƲƿřƱŵƺưƳƖƠţźƯŢƸūƹźƴƿřŻřŢſř ƲǀƴģŚƯřŵƺƃƾƯƵŵŚƠŤſřšŚůLjƇřƾųźŝƭŚŬƳřƽřźŝŢǀſŚƀů ŢƸūŚƷŵźĪƿƹŹźĮƿŵŻř>ç@ŢſřƾƫŚƘƠƳřŵźĪƿƹŹĨƿƽŵźĪƿƹŹ żǀƳƵŶƳźǀĭƂǀěƽŚƷŵźĪƿƹŹŌżūƶĩƱŚƴǀưƏřƭŶƗƲƿřŚŝŵŹƺųźŝ
ƩŶƯŹŵƱōƽźǀĭŹŚĪŝƹƱōħŹŵƱŚƴǀưƏřƭŶƗƲƿřƲŤƟźƿŸěŢſř
ŢſřƵŶƃƵŵŚƸƳŚƴŝŽŚſřƲƿřźŝżǀƳƾƟŵŚƈţƽżƿŹƶƯŚƳźŝƶĩŢſř
ƩŶƯƽřźŝƱřƺţƾƯřŹƽŹřŶƿŚěƕƺƳƹŵơƺƟƵŶƃŭźƐƯŜƫŚƐƯƶŝ ŵƺưƳƞƿźƘţƢǀƤŰţƲƿřƾƟŵŚƈţƽżƿŹƶƯŚƳźŝ
ŲſŚěƽŹřŶƿŚěĨƿƕƺƳƽŹřŶƿŚě
ƎƿřźƃƶŝƶūƺţŚŝƶĩŢſřŲſŚěƽŹřŶƿŚěƽřŹřŵƽřƶƴǀƸŝśřƺū
ƶƴǀƸŝƶŝĨƿŵżƳmM ƹsS ƽŚƷƺƿŹŚƴſƶǀƬĩƽřźŝƾĮƴǀƸŝ
ŶƃŚŝ
ƩŶƯƽŹřŶƿŚěƹŵƕƺƳƽŹřŶƿŚě
ƎƿřźƃƶŝƶūƺţŚŝƶĩŢſřƩŶƯƽŹřŶƿŚěƽřŹřŵƽřƶƴǀƸŝśřƺū
ŚŞƿźƤţ mM ƹ sS ƽŚƷƺƿŹŚƴſƶǀƬĩƽřźŝƱŵƺŝƶūƺƯ
ŶƃŚŝƶūƺƯ
ƹŚƋŚƤţƽřźŝƱŚƴǀưƏřƭŶƗƱŚǀŝŢƸūŶƃƱŚǀŝżǀƳLjŞƣƶĩŹƺƐƳŚưƷ ƹŚƋŚƤţƽřźŝƾƠƬŤŴƯƽŚƷƺƿŹŚƴſƶĩŢſřƵŶƃƉźƟŢǀƟźƓ
T
 xkj æjk1
ŚŝŢſřźŝřźŝŶƳƺƃƾƯũŹŚųƭřkŻƹŹŹŵƶĩƾƳřźƟŚƀƯƪĩŵřŶƘţƹ
k1
 xik çi0
ƶƿŚěŢƫŚůŹŵƾƟŵŚƈţƽżƿŹƶƯŚƳźŝƩŶƯè
řŹƱřźƟŚƀƯũƹźųƹŵƹŹƹƵŶƃƞƿźƘţƽŚƷźǀƜŤƯƶŝƶūƺţŚŝ źƔƳŹŵƶĪŞƃŹŵƵźĭĨƿƾūƹźųƹƽŵƹŹƹƱŚƿźūšŹƺƈŝƱřƺţƾƯ
ŵřŶƘţ k,k 1,….,k 1,ƆŴƄƯŻƹŹƽřźŝæŹřŵƺưƳŢƟźĭ ŶƴŤƀƷƶŞſŚŰƯƪŝŚƣźƿŻšŹƺƈŝƪŤƷŹŵźƋŚůƱřźƟŚƀƯ

