An Algorithm for Large Scale Capacitated Location Problem in Networks

Copyright © IFAC Large Scale Systems, Beijing, PRC, 1992

AN ALGORITHM FOR LARGE SCALE CAPACITATED LOCATION PROBLEM IN NETWORKS G.R. Mateus 1 and Z.K.G. Patrocinio, Jr. Universiciade Federal de Minas Cerais. Deparlam£f\JO de Ciencia cia MC. Brazil

Compu1a~ao.

30161 - Belo Horizonle.

Abst.ract. Th e Capacit.ated Locat.ion Probl em in Networks co nsiders a set 5 of cand idat.e places to locate facilities, a set T of inte rmediate or transshipment nodes and a set D of demand nodes. The problem cons ists of locating a set !\' ~ 5 of facilities to serve the demand nodes, so t.hat the total cost, including fix ed and variable costs, is minimized. We propose a bran ch-and-bound a lgorit.hm based on t.he Lagrangean relaxation of the mixed-integer mod el, so lved by a subgradient procedure. We develop and compare two lower bounds. \Ve apply a set of redu ct. ion t.est.s and penalties to allow an increase in th e comput.at.ional effici ency. Wc give comput.at.ional res ult.s o n t.est problems. Keywords. Di stribut.ion net.works : large sca le systems; opt. imizat. ion; integer programming; computat.ionalmdhods. INTRODUCTION

artificial arc (i, 0), Vi E D; and be a 0/1 variabl e that aSSllmes t.h e value if t.he facility i is open, a nd zero ot herwise.

Yi

Th e Capacit ated Locat.ion Problem in Networks (CLPN) cons id ers a set 5 of candidate sites to locate faciliti es, a set T of intermed iate or transshipment nod es and a Sf't. D of demand nod es . Each facility i E 5 has a maximum capacit.y Si > 0 and a fixed cost. J;. Eac h node i E D has a demand d; > O. The set of nodes N = 5 u T U D is connected by a set of directe d arcs A. Each arc (i, j) E A has an uppe r bound llij, a low er bound i ij on the total flow and a unit. cost Cij. T he problem consists of locating a set !\' C S, 151 ::; P, of facilit.i es to serve th e demand nodes, so t hat the total cost, in cludin g fix ed and variable costs, is minimized. Let

The problem may be forlll ul ated mathemat.ically as : min

Pu

o Xij

Xoi

Xi o

CijIij

+L

(1)

! iYi

lES

subject to

L (i,j) Ef+(i)

Iij -

L

0, Vi E NU {o}(2)

l :li

(UlEr-(i)

l ij ::; J' ij

<

0::;

Ioi ::;

di

::; Li o

PI::;

LYi::;

llij,V(i , j) E A(3) SiYi,Vi

E 5 (4)

ViED

(5)

(6)

iES

r+(i) be the set of outgoing arcs in nod e i; r- (i) be th e set of incoming arcs in nod e i; PI

L (i,j)EA

Yi

be the lower linut on the number of open facilities, PI ::; Pu; be the upp er limit on the numbe r of open facilities, Pu ::; P; be an artificial nod e; be the amount of flow through the arc (i,j) E A ; be the amount of flow t.hrough the artificial arc (0, i) , Vi E 5; be t.he amount. of flow through t he

E

{O , I}, Vi E 5 (7)

For all a rtifi cial arc (0, i) , i E S, we set. Coi = 0 , and 0 ::; X oi ::; Si· On t.he ot. her ha nd , for a ll arc (i, 0), i E D , we set Cio = 0, and d i ::; Xi o ' The objective function (1) minimizes the variable and fixed costs. Constraints (2) are the usual net wor k flow conservation at each node. Constraints (3) provide bounds on the amount of flow through each arc (i , j). Conditions (4) express the fact that the flow through an arc (0 , i) must be zero if th e facilit.y i is not included in the solution, ot herw ise the flow is limited by capac it.y of th e facility. Constraints Ci) impose th e demand

