LECTURE 4 QUANTITATIVE MODELS FOR FACILITY LOCATION: SERVICE FACILITY ON A LINE OR ON A PLANE

Transcription

1 LECTUE 4 QUANTITATIVE MODELS FO FACILITY LOCATION: SEVICE FACILITY ON A LINE O ON A PLANE Learg objectve 1. To demostrate the quattatve approach to locate faclt o a le ad o a plae

2 6.10 Locatg Faclt o a le The target group.e. group of customers s cocetrated throughout a le wth dfferet customer dest. The problem s smlar to a stuato where a grocer retaler wats to ope a faclt o a log street where ma resdets sta. The retaler wats to mmze the average walk b the customers from awhere the street. The resdetal area has all kd of home facltes (mult-stored to dvdual houses) whch leads to dfferet customer dest throughout the street. Cosder the followg otato to locate sgle faclt o a le : Ste of the faclt or possble locato where a small retaler ca ope a small outlet. X : Locato of the th demad pot from the org (A oe ed of the street ca cosder as org). W : elatve weght of demad attached to the th locato based o customer dest. The objectve fucto Z s to mmze the dstace travelled b the customers ether sde of the street. Cosder the west ed of the street to be org as show Fgure We wat to locate o X where t rages from 0 to.

3 600 Custo mer 500 Dest Customer Dest (No. of customers) w s FIGUE 6.17: CUSTOME DENSITY ON A STEET The faclt locato problem ca be descrbed as followg. M mze Z = w ( ) + w ( ) = 0 = To fd the mmum Z dfferetate Z wth respect to ad set equal to 0 as gve below. dz d = w w = = 0 = or w = w = 0 = 0

4 The optmum Z wll be acheved f s located at the meda wth respect to the dest dstrbuto of customers or resdets resdg o a street betwee =9 ad =10 as show Fgure w = 2900 w = 2900 = 0 = Customer Dest ( No. of Customers) FIGUE 6.18: FACILITY LOCATION AT THE MEDIAN WITH ESPECT TO CUSTOME DENSITY DISTIBUTION 6.11 Locatg a faclt o a plae wth ectlear Metrc A faclt ca be located o a plae a smlar fasho as we dd o le wth ol dfferece s the cosderato of ad coordates. The problem o a plae ca be descrbed as

5 ( ) M Z = w + w Where = 1 th, = coordates of the customer demad pot, = coordates of the ew faclt Eample: Locatg MED hosptal a ct Cosder a problem of locatg a hosptal a rectlear ct wth the estg demad coordates ad correspodg dest weghts as gve Table 6.5. The ct s dvded to small 5 regos where demad s cocetrated ad cosdered to be at oe pot each rego. The regos ca be plotted o - coordate as show Fgure TABLE 6.5: THE DATASET FO ALL FIVE EGIONS AE AS GIVEN BELOW Customer Demad ego X - Coordate Y- Coordate Weght Cumulatve weght A B C D E Where to locate?

6 Y - Coordate , , , , , X - Coordate FIGUE 6.19: LOCATION OF CUSTOME DEMAND EGION To mmze the total dstace travelled b the customers a metropolta rego (mostl see urba areas), cross meda approach s used. Calculate the meda usg the weght at each demad rego Meda = = = Now, we wat to determe the coordated X, Y for the ew faclt,.e. hosptal. Fd the X coordate meda for to locate Step 1:- Add the values of w the -drecto whle gog from west to east ad east to west.

7 Step 2:- Lst dow the demad rego as the appear ascedg order (descedg order) from west to east (east to west). Step 3:- The weghts attached to each pot s summed ascedg order (descedg order) for west to east (east to west) utl the meda value.e s reached or eceeded. West to East Customer rego Weght Cummulatve weght D A B C 300 [ meda1900] E 1200 East to West Customer rego Weght Cummulatve weght E C B [ meda 1900] A 500 D 800

8 epeat these steps for North to South ad South to North North to South Customer rego Weght Cummulatve weght D C B E 1200 [ meda 1900] A 500 South to North Customer rego Weght Cummulatve weght A E B C 300 D 800 [ meda1900] So, pot B.e (800, 500) s the ew locato for a hosptal. The result of cross meda approach ma gve a pot o a plae or a le segmet whch wll mmze the travel dstace covered b the customers Locatg a sgle faclt o a plae wth Eucldea dstace metrc I most real cases, the dstace betwee facltes s measured Eucldea metrc. The same objectve of mmzg the travel dstace b the demad pots followg Eucldea metrc ca be wrtte as

9 Mmze Z = 1 w As we dd the rectlear case, to fd optoal soluto we wll take partal dervatve of Z wth respect to ad coordates ad settg them equal to zero z z w w 0 0 For coordate w w w We ca wrte, S, for = 1, The optmal -coordate for, * 1 Smlarl, we ca wrte -coordate of ew faclt 1 s, S as,

10 * 1 1 s s,, We ca see from the ad epressos that ad appear both the sdes of equalt ther respectve equatos. So we caot get a drect soluto to the problems. We ca use a tatve wa or umercall solve the problem b gvg some tal values of ad o the rght had sde of the respectve equatos. Calculate the revsed values (left had sde) of ad. Now substtute the reversed values to the left had sde ad repeat the procedure utl the dfferece betwee successve values of ad s eglgble. How to start wth ad? Oe tutve wa to talze the problem s to use ceter of gravt method. The ceter of gravt s the average locato of the weght of a object. We ca use the average weghted locato of ew faclt as a startg soluto to fd the eact locato of. Ceter of gravt If there are demad regos ( =1,...) wth coordates ad wth weghts or demad dest at each rego represeted b ω,the ceter of gravt gve average weghted coordate. The ceter of gravt coordates g ad g wll ot mmze the dstace travelled b the customers but wll gve average of weghted customers located dfferet regos wth varg demad dest.

11 g 1 1 g 1 1 We ca use g ad g values as a startg value of ad sgle faclt locato problem o a plae wth Euclda dstace. Cosder the same eample of MED hosptal. We wll fd the locato of hosptal f we use Euclda dstace metrc. Frst we wll fd the ceter of gravt coordates to get tal values of ad g = (500*400) + (800*1000) + (1100*300) + (200*800) + (1300*1200) = 803 g = (200*500) + (500*1000) + (800*300) + (900*800) + (300*1200) = 505 We wll substtute g for S(, ) ad calculate *. The * value ca aga be substtuted as (, ) * wll repeat the same to determe S ad repeated tll the * ad gap becomes eglgble. We

12 Iteratos are eplaed Ecel Worksheet It wll gve fal values as * * = 800 = 499 The faclt should be located at (800,499) coordates.