THE PROPORTION OF POPULATION METHOD AND OTHER COST ALLOCATION METHODS FOR THE JOINT WATER SUPPLY

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1 THE PROPORTION OF POPULATION METHOD AND OTHER COST ALLOCATION METHODS FOR THE JOINT WATER SUPPLY Hug-Mg Peter Wu Departmet of Iovato Maagemet, Alethea Uversty, Pe-Shh Lao, Matou, Taa, Tawa Chag, Shu Chao Departmet of Busess Admstrato, Natoal Tawa Uversty of See ad Tehology, No. 43 Keelug Road Se. 4, Tape, Tawa Ths artle uses the ot water supply ase to ompare the ost alloato methods based o separable osts ad o-separable osts: the egaltara o-separable (ENSC) method, the separable osts remag beefts (SCRB) method, the mmum osts remag savgs method (MCRS), o-separable ost gap method (NSCG), ad a ew method, whh s alled o-separable proporto of populato (NSPP). We foud the SCRB, MCRS, ad NSCG method ode the ovex ost game ase. The ENSC ad NSPP method also get the lose result. ButENSC method s result s aïve. Whle the detal ost formato s ot avalable, the ew method NSPP s a good hoe beause the method does ot eed the detal ost formato, whh s the ost game lke MCRS ad NSCG eed. 1. Itroduto To derease the water supply ost, the ot water supply are usually set beause of eoomes of sale [1]. I the real world, there may ot have a regoal authorty that ould help to buld ths kd of effet system, f they do ot have far ost alloato. If the dvdual do ot pay a far ost, they may dede to buld ther ow water supply plats, ad a bg loss effey wll take plae. Ths shows ost alloato s very mportat. The purpose of ths artle s usg the ot water supply ase to ompare the ost alloato methods based o separable osts ad o-separable osts: the egaltara o-separable (ENSC) method, the separable osts remag beefts (SCRB) method, the mmum osts remag savgs method, oseparable ost gap method (NSCG), ad a ew method, whh s alled o-separable proporto of populato (NSPP). Ths study s orgazed as follows. Frst, we trodue ths paper. Seod, the seleted lterature revew s show. The thrd seto explas the theory ad model. The ext seto outles a empral example ad dsusso. The fal seto oludes ths paper. 2 Seleted Lterature Revew There are some wdely used methods from ooperatve game theory for the ost alloato problem. They are the Shapley value [2-3], the uleolus [4-5], varats of the uleolus [6, 1], ad the ore ad varats of the ore [6-8]. The ooperatve game s also used as the theoretal bass for the alloato methods based o separable ad o-separable osts [9]. I the ooperatve game, the separable osts are defed as the dfferee ost betwee the player who os the oalto ad who does ot o the oalto [10]. If every player has bee assged ts ow separable ost, the those remag osts, whh are ot separable osts, are aled the o-separable osts. Some ost alloato methods are based o separable ad o-separable osts. They are the egaltara o-separable ost (ENSC) method, the separable osts remag beefts (SCRB) method, the mmum osts remag savgs method (MCRS), ad the o-separable ost gap (NSCG) method. 1