1 Visiting Researcher at the l'niversit.y o f Ottawa Canada. Researc h suppor t ed by t.h" CNPq - Brazil.

63

requirements and constraint (6) assures that the number of open facilities must be between PI ~ Pu and Pu ~ p . Finally (7) denotes the integrality constraints. The CLPN is a NP-complete problem that arises in a variety of settings. We are applying the model in the switching center location context, but the model can still be extend to consider additional constraints as in Mateus , Costa and Luna (1992) . Moreover, a host of network models can be viewed as special cases of the problem above. For example, in the Capacitated Location Problem (CLP) (Sa, 1969; Akinc and Khumawala, 1977; Christofides and Beasley, 1983; Beasley, 1988; Mateus and Bornstein, 1991) the set T of intermediate nodes is empty and the bounds on the arcs are relaxed . In addition, the problem is restricted to a Transportation Problem setting values 0 or 1 for the location Yi variables. In the Uncapacitated Location Problem (Cornuejols, Fisher and Nemhauser , 1977; Erlenkotter, 1978; Krarup and Pruzan, 1983; Boffey and Karkazis, 1984) the set T is also empty and the supply capacities and the upper and lower bounds on the arcs are disregarded. If the set of location Yi variables are fixed at 0 or 1 in the CLPN , then the remaining part of the problem to be solved is a Minimal Cost Network Flow Problem (MCNF) . There are many algorithms to solve CLP. They can be grouped in three distinct groups based on the technique used in the solution. Van Roy (1986) developed cross decomposition algorithm motivated by Benders' decomposition and column generation . Enumeration algorithms based on branch-and-bound procedures were proposed by Sa (1969), Akinc and Khumawala (1977) , Nauss (1978), Christofides and Beasley (1983) , Beasley (1988). Heuristics for the CLP were presented by Baumol and Wolfe (1958), Jacobsen (1983), Mateus and Bornstein (1991). On the other hand , CLPN has not been explored extensively, (Patrocinio Jr . and Mateus, 1991; Mateus, Costa and Luna, 1992; Mateus and Thizy, 1992) . This paper focuses on an enumeration strategy for CLPN . We propose a branch-and-bound algorithm based on the Lagrangean relaxation of the mixed integer model, solved by a subgradient procedure. We develop and compare two lower bounds based on Lagrangean relaxation of the problem . The relaxed problems can be easily solved . We report computational results on test problems .

LOWER BOUNDS The branch-and-bound algorithm is a binary depth-first tree search procedure. The branching strategy chooses a facility to define the division of the problem. This means that when a site i is considered, we branch on Yi and set Yi = 1, the facility is considered open, and set Yi = 0, the facility is closed, determining two subproblems or nodes of the enumeration tree . For each node of the tree, an integer solution is an upper bound for the CLPN and a solution of the relaxed problem is a lower bound for the subproblem . We first introduce two lower bounds considering the lower bounds on the arcs . But, we suppose that lij = 0 for all (i, j) E A in the computational results presented in the last section . After , we discuss reduction tests and penalties . They are applied to fix open or closed a set of free facilities, reducing the tree search . The subgradient procedure updates the set of Lagrangean multipliers, thus yielding tighter lower bounds. First Lower Bound A first lower bound can be obtainel;! by relaxing the lower and upp er bounds on th e arcs and the demand constraints . Suppose we partition the set of facilities S into 3 subsets :

Ko the set of facilities i E S that has been closed (Yi = 0). Initially Ko = 0; K 1 the set of facilities i E S t.hat has been opened (Yi = 1). Initially Kl = 0; and K 2 the set of facilities i E S that has not been fixed (open or closed) yet (free faciliti es) . Initially f{2 = S.

Let r , v and W be vect ors of nonnegative Lagrangean multipliers. We associ ate r and v with constraints (3) on the upper bound and on the lower bound respectively, and associate w with constraints (5) on the lower bound. In this way, we obtain a Lagrangean relaxation of the problem which is given by the program :

L(r ,v,w)

=

minr,v , w~o

L(i ,ilEA (Ci j

-Vij )Xij -

+ rij

LiED XioWi o

-

+

+ LiE(K,uK 2 ) J;Yi + LiED diWi o +

+ L(i,j)EA lijVij 64

- Uijrij

(8)

s.t. (2),(4), (6) and Vi = 1,Vi E K1,Vi E {O, l},Vi E K 2 .

• If C(i,o) ~ 0 then set tl.Jlow = set tl. Jlow = O.