ƭřkŻƹŹŹŵũƹźųƹŵƹŹƹƱŚƿźūæŹřŵƺưƳ
ƭřKŻƹŹŹŵƪŤƷŹŵźƋŚůƱřźƟŚƀƯŵřŶƘţ è
k1 TTk1kT
 xij   xkj  xik  xij
i0 jkjk1i0i0 jk1
šŹƺƇƾūƹźųĢǀƷźƠƇŻƹŹŹŵšŚǀƋźƟŽŚſřźŝźĮƿŵƝźƏŻř ŵŹřƹƲƿźƟŚƀƯřŸƫŢſřźƿŸěƱŚĪƯřƱřźƟŚƀƯŵƹŹƹŚƸƴţƹŵźǀĭƾưƳ
ŶƳƺƃƾƯũŹŚųƪŤƷŻřTŚţæƽŚƷŻƹŹŻřƾĪƿŹŵŻƹŹƲƿřŹŵƵŶƃ ŚŝŢſřźŝřźŝŻƹŹƲƿřŹŵƽŵƹŹƹƱřźƟŚƀƯŵřŶƘţƲƿřźŝŚƴŝ
T
x0j éj1
ŭźƃƶŝƶƿŚěŢƫŚůŹŵƾƟŵŚƈţƽżƿŹƶƯŚƳźŝƩŶƯŜǀţźţƲƿřƶŝ >æ@ŹŵƵŶƃƶŗřŹřƩŶƯƶǀŞƃŢſřźƿŻ
T1T
Max Rij xij ê
i0 ji1
S.T: ë
kT xij  C
i0 jk1
T
x0 j  C ìj1
æêƾƠŬƳƽŶƸƯƹŽŹŶƯŶưŰƯƪŤƷŶƯōŹŵŹřŶƿŚěƽŻŚſƶƴǀƸŝƾƟŵŚƈţƽżƿŹƶƯŚƳźŝ

Ţſř
ŢưƀƣƹŵƽřŹřŵæçƩƹřƝŶƷƖŝŚţŵƺƃƾƯƵŶƷŚƄƯƶĩŹƺƐƳŚưƷ ƹƱřźƟŚƀƯƶŝƾƷŵžƿƹźſŻřƪƇŚůŵƺſźĮƳŚǀŝƩƹřŢưƀƣŢſř ƽřźŝŶƯōŹŵƎſƺŤƯŻřƝřźŰƳřƢƬƐƯŹŶƣƎſƺŤƯźĮƳŚǀŝƭƹŵŢưƀƣ ŶƯōŹŵƝLjŤųřƶĤƳŚƴģƶĩŢſřƾƸƿŶŝŢſřƞƬŤŴƯƽŚƷƺƿŹŚƴſ ŹřŶƤƯ ƶưƿźū ƱřżǀƯ ŶƃŚŝ ŵŚƿŻ ƞƬŤŴƯ ƽŚƷƺƿŹŚƴſ Żř ƶƬƇŚů
ƩŚŞƳŵƶŝƩŶƯƹźƴƿřŻřŶŝŚƿƾƯƂƿřżƟřŶƯōŹŵƎſƺŤƯŻřƶŤƟŚƿƂƷŚĩ
ŜƿźƋ ƱřƺƴƗƶŝ żǀƳŜƿźƋ Ţſř ƝLjŤųř Ʋƿř Ʊŵźĩ ƶƴǀưĩ ŶƃŚŝźŤƄǀŝŢƿźƿŶƯƽźƿŸěƦƀƿŹƱřżǀƯƶģźƷŢſřƽżƿźĭĨƀƿŹ
ƝŶƷƖŝŚţƲƿřŜǀţźţƲƿřƶŝŵƺƃƾƯśŚŴŤƳřźŤƨģƺƧŹřŶƤƯ ƭƹŵƝŶƷƖŝŚţŢſřŲſŚěƽŹřŶƿŚěƱřżǀƯƱŵźĩźŨĩřŶůƩŚŞƳŵƶŝ ŻřƝřźŰƳřƱřżǀƯźĮƳŚǀŝƩƹřŢưƀƣŢſřŢưƀƣƹŵƽřŹřŵżǀƳæèźĮƳŚǀŝżǀƳƭƹŵŢưƀƣƹƵŵƺŝsS ƞƬŤŴƯƽŚƷƺƿŹŚƴſƽŚƋŚƤţ
ŜƿźƋŻřŢſř mM ŚƷƺƿŹŚƴſƲƿřŢǀƟźƓŻřƝřźŰƳřƱřżǀƯ
ŵƺƃƾƯƵŵŚƠŤſřƝřźŰƳřƹŵƲƿřƱŵźĩƱŚƀĪƿŢƸūżǀƳ ƆŴƄƯƕƺƣƹƩŚưŤůřƽřŹřŵŚƷƱōŻřĨƿźƷƶĩŵŹřŵŵƺūƹŢǀƟźƓ
ƶĪƿŹƺƐŝŶƴŤƀƷ
S Ps 1æås1
M m 1ææ
m1
ƪŤƷŶƯōŹŵŢƿźƿŶƯƾƟŵŚƈţƽżƿŹƶƯŚƳźŝƩŶƯŢƠĭƱřƺţƾƯƹźƴƿřŻř ƩŶƯƽŹřŶƿŚěƹŲſŚěƽŹřŶƿŚěŻřŶƴţŹŚŞƗƶĩŵŹřŵƾěŹŵřŹƝŶƷƹŵ ƞƬŤŴƯƽŚƷƺƿŹŚƴſƽřźŝŚƷŢưǀƣƽƹŚƀţƉźƟƶŝƶūƺţŚŝƩŚů
šLJŵŚƘƯšŹƺƈŝƪŤƷŢƿźƿŶƯƾƟŵŚƈţƽżƿŹƶƯŚƳźŝƩŶƯŚƋŚƤţ
ƆŴƄƯƽŚƋŚƤţŻřƝřźŰƳřƶưƿźūƹƽżƿźĭĨƀƿŹƱřżǀƯźĮƳŚǀŝ
2094627-119249P