2 Dresse ad Ts [9] ompared these four dfferet ost alloato methods based o the ot osts of water resoures proets. They argue that: All these methods, exept the ENSC method, a be desrbed wth the ad of lower ad upper bouds for the ore of the volved ost game. For ovex ost game, these three methods use the same bouds for the ore ad hee ode, but ther ost alloato does ot eessarly belog to the ore. The separable osts remag beefts (SCRB) method s used multpurpose water reservor proets the U.S. [11]. The method was wdely used multpurpose water developmet proets the 1980s. The dsadvatage of the SCRB method s t oly aalyzes oaltos of sze 1, N 1, ad N. All other formato s gored. The bouds, upper ad lower, for the ore are made by smple formulas. The mmum osts remag savgs method (MCRS) s proposed by Heaey ad Dkso [7]. It s a geeralzato of SCRB. The bouds for the ore are obtaed by solvg several lear programs. Therefore, the MCRS method, those bouds are as sharp as possble [9]. The egaltara o-separable ost (ENSC) method s a ave method. Ths s beause ENSC smply equally assgs the o-separable ost, whh should be proportoed equaly. The o-separable ost gap method (NSCG) s derved by the τ value, whh s trodued by Ts [12]. The method gets the bouds of the ore by smple formulas ad argues the alloato of the o-separable ost s ot oly based o the remag alterate osts of the oe-perso oaltos, but also eeds to be based o the remag alterate osts of other oaltos. Heaey ad Dkso [7] argue... f the problem of regoalzato of sewage treatmet faltes amog N tes s beg examed, the optmal soluto may all for oe large plat. I order to realze these savgs, the N ommutes must somehow apporto the ost of ths regoal falty amog themselves a far maer [7, p. 476]. That tells us the mportae of ost alloato. I ths paper, they also ompare several rtera of ost alloato: (1) the Shapley value; (2) the uleolus; (3) the separable osts, remag beefts (SCRB); ad (4) the mmum ost, remag savgs (MCRS) soluto. The Shapley value may fall outsde the ore (whh s a very udesrable property) whe the game s oovex. A dsadvatage of the uleolus s that solvg N 1 lear programs to get the uleolus value s rather tedous. For the SCRB method, oly oaltos of sze 1, N 1, ad N are aalyzed (other formato s gored).i otrast, all other termedate oalto formato s used the MCRS method [7]. Therefore, Heaey ad Dkso [7] argue The MCRS method s preferable to the SCRB method beause t uses bouds whh are feasble [7, p. 481]. Whe eoomes of sale are preset, ooperatve game theory s a good method to hadle the ost alloatos regoal water qualty modelg systems. However, the ooperatve game theory has a dsadvatage: the real world, there s ofte ot eough formato o osts to apply the theory, espeally whe the rego s bg ad the players are may [6, 13]. Aother dsadvatage s t s dffult to hoose the ost alloato method beause o oe method s uversally preferred [13, 6]. 3. Theory ad Model 3.1 Theory of N-Perso Cooperatve Game Cooperatve game theory a be systematally appled to make ooperatve desos [1]. Therefore, faress must exst betwee the proet members. If oe game s seekg the mmum ost, we all ths game a ost game. Heaey ad Dkso [7, p. 477] wrote: A N-perso game (N, ) haraterst futo form ossts of a set N = 1, 2,..., of players alog wth the haraterst futo, whh assgs the real umber (S) to eah oempty subset S of players. Cost games are sub-addtve,.e. (s) + (T) (S T) for S T = φ S, T N. (1) Where φ s the empty set, S ad T are ay two subsets of N. 2

3 There are three geeral axoms a ost game to set the far soluto [7, p. 478]. I the frst, the osts assged to the th group, x, must be o more tha ther osts whe they at depedetly, that s, x () N. (2) I the seod, the total ost (N) must be apportoed amog the groups, that s, x ( N) (3) N If the above two equatos are satsfed, we al these soluto mputatos. I the thrd, the rtero s exteded from the frst equato. That meas the ost to eah member must less or equal to the osts they would reeve ay oalto S otaed N, that s, x ( s) s N (4) s All solutos satsfed the above three equatos. It makes the ore of the game. Heaey ad Dkso [7, p. 478] argue: For sub-addtve games the set of mputatos s oempty, but the ore may be empty. A ost game has a ovex ore f (s) + (T) (S T) + (S T) for S T = φ S, T N (5) I geeral, the more attratve (lower ost) the game s, the greater the hae that the ore s ovex. Coversely, the less attratve the game s, the greater the hae that the ore s empty. 3.2 Cost Alloato Methods Based o Separable ad No-separable Costs For the ost alloato methods based o separable ad o-separable osts, we use the ENSC, SCRS, NSCG, MCRS methods, ad the NSPP method, whh s the method proposed by ths paper. I ths seto we wll defe what s separable ad o-separable ost appled ost alloato. The, the above fve methods wll be explaed. I the ooperatve game, the separable osts are defed as the dfferee ost betwee the player who os the oalto ad who does ot o the oalto [10]. It s the same as margal ost ad we a defe the separable ost of player a ost game (N; ) by SC : N N (6) If every player has bee assged ts ow separable ost, the those remag osts, whh are ot separable osts, are aled the o-separable osts. It a be defed as NSC N SC : (7) 1 To dede the alloato of the o-separable ost amog the plays, a real umber, =1,2,3,,, a be hose by the dfferet ost alloato methods. The egaltara o-separable ost (ENSC) method s to hoose =1 for all N. Ths meas ENSC smply equally assgs the o-separable ost, whh should be proportoed equaly. Thus ths ost method for the ost alloato to player s gve by 1 ENSC SC NSC (8) The separable osts remag beefts (SCRB) method s used multpurpose water reservor proets the U.S. (Iteragey Commttee o Water Resoures, 1958) ad t was wdely used multpurpose water developmet proets the 1980s. The method hooses m( b, SC ) for all N, where b meas the beeft to play the game (N; ) by atg depedetly. SC s the dferee betwee eah player s wlges to pay ad the separable ost already 3

