Facility location I. Chapter 10. Facility location Continuous facility location models

Transcription

1 Faclty locaton I. Chapter 10 Faclty locaton Contnuous faclty locaton models Sngle faclty mnsum locaton problem Sngle faclty mnmax locaton problem

2 Faclty locaton Factors that nfluence the faclty locaton decson: Transportaton (avalablty, cost) Labor (avalablty, cost, sklls) Materals (avalablty, cost, qualty) Equpment (avalablty, cost) Land (avalablty, sutablty, cost) Market (sze, potental needs) Energy (avalablty, cost) Water (avalablty, qualty, cost) Waste (dsposal, treatment) Fnancal nsttutons (avalablty, strength) Government (stablty, taxes, mport and export restrctons) Exstng plants (proxmty) Compettors (sze, strength and atttude n that regon) Geographcal and weather condtons

3 Faclty locaton Faclty locaton problem Ste pre-selecton (qualtatve) Pre-selected stes evaluaton (quanttatve) Factor Ratng Cost-Proft-Volume analyss Contnuous faclty locaton problem Faclty locaton models Choce of ANY ste n the space The sole consderaton s transportaton cost

4 Contnuous faclty locaton problems For the new faclty we can choose ANY ste n the space For the exstng related facltes (supplers, customers, etc.) we know the coordnates (x,y) and the flows (cost) between them and the new faclty The sole consderaton s transportaton cost Faclty locaton models have numerous applcatons New arport, new hosptal, new school Addton of a new workstaton Warehouse locaton Bathroom locaton n a faclty etc.

5 Contnuous faclty locaton problems Dstance measures Rectlnear dstance Along paths that are orthogonal or perpendcular to each other x 1 x 2 + y 1 y 2 A B Eucldean dstance Straght lne between two ponts 2 D x x y y2 Flow path dstance Exact travelng dstance between two ponts A A B B

6 Rectlnear Faclty Locaton Problems Example Determne a new locaton of a warehouse n Montreal area whch provdes materals to 5 dfferent companes Locaton of these companes (a, b) and the materal movement between the new warehouse and the exstng facltes (w) are provded: Where should the new warehouse be located? a b w

7 Rectlnear Faclty Locaton Problems Varous objectves can be used Mnsum locaton problem Mnmzng the sum of weghted dstance between the new faclty and the other exstng facltes Mnmax locaton problem Mnmzng the maxmum dstance between the new faclty and any exstng faclty

8 Sngle-faclty mnsum locaton problem Objectve functon: mn f ( x) m 1 w d( X, P ) Dstances n rectlnear models: d( X, P) x a y b Where X = (x, y) P = (a, b ) w d(x, P ) Locaton of new faclty Locatons of exstng facltes weght assocated wth travel between the new faclty and exstng faclty dstance between the new faclty and exstng faclty

9 Sngle-faclty mnsum locaton problem Fnd the x and y values for the new faclty that satsfy the gven objectve mn f ( x) m 1 w 1 Apply these rules to fnd the optmum value of x: 1. X-coordnate of the new faclty wll be the same as the x-coordnate of some exstng faclty 2. Medan condton: Selected X coordnate cannot be more than half the total weght whch s to the rght of x, or whch s to the left of x. Same rules apply n selecton of the optmum value of y x a m w y b

10 Sngle-faclty mnsum locaton problem Procedure 1. Fnd x-coordnate: Order the facltes based on the ascendng order of ther x-coordnates Calculate partal sum of weghts Fnd the faclty for whch the partal sum frst equals or exceeds one-half the total weght The x-coordnate of the new faclty wll be the same as the x-coordnate of ths faclty 2. Fnd y-coordnate Repeat the same for y-coordnate

11 Sngle-faclty mnsum locaton problem Alternate stes If we cannot place the new faclty on the selected locaton, then alternate stes could be evaluated by computng the f(x,y) values for all the possble locatons and chose the locaton that gves the mnmum f(x,y) value. mn f ( x) m 1 w x a m 1 w y b

12 Sngle-faclty mnsum locaton problem Example A new locaton for a manufacturng faclty s beng consdered. The faclty has frequent relatonshps wth ts fve major supplers and snce the suppled materal s bulky and transportaton costs are hgh the closeness to the fve supplers has been determned as the major factor for the faclty locaton. The current coordnates of the supplers are S1=(1,1), S2=(5,2), S3=(2,8), S4=(4,4) and S5=(8,6). The cost per unt dstance traveled s the same for each suppler, but the number of trps per day between the faclty and each of ts supplers are 5,6,2,4 and 8. Fnd a new locaton for the faclty whch mnmzes the transportaton costs Calculate total weghted dstance for the new locaton. If the faclty cannot be placed n the optmal locaton, fnd the second best alternatve ste out of (5,6), (4,2) and (8,4).