P

4253235-119249

2798421493899

3849543493899

Max Z1  S Ps  T1T Rij xS T1TS T1Tsæçs sRijs xij PsRij xij
ij
s1i0 ji1s1i0 ji1 s1i0 ji1
1604103-18493

Min Z2  S Ps T1 T wijUijMm M mC m  æè
s1i0 ji1s  xij  m1mC m1
ŢſřæìŚţæçŜǀţźţƶŝƶĩƵŵƺŝƾƠƴƯźǀƛƾƳŻƹƽŚƷźŤƯřŹŚěwij ƹƽŚƷźŤƯřŹŚě
ƾƳŚƯŹōƽżƿŹƶƯŚƳźŝæƶǀƌƣ
MinZ 

f x g

æí
s.t.
X  F æî
šŹƺƇƶŝƱřƺţƾƯřŹŢſřƶūƺƯƽŚƷśřƺūƶƗƺưŬƯFƱōŹŵƶĩ ŵźƧƾƐųźƿŻ
Min Z  f x g  2 çå
S.T.
g  f x 0 çæ
 0çç
X F çè
šŚŞŧř
ŹŵŢŞŨƯŜƿźƋĨƿŵƺƃƾƯƵŶƷŚƄƯƭƹŵƩŶƯŹŵƶĩŹƺƐƳŚưƷ ŢſřƽŻŚſƮưǀƴǀƯšŹƺƈŝƝŶƷƖŝŚţƾƟźƏŻřŢſřƝŶƷƖŝŚţ
kT
xij  C max k 1,…,T æé
i0 jk1
T
x0 j  C max æê
j1
xij Uijmax i  0,…,T 1j 1,…,T æë
xij  0 i  0,…,T 1j 1,…,T æì
ƩŶƯƽŹřŶƿŚěƱŵźĩźŨĩřŶůƾěŹŵżǀƳƭƹŵƝŶƷƖŝŚţƲƿřźŝŚƴŝ ƾƬŞƣƩŶƯƶŝŚƄƯƩŶƯƲƿřŹŵƵŶƃƶŗřŹřƽŚƷŢƿŵƹŶŰƯźƿŚſŢſř
źŨĩřŶůƞƬŤŴƯƽŚƷƺƿŹŚƴſŵƺūƹƪǀƫŵƶŝƶĩšƹŚƠţƲƿřŚŝŢſř
ƹUijmax źƿŵŚƤƯŜǀţźţƶŝU ij ƹCƽŚƷźŤƯřŹŚěƽřźŝƲĪưƯŹřŶƤƯ
ŵŵźĭƾƯŹƺƔƴƯCmax
ŢƸūŢſřƾƐųźǀƛƾƫŶƯơƺƟƵŶƃƶŗřŹřƩŶƯƶĩŢſřŮƋřƹ
>è@ƾƫƹƺƿƎſƺţƶĩźƿŻƶǀƌƣŻřƱřƺţƾƯƩŶƯƲƿřƱŵźĩƾƐų ŵƺưƳƵŵŚƠŤſřŢſřƵŶƃƶŗřŹř
ƾƠŬƳƽŶƸƯƹŽŹŶƯŶưŰƯƪŤƷŶƯōŹŵŹřŶƿŚěƽŻŚſƶƴǀƸŝƾƟŵŚƈţƽżƿŹƶƯŚƳźŝæë
ŢƿŵƹŶŰƯƶŝƶūƺţŚŝŶƃŚŝ f x g 0ƶĤƳŚƴģźĮƿŵƝźƏƾƴƘƿŢŞŨƯŹřŶƤƯƲƿźŤưĩƶĩŢſřƲƿřśƺƬƐƯŢƫŚůƲƿřźŝŚƴŝ
g  f xŚŝŢſřźŝřźŝƽřźŝƲĪưƯŹřŶƤƯƲƿźŤưĩçæçæŢƿŵƹŶŰƯƶŝƶūƺţŚŝźĮƿŵƝźƏŻřŵźǀĮŝřŹźƠƇŹřŶƤƯ
Z  f x g 2g  f x Z ƹźƴƿřŻřŶƃŚŝ f x g 0 ƶĤƳŚƴģ ƩŚů Ţſř  g  f x
ƶŝæíçèƾƟŵŚƈţƾƐųźǀƛƽżƿŹƶƯŚƳźŝæƶǀƌƣŻřƵŵŚƠŤſřŚŝŢſř Źřźƣźŝ  źƿŵŚƤƯ ƶǀƬĩ ƽřŻř ƶŝ çæ ŢƿŵƹŶŰƯ
ŵŵźĭƾƯƪƿŶŞţƾƐųƩŶƯƹźƴƿřŻřŵźǀĭƾƯřŹźƠƇŹřŶƤƯƾƴƘƿƲĪưƯŹřŶƤƯƲƿźŤưĩƲƿřźŝŚƴŝ Żř ŵƺŝ ŶƷřƺų Z  f x g 20 Z ŢƫŚů Ʋƿř Źŵ
S T1TS Ps Ti01 jTi1Rij xij sS1 Ps Ti01 jTi1Rijs xij   2s  ss
Max Z1 s1 Ps i0 ji1Rij xijs1
 S Ps T1 T wij Uijs  xij  2ijs M m Cm mM1mCm   2m 
Min Z
2
s1i0 ji1 m1
St : çéçê
kT
xij  C max k 1,…,T 1
i0 jk1

T
x0 j  C max
j1

xij Uijmax i  0,…,T 1j 1,…,T

ST1TT1T
Ps Rijs xij s  Rijs xij s 1,…,S
s1i0 ji1i0 ji1
çë
çì
çí
çî
èå
èæ
èç
xij ijs Uijs i  0,…,T 1j 1,…,T s 1,…,S