4 b, usually exeed the alterate ost, alloated. Se the beefts, of the ost alloato to player s usually defed by SCRB SC SC SC SC NSC 1, the SCRB method (9) 1 The SCRB method oly aalyzes oaltos of sze 1, N 1, ad N. All other formato s gored. Dresse ad Ts [9] proposed the remag alterate osts of other oalto of the o-separable ost eed to be luded. Therefore, they trodue the ost gap futo g. For eah oalto S, the ost gap of S the ost game (N; ) s defed as S S SC g f S s Se the ost gap futo s oegatve, we assume g S 0 ad :0 g (10) S N. Let T be a oalto that player belogs to. I ay ost alloato, player wll reet payg more tha SC g T.Ths s beause player wll threate to set up the oalto T where all the members exept player wll be harged oly ther separable osts, whle player wll be harged the remag ost whh s C T SC equalg T () SC g T.The above argumet apples to ay oalto T ludg player. Therefore player wll ot pay more tha the amout m SC g T equals SC m g T T ; T a ost game (N;) by T ; T whh. From the above argumet, we a defe the oesso amout of player T m g (11) T ; T The above equato shows the oesso amout of player a be see as maxmal otrbuto to the o-separable ost NSC(). Therefore, we assume the total of the maxmal otrbuto s ot less tha the o-separable ost. That s NSC 1. The o-separable ost gap (NSCG) method uses the players proportoal oesso amout to alloate the o-separable ost. Therefore, f g S 0 S N ad NSC 1 NSCG are satsfed. The NSCG method of ost alloato for player s defed as SC NSC 0 SC 1 NSC 0 1 NSC (12) 4

5 The MCRS method, whh s proposed by Heaey ad Dkso [7], s a geeralzato of SCRB. The lower ad upper bouds used the method for the ore are as sharp as possble. If games (N;) have a oempty ore, we fd the upper ad lower bouds o eah x,, ost alloato of player, by subet to: Maxmze or mmze x (13a) x () N (13b) s x ( s) s N (13) N x ( N) (13d) x 0 N. (13e) Therefore MCRS method of ost alloato for player s defed as where MCRS ( ) x m u ( ) ( N) N x m 1 max m max m x x x x u ( ) (14) N For the NSPP method, the o-separable ost s proportoed by theplayer s shareof ther populato. Thus the NSPP method of the ost alloato to player s defed by where NSPP SC Q Q NSC 1 (15) N Q s the quatty per aptal. The advatage of ths method s that t does ot eed detal formato (formato of all the subgroups), whh the MCRS ad NSCG eed, of the ooperatve game. 4. Empral Example ad Dsusso 4.1 data The data used the paper s the example gve by Youg [6, p. 464]. I the example, there are three eghborg mupaltes 1,2, ad 3. They buld a ot water supply falty to supply themselves wth mupal water. To ompare the above fve ost alloato methods, we osder these three mupaltes ost game (N;) wth C(1)= 6.5 C(12)= 10.3 C(123)= 10.6 C(2)= 4.2 C(13)= 8.0 C(3)= 1.5 C(23)= 5.3 5