13 Sngle-faclty mnsum locaton problem Example Suppler Relatonshp wth the faclty (trps per day) w Fnd x-coordnate: Order the supplers based on the ascendng order of ther x-coordnates Calculate partal sum of weghts Fnd the suppler for whch the partal sum frst equals or exceeds one-half the total weght The x-coordnate of the new faclty wll be the same as the one of ths suppler Suppler Relatonshp wth the faclty (trps per day) Half the total weght: ( )/2 = 25/2 =12.5 Rule 1: here the partal sum frst equals or exceeds ½ the total weght of the supplers (here S2)

14 Suppler Relatonshp wth the faclty (trps per day) Repeat for y-coordnate: Rule 1: here the partal sum frst equals or exceeds ½ the total weght y-coordnate of the supplers (here S4) Faclty (5, 4) If the partal sum exactly equals ½ the total weght, then the soluton ncludes all ponts between the coordnate where the equalty occurred and the next greater coordnate

15 ) 4 4 4( 5 8) 4 2 2( ,4) ( f The total weghted dstance between the new faclty and ts supplers can be found as: m m b y w a x x f 1 1 w ) ( The best locaton for the new faclty corresponds to the coordnates x = 5 and y = 4 (5, 4) Faclty a b

16 Sngle-faclty mnsum locaton problem Example O 5 f(8,4) = = 132 O 6 f(5,6) = = 115 O 7 f(4,2) = = 114 If these are the only optons avalable, then we would select the locaton 7 to place the new faclty

17 Sngle-faclty mnsum locaton problem Iso-cost contour lnes Iso-cost contour lnes Desgnate movement that does not change the value of the objectve functon Can help n determnng an approprate locaton for a new faclty.

18 Sngle-faclty mnsum locaton problem Iso-cost contour lnes Procedure: 1. Plot the locatons of exstng facltes 2. Draw vertcal and horzontal lnes through each exstng faclty 3. Sum the weghts for all exstng facltes havng the same x-coordnate and enter the total at the bottom of the vertcal lnes. Do the same for y coordnates 4. Calculate net pull for each canddate x- coordnate. (pull to the rght s postve and pull to the left s negatve). Do the same for y coordnates 5. Determne the slope for each grd regon enclosed by the canddate coordnates The slope equals the negatve of the rato of the net horzontal pull and the net vertcal pull 6. Construct an so-cost contour lne from any canddate coordnate pont by followng the approprate slope n each grd.

19 1. Plot the locatons of exstng facltes 2. Draw vertcal and horzontal lnes through each exstng faclty Weghts are gven n RED, and the coordnates of the exstng facltes are gven n BLACK. Accordng to the mnsum algorthm the ntersectons of the lnes are consdered as the canddate locatons of the new faclty.

20 3. Weghts are placed on x and y coordnates

21 Weghts Y Weghts 4. Calculate net pull for each canddate x- coordnate. Pull to the rght s negatve and pull to the left s postve. Do the same for y coordnates, where pull up s negatve and pull down s postve = -3 M1, 1 M3, 8 M4, = M2, 2 M4, X Sum of all the weghts =-3

22 Net horzontal and vertcal forces for regons defned by canddate coordnates Weghts M3, M4, 6 4 Y 5 4 M4, M2, 2 M1, X Weghts

23 Sngle-faclty mnsum locaton problem Iso-cost contour lnes 5. Determne the slope for each grd regon enclosed by the canddate coordnates. The slope equals the negatve of the rato of the net horzontal pull and the net vertcal pull _ Horzontal pull Vertcal pull

24 6. Construct an so-cost contour lne from any canddate coordnate pont by followng the approprate slope n each grd. -15

25 Sngle-faclty mnsum locaton problem Iso-cost contour lnes Sample so-contour lnes:

26 Sngle faclty mnmax locaton problem The objectve s to mnmze the maxmum dstance between the new faclty and any exstng faclty The objectve functon: Mnmze f(x) = max[( x a + y b ), ={1, 2,.. M}] Procedure: To obtan a mnmax soluton, let c 1 = mnmum (a + b ) c 2 = maxmum (a + b ) c 3 = mnmum (-a + b ) c 4 = maxmum (-a + b ) c 5 = max (c 2 -c 1, c 4 -c 3 ) Optmum soluton for the new faclty locaton s on the lne segment connectng the ponts X 1* (x 1*, y 1* ) and Y 2* (x 2*, y 2* ) X 1* (x 1*, y 1* ) = 0.5(c 1 -c 3, c 1 +c 3 +c 5 ) Y 2* (x 2*, y 2* ) = 0.5(c 2 -c 4, c 2 +c 4 - c 5 ) Max dstance equals c 5 /2

27 Sngle faclty mnmax locaton problem Example A company whch has already eght facltes ntends to buld another one and s currently lookng for the most convenent locaton. It was determned that the most approprate place s the one whch s closest to the exstng facltes. The locatons of the current facltes are gven below. Fnd the best mnmax locatons for an addtonal faclty. What wll be the maxmum dstance to any other faclty?

28 Sngle faclty mnmax locaton problem Example a b a + b -a + b Optmal locaton for the new faclty s on the lne connectng these two ponts: X 1* (x 1*, y 1* ) = 0.5(c 1 -c 3, c 1 +c 3 +c 5 ) = ½(6, 10) = (3, 5) Y 2* (x 2*, y 2* ) = 0.5(c 2 -c 4, c 2 +c 4 - c 5 ) = ½(12, 4) = (6, 2) c 1 = mnmum (a + b ) c 1 = 0 c 2 = maxmum (a + b ) c 2 = 16 c 3 = mnmum (-a + b ) c 3 = -6 c 4 = maxmum (-a + b ) c 4 = 4 c 5 = max(c 2 -c 1, c 4 -c 3 ) c 5 = 16

29 Sngle faclty mnmax locaton problem Example P 5 P 8 (3, 5) P 2 P 6 P 7 P 4 (6, 2) P 3 P 1 Max dstance equals c 5 /2 = 16/2 = 8 The pont (3,5) s 8 dstance unts away from P 1,P 3, P 4 and P 8, the pont (6,2) s 8 dstance unts away from P 1,P 5 and P 8 and the remanng ponts on the lne segment are 8 dstance unts away from P 1 and P 8

30 Next lecture Faclty locaton II. Locaton allocaton model Plant locaton model Network locaton problems