M mCm m  C m m 1,…,M
m1

xij  0 i  0,…,T 1j 1,…,T ƪŤƷŹŵƶǀƫƹřƵŶƃƽżƿŹƶƯŚƳźŝƱŚƯŻŻřźŤƄǀŝŚƿźŤưĩŢſřƲĪưƯƹƭŚĮƴƷŵƹŻũƹźųŚŝƾƟŵŚƈţƽżƿŹƶƯŚƳźŝƩŶƯê
ƶŝźŬƴƯŢƿŵƹŶŰƯƲƿřƽŻŚſŵřŻōŚŤƘǀŞƏŶƴƃŚŝƶŤƃřŵŹƺƌůƞƗŚƌƯƹŹŻŹƹƭŚĮƴƷźƿŵ
ŻřƱōƱŶƃźţƵŶǀĤǀěƹŢƸūĨƿŻřƽżƿŹƶƯŚƳźŝƩŶƯƱŶƃźţƾƘƣřƹ
ƱřźƟŚƀƯƶĩŢſřƵŶƃƉźƟƾƬŞƣƵŶƃƶŗřŹřƽŚƷƩŶƯŹŵ ƱřżǀƯƶŝźŨĩřŶůźƟŚƀƯźƷƶĩƮǀƴƧƾƯƉźƟŵŵźĭƾƯźĮƿŵŢƸū
ƱŚĪƯřƲǀƴĤưƷƹŶƳŹřŵŹƺƌůƪŤƷŹŵƶǀƫƹřƶƯŚƳźŝƢŝŚƐƯŚƤǀƣŵ ŶƷŵƾƯƂƷŚƧŚƿƂƿżƟřźǀǀƜţřŹŵƺųŢƯŚƣřƶƯŚƳźŝŻƹŹƹŵ
ƹźƴƿřŻřŢƀǀƳƶƳƺĮƴƿřŢǀƘƣřƹŹŵƶĪǀƫŚůŹŵŵŹřŶƳŵƺūƹƞƗŚƌƯƹŹŻŹ
ƹåæŚŝźŝřźŝŜǀţźţƶŝŻƹŹƹŵƹƦƿšŶƯƶŝƶƯŚƳźŝźǀǀƜţƩŚưŤůř
ƱřźƟŚƀƯƶĩŵƺƃƾƯƶŤųřŵźěƾƿŚƷŢƫŚůƾſŹźŝƶŝŢưƀƣƲƿřŹŵ źƿŻƽŚƷźǀƜŤƯƞƿźƘţƶŝŢƫŚůƲƿřŹŵŵƺƃƾƯƉźƟååê
řŹřŵřŹƂƿƺųŢƯŚƣřšŶƯƱŵƺưƳźţƾƳLJƺƏŚƿƹƱŵźĩźţƵŚţƺĩƱŚĪƯř
ƮƿŻřŵźěƾƯ
ŵƺūƹƪŤƷŢƿźƿŶƯƽřźŝƞƗŚƌƯƹŹŻŹƱŚĪƯřƲǀƴĤưƷƹŶƴŤƀƷ
ŻřƭřjŻƹŹŹŵƹŵŹřƹƭřiŻƹŹŹŵƶĩƾƳřźƟŚƀƯƾƘƣřƹŵřŶƘţYijŵŹřŵ
ŶƳŵźĭƾƯũŹŚųƪŤƷ
ƹiŻƹŹŹŵŵƹŹƹŶƈƣƶǀƫƹřƶƯŚƳźŝƢŞƏƶĩƾƳřźƟŚƀƯŵřŶƘţEiejƭŚĮƴƷźƿŵƭŚĮƴƷŵƹŻũƹźųæê
ŵŹřƺƯŻřƽŹŚǀƀŝŹŵŶƃŭźƐƯżǀƳšŚǀƋźƟŹŵƶĩŹƺƐƳŚưƷ ŶƳřƵŶƃũŹŚųźţŵƹŻƾƫƹŶƳřƶŤƃřŵřŹjŻƹŹŹŵũƹźų
ƵŶƃƶƯŚƳźŝźǀǀƜţŹŚģŵƪŤƷŹŵŢƯŚƣřŢƸūƵŶƃƶŤƟźƿŸěƱřźƟŚƀƯ
æìƾƠŬƳƽŶƸƯƹŽŹŶƯŶưŰƯƪŤƷŶƯōŹŵŹřŶƿŚěƽŻŚſƶƴǀƸŝƾƟŵŚƈţƽżƿŹƶƯŚƳźŝ