This problem can be easily solved in two steps. In the first step, considering (Cij + rij - Vij) and (-Wio) as the cost values for the arcs (i,j) E A and for the arcs (i, 0), i E D. respectively, do the following:

Si

else

• Set x~i = x~i + tl.Jlow, • For each arc (p, q) that belongs to the shortest path from the facility i to the artificial node 0, do X;q = X;q + tl. Jlow. Hence the optimal solution of the Lagrangean relaxation is (yr , xr) as defined above and the value of the lower bound L(r, v, w) is given by:

• For each facility i E (KI UK 2 ) find the shortest (least costly) path from i to the artificial node 0. This can be done by solving only one shortest path problem. Let C(i,j) be the cost of the shortest path from i to j;

L(r, v,

w)

=

min LiEM

Q:i

+ LiED diWio +

+ L(i,i)EA(lijVij - Uijrij)

(10)

• For each facility i E (Kl U K 2 ) calculate its contribution to L(r, v, w), which is given by: Q:i

Second Lower Bound

= J; + min{O, C(i, o)sd

(9)

We also propose a second lower bound relaxing the lower and upper bounds on the arcs and the supply capacities instead of demand constraints. Let r and v the set of multipliers associated with constraints (3) and t the set of multipliers for constraints (4). The relaxed problem is given by:

Thus, the Lagragean relaxation problem reduces to: L(r, v,

w)

=

minr,v,w~o LiE(K,UK 2 ) Q:iYi LiED diWio

+ L(i,j)EA lijvij

+

- Uijrij

s.t. (4), (6) and Yi = I,Vi E Kl,Yi E {O, I},Vi E K 2 .

L(r , v, t)

=

minr,v,t~O L(i,j)EA (Cij

(a) Set Yi

= 1 for all facility i E K

+ LiEK 2 (fi - Siti)Yi + + LiEK,

PI -

( cl) If PI

> IKII, then set Yi = 1 for IK 11 facilities in this list.

= Pu

J; -

siti +

+ L(i,j)EA lijVij

l .

(b) Form a list of the facilities i E K 2 arranged in ascending order of Q:i.

(c) If PI

-

+ LiEK,UK2 xoiti +

-Vij )Xij

This 0-1 problem can be solved by inspection as follows:

+ rij

- Uijrij

(11)

s.t. (2),(5), (6) and Yi = 1, ViE K 1 , Yi E {O, I}, ViE J( 2. The relaxed problem can be decomposed in two subproblems:

the first

(PI ):

go to Step f.

mm

(e) Proceed along the list from the (PI + I )th facility, setting Yi = 1 until either we have opened Pu facilities or the values Q:i become positive.

L

(Cij +rij -Vij )Xij

s.t . (2), (5) and

(P2) :

(f) Let M = {i E (KI U K 2 ) 1 Yi = I} be the set of facilities that are open in the current solution, if = (Yi) be the set of values decided above and xr = (xij) be the corresponding values for the flow on arcs.

L

+

xoiti

(12)

iEK,uK,

(i.j)EA

min

L

Xij ~

0, V(i,j) E A. (13)

(fi - Siti)Yi

iEK,

s.t. (6) and

(g) Set xij = O,V(i,j) E A,x~i = O,Vi E (Kl U K2)' xio = 0, Vi E D.

Yi

E {O, I}, Vi E K2

Problem (Pl) :

(a) Find the shortest path from node j E D.

(h) For each facility i E M, do as follows:

65

S

to all demand

(b) For each j E D, send the flow dj from the nearest facility i along the minimal path to j.

Penalties to Open Facilities Replacing the last facility k selected in M, k

f!.

Kt, by another facility i E (K2 - M), yields a (c) Put the facility i in the set M 1 •

change of cost given by:

= ~i =

~i

Problem (P2) :

(b) Arrange the facilities

E K2

In

maxl:E(M-K,)

0i 0i -

if

Ok,

IMI = Pu

od, iflMI f. Pu

max {O, maxl:E(M-K,)

If ~i + L(r, v, w) > LS, then Yi = 0; thus, set Ko <- Ko U {i}, K2 +- K2 - {i} .

ascending

order of (fi - sit;) .

= 1 for i E K 2 , M2 = M2 U {i} , IM21 = PI·

(c) Set Yi

Penalties to Close Facilities

until

Replacing a facility i E (M - Kt), by a facility k E (I{2 - M) that has not be en selected and has the smallest cost in (K2 - M), yields a change of cost given by:

(d) Proceed along th e list, setting Yi = 1 until eith er we have open ed Pu fac ilities , or (fi s it;)

:S O.