6 The aual rates of water use of supplyg water the three mupaltes gve the table 1 a be used to ompute the proporto of the populato. Table 1. Aual Rates of Water Use of Supplyg Water Three Hypothetal Mupaltes Mupalty Use per Capta, m 3 A 140 B 120 C 120 * The above formato s from Youg et al [6, p. 464], table Results ad Dsusso The results of ost alloato are show table 2. We use software GAMS to solve ay methods wth lear programs. From table 2, we ot oly a see the ost alloato of ay mupalty s less tha the ost they work dvdually but also all the ost alloato s the ore (see the equato (2),(3), ad (4)). Ths meas the oalto s stable ad o oe wll leave the grad oalto (3 mupaltes) to form the subset of the 3 mupaltes. Therefore, the oalto s effet ad equal. The example used the paper s a ovex ost game (see the equato (5)), whh has very e propertes the game theory lterature. I table 2, the SCRB, NSCG, ad MCRS method ode. That s beause they may use the same bouds the ovex ost game. The ENSC ad NSPP method show a lose result wth the above three methods. But the ENSC s less mportat, se t smply assgs the oseparable ost. For the ew method NSPP, t may a good hoe. Although the NSCG ad MCRS methods use the formato of all oaltos ad get the sharper bouds for the ore, we may ot get all the formato, for example, there are too may players. TABLE 2.Comparatve Results for these fve Cost Alloato Methods Mupalty Total ENSC SCRB NSCG MCRS NSPP (Ut: dollars 10 ) 5.Coluso I the ot water supply ase, all these fve ost alloato methods based o separable osts ad oseparable osts solve the effey ad equty problem. Usg the ovex ost game, we foud the NSPP method proposed the paper has stable ost alloato the ore. The method s a good hoe whe the oalto formato a ot be too detaled. For the future study, we may use the o-ovex ost game ad may players (greater tha four) to test the NSPP method to see whether the method stll gves a good result. 6

7 Referees 1. R. P. Leao, ad C. A. Davos, Cost Alloato of Multagey Water Resoure Proets: Game Theoret Approahes ad Case Study. Water Resoures Researh 31: , S. C. Lttlehld ad G. Owe, A Smple Expresso for the Shapley Value a Speal Case, Maage. S., 20, , E. Loehma, J. Orlado, J. Tshrhart, ad A. Whsto. Cost Aloato for a Regoal Wastewater Treatmet System, Water Resoure Researh, 15(2), , X. C. Lttlehld, A smple expresso for the uleolus a speal ase, It. J. Game Theory, 3: 21-29, M. Suzuk ad M. Nakayama, The Cost Assgmet of the Cooperatve Water Resoure Developmet: A Game Theoret Approah, Maage. S., 22(1), , N. O. Youg, ad T. Hashmoto, Cost Alloato Water Resoures Developmet, Water Resoure Researh, 18, , J. P. Heaey ad R. E. Dkso, Methods for Appotg the Cost of a Water Resoure Proet, Water Resoures Researh, 18(3), , Jue, H. M. P. Wu ad K. D. Wllett, Eoomes of Sale Ca Be Implemeted by Cooperatve Game: The Cha Model Case, Joural of Evrometal Systems, 31(1), , T. S. H. Dresse ad S. H. Ts, The Cost Gap Method ad Other Cost Alloato Methods for Multpurpose Water Proets, Water Resoure Researh, 21(10), , Federal Iteragey Rver Bas Commttee, Proposed Prates for Eoom Aalyss of Rver Bas Proets, Tehal Report, Washgto, DC, Iter-Agey Commttee o Water Resoures, Proposed Prates for Eoom Aalyss of Rver Bas Proets, Tehal Report, Washgto, DC, S. H. Ts, Bouds for the Core ad the τ Value, Game Theory ad Mathematal Eooms, O. Moeshl ad D. Pallashke (eds.). Amsterdam: North-Hollad, , R. J. Gglo ad R. Wrghtgto, Methods for Apportog the Costs Amog Partpats Regoal Systems, Water Resoure Researh, 8(5), ,