ƲƿřŻřƵŵŚƠŤſřŚŝƱřƺŤŝŜǀţźţƲƿřƶŝŚţŢſřƱōŢǀƟźƓŻřƂǀŝ
ŵřŵƂƷŚƧřŹƵŶƃƹŹŻŹƽŚƸƣŚţřƱŵƺŝƾƫŚųƽĦţřźŤſř ĨƿƽřźŝƞƗŚƌƯƹŹŻŹƱřżǀƯŶƃŭźƐƯšŚǀƋźƟŹŵƶĩŹƺƐƳŚưƷ ƱōŹŵƽŵƹŹƹƱřźƟŚƀƯŵřŶƘţŻřƽŵƺƘƇźǀƛƾƘŝŚţkƆŴƄƯŻƹŹ
ƽřźŝƞƗŚƌƯƹŹŻŹŢǀƟźƓ Ok ƶĤƳŚƴģźĮƿŵšŹŚŞƗƶŝŢſřŻƹŹ
ƮƿŹřŵŶƃŚŝƭřkŻƹŹƽřźŝ

Ok  hIk èë

2158608-25570

ƾƘŝŚţhƹkŻƹŹŹŵƱřźƟŚƀƯŵƹŹƹƎſƺŤƯƱřżǀƯ Ik ƱōŹŵƶĩ
h ƽřźŝ ƱřƺţƾƯ ƶĩ ƾƘŝřƺţ Żř ƾĪƿ Ţſř Ik Żř ƽŵƺƘƇźǀƛ
ŢſřèìƶƫŵŚƘƯšŹƺƈŝŢƟźĭźƔƳŹŵ
22545880774

93686980774

hIk C Ik èì

ƪŝŚƣƾƫŚųŢǀƟźƓŹŵƶĩŢſřƾƳŻƹƾŞƿźƋ ƱōŹŵƶĩ
ŵƺƃƾƯśźƋƾƴǀŝƂǀě
ŻƹŹĨƿŹŵƪŤƷƽřźŝƹŹŻŹƪŝŚƣŢǀƟźƓơƺƟŜƫŚƐƯƶŝƶūƺţŚŝ ŢſřƶŞſŚŰƯƪŝŚƣźƿŻšŹƺƈŝkƆŴƄƯ
STs 
C  Ok  C C  Ps .Ykj 
s1 jk1èí
1C S T Ps .Ykjs
s1 jk1
ƹƭŚĮƴƷźƿŵƭŚĮƴƷŵƹŻũƹźųŚŝƾƟŵŚƈţƽżƿŹƶƯŚƳźŝƩŶƯƲƿřźŝŚƴŝ ŢſřźƿŻšŹƺƈŝƾƐųŢƫŚůŹŵƞƗŚƌƯƹŹŻŹ
ST 1 TS T 1 T
Max Z1   Ps  RijsYij  Ps  RijsYij
s1i0 ji1s1 i0 ji1

Min Z 2  s 1 Ps i 0ji 1wij U ijs  Yij  2ijs  m 1m C m  m1mC m
  
S.T .