= ~i =

L(" ,v,t)

+ minkE(K2-M) Ok, if IMI = PI -Oi + min {O, minkE(K2-M ) Ok} , iflMI f.

~i

The value of the objective fun ction for problem (PI) is given by z(P1) = LiEMl LjED C(i , j)d j and z (P2) = LiEM 2 fi -Siti· Then , the solutions of both subproblems yi eld th e lower bound:

-Oi

If~i+L(r,v , w)

1. . . 1

<-

> LSthen Yi

KI U {i}, K2

<-

1; thus , set

= '\' ' J')d j + L,iEM , '\' L,jED C( l, + LiEM 2 fi - s iti + + L(i ,j) EA lijv i j - Hijl'ij

Penalty on the Number of Open Facilities

(14)

Let P be the number of open facilities. Replacing the facilities i E (M - K I) by (p - IK 11) facilities k E K 2 with smallest cost Ok, yields a change of cost given by:

Thus, for each set of La grangean multipliers (r ,v, w) or (r ,v ,t) , we obtain a lower bound as defined in (10) or (14). Therefore , the tightest lower bound which can be a chieved is th e solution of the Lagrangean dual problem:

~i

L

=

max L(r , v , w)

or

max L(r , v, t)

r ,v, t~O

L

Ok -

p-IKtI smallest ex. , kEK2 " ,v, w~o

=

K2 - {i}.

OJ

iE(M-K,)

(16)

(15) If .6. i + L(r , v, w) then if P else if P

PROBLEM REDU CTION In this sec t.ion, we improv e a set of redu ction tests and penalties to redu ce the probl em si ze and to allow an increase in the computational effici ency as in Mateus and Thizy (1992) . Some tests are extensions of t.hose proposed to the CLP by Christofides and Beasley (1983), Jacobsen (1983) , Mateus (1986), Beasley (1988), Mateus and Bornstei n (1991). Let LS be an uppe r bound on th e optimal solution of CLPN. We pres ent some tests based on the first lower bound. but th ey can also be defined for th e second on e.

> LS

= PI, set PI = Pu, set Pu

<-<--

+ 1, Pu - l.

PI

Test for ope ning a facility Let h(K), f{ ~ 0 be the optimal value of CLPN with the set K of facilities open and all other facilities closed. It is easy to see that h(K) is also the optimal value of a problem (MCNF) where the supply nodes belong to the set K . Thus we can define for each i E K 2 :

and if .6. i (1992) .

66

~ fi

then Yi

= 1,

Mateus and Thizy

PI

(9) If V = R = W

However, if we intend to use this test for all IK21 + 1 problems MCNF which can be too expensive computation ally. We can overcome this computational burden by using a lower bound Ai to approximate ~i' This lower bound can be calculated by relaxing the capacities of the facilities i E (K 1 u J( 2 - {i}) and the bounds imposed on the flow along the arcs . Toward this end, we define:

i E K 2 , we have to solve

twhere 0

< _ Si

,

0

+ t Vij}, V(i , j)

Vij

=

max{O, Vij

rij

=

max{O , rij +tR;j},V(i , j) E A

Wio

=

max{O, Wi o + t Wi o }, Vi E D

EA

D

In deciding a value for p (equation(20)) we follow the approach of Fisher (1981 , 1985) in setting p = 2 initially. At Step 12, we half p if the bes t lower bound (LI) do not increase within 30 subgradient iterations, following in this case wh at has been done by Beasley (1988) . In our algorithm we only consider that the lower bound has in crease d if a reduction in the duality gap of 0.01 % is achieved . After some tests, we also found useful to lIse a target value in equation (20) 10% above the upper bound (LS) .

SUBGRADIENT PROCEDURE The objective of the subgradient procedure is to compute a set of Lagrangean multipliers that maximizes the optimal value of the relaxed problem , thus leading to a tighter lower bound . We adopt the following procedure: (0) Set : p = 2 , LI = - (Xl , /\0 = 0 , ]{1 0, K2 = S . Initi ali ze the multipliers : rij O,V(i,j) E A , Vi j = O,V(i , j) E A , Wjo miniE(K,UK2 ) C(i , j) , Vj E D .

BRANCH-AND-BOUND

(1) Find an upp er bound , LS, using a heuristic or LS = +00 .

In order to solve the CLPN , we adopt a bina ry depth-first tree search procedure. For the first lower bound , th e bran ching strategy used is to pick the facilit y i E J{ 2 corresp onding to

(2) Solve the relaxed problem and calculate L(r, v , w), (or L( r, v, t) .

(3) Update th e maximum lower bound , LI maxi LI , L(r, v , w) } .

=

(21 ) For the second lower bound the bra nchin g stra tegy is based on the minimal

(4) Using the set of selected facilities M , find a feasible solution whose total cost is denoted by ZLS th e upp er LS = mini LS , Z LS} .

2.

(13) If p < 0.001 then STOP else go to Step 2.

< x'1] · < d]·, VJ' E _

(5) Update

~ p ~

(12) Update p if it is required

jED

s.t . "0 ].E D x't]·

(20)

11V1I2 + IIRI12 + IIWII 2

(11) Update the multipliers (r, v, w) :

( 19)

Oij x:j

LS-L(r,v,w)

- p

min {max{C(k , j)-C(i, j),O}} kE(K,uK,-{i}) (18) and consider the following problem :

L

STOP .

(10) Calculate the step size t :

Oij=

Ai = max

= 0 then

(Ji - Sit;) , Vi E

bound,

]{2

(22 )

We branch by ope ning that facility. The details of the tree sear ch proce dure are given below.

(6) If LS = LI then STOP . Initial Tree Node

(7) Carry out the reduction tests and update Ko , J{l and K2 if necessary.

At the initi al tree node we first attemp t to reduce the problem size by carrying out the test for opening a facility. Noti ce th at some facilities may have been fix ed open at th e end of the problem redu cti on step.

=

(8) Calculate the subgradi ents V = (Vij) , R (R;j) and W = (Wi o ) : Vij = iij xfj ,V(i , j) E A, Rij X~j - lLij, V(i , j) E A , Wi o = di - xio,Vi E D .

=

67

Then, we first create a list of free facilities (i E K 2 ) sorted into descending order of ~i (equation (19)) in order to generate an initial feasible solution corresponding to an upper bound on the problem, and proceed as follows: (a) set j

(d) we do not attempt to find a feasible solution from the Lagrangean solution (yr, xr) . Finally we can backtrack in the search tree if the best lower bound (L1) becomes greater than the upper bound (L5) during the subgradient iterations, or if all free facilities are fixed (open or closed).

= 1 and J{i = J{l,

(b) solve the MCNF associated with the set in order to update L5,

J{i

(c) pick the next facility in the list, namely i, (d) let j = j

+ 1 and

J{i

=

J{i- 1

Terminal Nodes

U {i},

As we mentioned before, once all facilities are open or closed, the CLPN reduces to MCNF, where the sources are the set of open facilities. It is well known that the solution of MCNF is relatively time-consuming, especially for large problems. Hence to avoid unnecessary computation and save time, we check, before solving the MCNF, if the incumbent set of open facilities has been encountered and solved before.

(e) repeat Steps b, c and d until we find that the solution value from Step b begins to increase. Note that all sets J{i found in this process can be saved to avoid solving a problem MCNF twice for the same set of facilities. The subgradient procedure is then carried out. At each subgradient iteration, we carry out the penalties. Each time p is halfed we recall the set of Lagragean multipliers associated with the best lower bound (L1) found so far . If any facility is closed, we carry out the reduction test for opening a facility. Finally we recalculate the Lagrangean solution (yr, xr) and attempt to find a new feasible solution for the problem. However , the subgradient procedure may not converge to an optimal feasible solution. We recall the set of Lagrangean multipliers associated with the best lower bound (L1), solve the Lagrangean relaxation with that set of multipliers and carry out all the reduction tests given in the section on problem reduction until no further reduction could be achieved.

COMPUTATIONAL RESULTS The algorithm presented here was programmed in PASCAL (Turbo Pascal 6.0) and the tests were executed on a microcomputer (386 DX - 25 Mhz). In order to measure the efficiency of our algorithm, we develop a set of tests, using a set of randomly generated problems. We proposed six groups and for each group eight different problems, see Table 1. Tables 2 and 3 summarize the results for the initial tree node and for all the tree search, applying the first lower bound for the test problems in group I. Tables 4 and 5 are similar to Tables 2 and 3 but they present the results for the second lower bound.