k T Yij  1 C max S T Ps .Ykjs k 1,…,T 1
i0 jk 1s1 jk 1

TST
Y0 j  1C max   Ps .Y0sj
j1s1 j1

xij Uijmax i  0,…,T 1

ST1TT1T
Ps RijsYij s  RijsYij s 1,…,S
s1i0 ji1i0 ji1
  2m  éå
éæ
éç
éè
éé
éê
éë
 
 Yij ijs Uijs i  0,…,T 1j 1,…,T s 1,…,S
M mCm m  C m m 1,…,M ST 1T M
ƹiŻƹŹŹŵŵƹŹƹŶƈƣƶǀƫƹřƶƯŚƳźŝƢŞƏƶĩƾƳřźƟŚƀƯŵřŶƘţVijv
ŶƳřƵŶƃũŹŚųźţźƿŵƾƫƹŶƳřƶŤƃřŵřŹjŻƹŹŹŵũƹźų
ƪŝŚƣźƿŻšŹƺƈŝVijv ƹEiej źƿŵŚƤƯƵŶƃƭŚŬƳřšŚǀƋźƟƶŝƶūƺţŚŝ ŶƴŤƀƷƶŞſŚŰƯ
0
 x
iej  0/1ij
E x
0/1ij
0/05 xij
j  i 1 j  i 2 j  i  3 j  i  3 e  j 1èèe  j 1 e  j 2
0
  x
ijv 0/1 xijij
V
0/1
0/05 xij j T j T 1 j T 2 j T 2 v  j 1
v  j 1èé
v  j 2
ũŹŚųjŻƹŹŹŵƹŵŹřƹiŻƹŹŹŵƶĩƾƳřźƟŚƀƯƾƘƣřƹŵřŶƘţƲƿřźŝŚƴŝ ŶƴŤƀƷƶŞſŚŰƯƪŝŚƣźƿŻšŹƺƈŝŶƳŵźĭƾƯ
j1j1
Yij  Xij  Eiej  Vikj èê
ei1ki1
ƞƗŚƌƯƹŹŻŹçê
ŵƺūƹŚŝŚƿƹŶƿŚưƳƾƯƺƜƫřŹŵƺųƹŹŻŹźƟŚƀƯŵŹřƺƯŻřƾųźŝŹŵ ƵƺƤƫŚŝŵƺſƂƷŚĩƶŝźŬƴƯźƯřƲƿřƶĩŶƿŚưƳƾưƳƶƘūřźƯƾƬŞƣƹŹŻŹ ƪŤƷƽŚƷƹŹŻŹŵřŶƘţƶƧŢſřƲƿřƞƗŚƌƯƹŹŻŹŻřŹƺƔƴƯŵŵźĭƾƯ
 S Ps T1 T RijsYij   2s  èî
s 1i 0 j i 1
 M

m1
ƾƠŬƳƽŶƸƯƹŽŹŶƯŶưŰƯƪŤƷŶƯōŹŵŹřŶƿŚěƽŻŚſƶƴǀƸŝƾƟŵŚƈţƽżƿŹƶƯŚƳźŝæí

j1j1éì
Yij  xij  Eiej  Vikj i  0,…,T 1j 1,…,T
ei1ki1
xij ,Yij  0 i  0,…,T 1j 1,…,Téí

jŻƹŹŹŵƹŵŹřƹiŻƹŹŹŵƶǀƫƹřƶƯŚƳźŝƢŝŚƐƯƶĩƾƳřźƟŚƀƯŵřŶƘţ xijsƭŚĮƴƷŵƹŻũƹźųŚŝƾƟŵŚƈţƽżƿŹƶƯŚƳźŝƩŶƯë
ŢſřsƽƺƿŹŚƴſŢưǀƣƢŝŚƐƯŚƷƱōƶŝƁƹźƟƹƵŶƃũŹŚųšƹŚƠŤƯƽŚƷŢưǀƣƹƞƗŚƌƯƹŹŻŹƹƭŚĮƴƷźƿŵ
ƵŶƃũŹŚųjŻƹŹŹŵƹŵŹřƹiŻƹŹŹŵƶĩƾƳřźƟŚƀƯƾƘƣřƹŵřŶƘţYijsƞƬŤŴƯƽŚƷƺƿŹŚƴſƶĩŢſřƵŶƃƉźƟƾƬŞƣƽŚƷƩŶƯƶǀƬĩŹŵ
ŢſřsƽƺƿŹŚƴſŢưǀƣƢŝŚƐƯŚƷƱōƶŝƁƹźƟƹ ƲƿřŹŵ Rijs  Rij ƾƴƘƿŶƴŤƀƷƱŚƀĪƿƁƹźƟŢưǀƣƽřŹřŵ ũƹźų Śŝ ƾƟŵŚƈţ ƽżƿŹƶƯŚƳźŝ ƾƴƘƿ ƩŶƯ ƲƿźţƪƯŚĩ ƲƿřźŝŚƴŝ ƽřŹřŵŚƷƺƿŹŚƴſƱōŹŵƶĩŵƺƃƾƯƶŤųřŵźěƾŤƫŚůƾſŹźŝƶŝŢưƀƣ
šƹŚƠŤƯŢưǀƣƽŚƷƺƿŹŚƴſŢŰţƞƗŚƌƯƹŹŻŹƹƭŚĮƴƷźƿŵƭŚĮƴƷŵƹŻƝŶƷ řŸƫ Rijs  Rijs’ i, j ƾƴƘƿ ŶƴŤƀƷ šƹŚƠŤƯ Ţưǀƣ
ŢſřźƿŻšŹƺƈŝƾƐųŢƫŚůŹŵƾƟŵŚƈţƮƛŹƾƬƗƺţŹŵƺţŢƫŚůŹŵŢưǀƣźƷŻřƁƹźƟƱřżǀƯƲǀǀƘţ
ŢſřŚƷƱōƱŵƺŝ ŶƳŵźĭƾƯƞƿźƘţźƿŻŭźƃƶŝƩŶƯƽŚƷźǀƜŤƯ
S T 1 Ts S P Ti01 jTi1RijsYijs sS1 Ps Ti01 jTi1RijsYijs  2s 
Max Z1 PsRijsYijs 
s1i0 ji1s1
2 Ss T 1 Tij  ijsijsijs Mm  mMmm m 
Min Z s 1 P i 0j i 1w U  Y  2   m 1 C   m1 C   2 
 