Intermediate Nodes At each subsequent enumeration tree node, we start with an initial set of Lagrangean multipliers equal to the set associated with the best lower bound found at the predecessor tree node. Then, following Beasley 's implementation (1988), the subgradient procedure of the initial tree node is replicated with the following differences:

The duality gap is given by 100(L5 - L1)/ L5. The total time presented in Tables 3 and 5 refer to the amount of time for the whole execution . Although we have proposed a great number of test problems, troubles with the memory size limit compromise the results for other groups. We are coding the algorithm in C and running on a Workstation. Better results have been obtained. In spite of hardware limitations, an analysis of the results point out the good performance of the algorithm. We are also comparing a third lower bound obtained by a new Lagrangean relaxation of the problem.

(a) we start with a value of p in equation (20) equal to 0.125, (b) only 30 subgradient iterations are carried out, with p being halved every 10 iterations, (c) the redution test for opening a facility is not used ,

68

TABLE 1 Number of Nodes for Each Problem Group Group I II III IV V VI TABLE 2 Problem 1-1 1-2 1-3 1-4 1-5 1-6 1-7 1-8

Number of Subgradient Iterations 35585 0 36919 35795 31287

Problem 1-1 1-2 1-3 1-4 1-5 1-6 1-7 1-8

1-1 1-2 1-3 1-4 1-5 1-6 1-7 1-8

Number of Facilities Fixed Open 9 9 9 8 8

Number of Transsphiment 5 15 30 50 100 250

Number of Free Facilities 6 0 3 4 3

Final

Final

PI

PtI

5 4 4 4 5

7 0

4 4

7 8

Total of Nodes 50 100 250 500 1000 2500

15 15 15 15 15

Duality Gap (%) 17.483 85.395 15.035 11.456 14 .628

Total Time (sec) 31522.08 12 .58 30827.48 32660.51 29315.17

15 15

14 .040 83 .156

33355.10 18 .79

Results for the Tree Search Procedure for the First Lower Bound Number of Total Total Time Number of Number of MCNFP of MCNFP's Open Facilities Time Tree Nodes Solved (sec) in Optimal Sol. (sec) 74 39.80 11 33369 .33 126 12.63 27 10.91 0 9 30988 .14 14 40 19.89 11 33089 .75 44.49 10 30 50 29496.47 14 29 27 .25 9

TABLE 4 Problem

Number of Customers 30 50 120 250 500 1250

Results of the Initial Tree Node for the First Lower Bound

35811 0

TABLE 3

Number of Facilities 15 35 100 200 400 1000

Number of Subgradient Iterations 1439 0 1658 330 330 2515 330 0

254 0

146 25

152 .02 17.09

37394 .38 18.84

10 8

Results of the Initial Tree Node for the Second Lower Bound Number of Facilities Fixed Open 9 9 9 8 8 6 7 8

Number of Free Facilities 6 0 3 4 3 9 7 0

69

Final

Final

PI

PtI

5 4 4 4 5 5 4 4

15 15 15 15 15 15 15 15

Duality Gap

(%) 11.156 1.458 7.705 7.035 5.728 15.734 10.867 0.288

Total Time (sec) 1452.95 13.34 1541.59 345.65 338 .56 2452 .64 342 .57 18.68

TABLE 5 Problem

1-1 1-2 1-3 1-4 1-5 1-6 1-7 1-8

Results for the Tree Search Procedure fo r the Secon d Lower B oun d Number of Tree Nodes 126 0 14 30 14 888 250 0

Number of MCNFP Solved 74

27 40 fiO 29 446 145 25

Total Time of MCNFP's (sec) 73.68 11.74 21.80 46.59 28.89 392.05 173.33 17.08

Number of Open Facilities in Optimal Sol. 11

9 11 10 9 8 10 8

Total Time (sec) 3358.42 13.45 1703 .35 766.81 520.36 15333.77 4384.15 18 .79

Krarup, J. , and P.M. Pruzan (1983). The simple plant location problem: survey and synthesis. European Journal of Operational Research , 12,36- 81.

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