S.T .
Sk T Yijs  1 C max S T Ps .Ykjsk  1,…, T  1
s1 i0jk 1s1 jk 1 éî
êå
êæ
STST
Y0sj  1C max Ps.Y0sj
s1 j1s1 j1
xijs Uijs i  0,…,T 1j 1,…,T s 1,…,S
ST1TT1T
Ps RijsYijs s  RijsYijs s 1,…,S
s1i0 ji1i0 ji1
Yijs ijs Uijs i  0,…,T 1j 1,…,T s 1,…,S
M
mCm m  Cm m 1,…,M
m1
SSj1j1
Yijs  xijs  Eiej  Vikj i  0,…,T 1j 1,…,T
s1s1ei1ki1
xs ,Y s  0 i  0,…,T 1j 1,…,T
ijij
ƽřƶƳƺĭƶŝƱřźƟŚƀƯƁźƿŸěƵƺŰƳƩŚŞƳŵƶŝƹŢſřŻƹŹźƷŹŵƞƬŤŴƯƩŚŨƯĨƿŹŵƩŶƯƽźǀĭŹŚĪŝì êçêè
êé
êê
êë
êì
êí

ŶƿŚưƳźŨĩřŶůřŹƱřźƟŚƀƯƶŝŢƯŶųƶŗřŹřŻřƪƇŚůŶƯōŹŵƶĩŢſřƮƿŻřŵźěƾƯƩŚŨƯĨƿƾſŹźŝƶŝơƺƟƩŶƯƽźǀĩŹŚĪŝŢƸū
šŹƺƈŝƵŻƹŹæåƵŹƹŵĨƿŹŵƪŤƷƽřźŝŚƋŚƤţƞƬŤŴƯƽŚƷƺƿŹŚƴſƽŚƷŢưǀƣŚŝŚƋŚƤţƞƬŤŴƯƽƺƿŹŚƴſéƽřŹřŵƪŤƷĨƿŶǀƴĩƉźƟ
ŢſřźƿŻ
æîƾƠŬƳƽŶƸƯƹŽŹŶƯŶưŰƯƪŤƷŶƯōŹŵŹřŶƿŚěƽŻŚſƶƴǀƸŝƾƟŵŚƈţƽżƿŹƶƯŚƳźŝ

æƵŹŚưƃƺƿŹŚƴſ èƵŹŚưƃƺƿŹŚƴſ
26 46 117 156 130 26 23 13 10 10 12 18 72 96 72 7 10 4 5 4
0 26 33 156 182 26 13 13 10



قیمت: تومان


دیدگاهتان را بنویسید