empty freight vehicle movements are at the center of its preoccupations, which ... notation of Krarup and Pruzan [17] and will call it the (SPLP) problem. 2.
Section V
Planning Models and Applications
Annals of Operations Research, 18 (1989) 279-302
279
MODELS FOR MULTIMODE MULTICOMMODITY LOCATION PROBLEMS WITH INTERDEPOT BALANCING REQUIREMENTS Teodor Gabriel C R A I N I C t, Pierre D E J A X 2 and Louis D E L O R M E 3 1 Universit$ du Quebec ~ Montreal et Centre de recherche sur les transports, Universit$ de Montreal, Canada 2 Ecole Centrale de Paris, France 3 Centre de recherche sur les transports, Universit~ de Montreal, Canada
Abstract We present the problem of locating vehicle depots in an intercity freight transportation system with the objective of satisfying client demand for empty vehicles, while minimizing depot opening and operating costs, bidirectional client-depot transportation costs, as well as the costs of the interdepot movements necessary to counter the unbalancing of demand. We propose a class of models which has the form of a two-echelon multimode multicommodity location formulation with interdepot balancing requirements. We analyse the models and their properties to determine their underlying structure and characteristics to be used in subsequent algorithmic developments.
1. Introduction Our research was initially stimulated b y the following industrial problem: A distribution or transportation firm delivers products to its clients using different types of vehicles of various transportation modes: railcars, trucks, containers, etc. Client demand being assumed known, the firm knows the exact number of loaded vehicles to ship to each client. After delivery and unloading by the clients, the empty vehicles are shipped back to a warehouse or depot. The same or other clients also need e m p t y vehicles for subsequent loaded shipping with their own products. They order these e m p t y vehicles from the depots and these demands are also supposed k n o w n b y the transportation firm. After loading, the vehicles are shipped to their destination (normally one of the firm's network nodes). This shipment is carried out directly, or via the originating or another depot. The transportation system may use several modes (rail, road, combined m o d e . . . . ) and the transportation characteristics such as cost, travel time etc., are supposed k n o w n for each one of them. A major complication of the p r o b l e m is d u e to the Mailing address: Teodor Gabriel Crainic, Centre de recherche sur les transports, Universit6 de Montr6al, C.P. 6128, Succursale A, Montrb.al (Quebec), H3C 3J7 Canada. © J.C. Baltzer A.G. Scientific Publishing Company
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regional imbalance in empty vehicle availabilities and needs throughout the network: some areas lack vehicles of a given type, while others have too many of them. This requires the shipment of empty vehicles between depots for network balancing purposes. The global problem is therefore that of the location of the empty vehicle depots in order to satisfy client demand and to minimize the total costs of depot opening and operation and vehicle transportation from clients to depots, depots to clients and between depots. This problem belongs to the general class of strategic problems of planning logistics and distribution networks. Yet, compared with the classical location-distribution problem, where movements are unidirectional from plants through warehouses to clients, the present problem owes its originality to the fact that empty freight vehicle movements are at the center of its preoccupations, which implies that back-and-forth traffic between depots and clients, as well as between depots, has to be explicitly considered. We have not found any papers dealing specifically with this problem neither in the empty vehicle management literature [11] nor among papers dedicated to location-distribution problems [17,5,1]. The functional form of the transportation costs determine the problem type: linear or not and, in the non-linear case, concave, convex or otherwise. Linear functions are normally used. Yet, concavity, for example, may facilitate the modeling of the economies of scale usually imbedded into transportation tariffs. The problem may also be either deterministic or stochastic, depending on hypotheses concerning the nature of demand, supply and transportation times. We are currently working on the linear deterministic version of the problem. These hypotheses allow the separate planning of loaded and empty movements. We therefore concentrate our attention on the empty vehicle management problem, since it strongly interacts with the location of the depots. Dejax et al. [9] first treated this problem as part of a global study of the logistics system of a large European container transportation and distribution company. Their approach is to sequentially solve a classical depot location problem (using Erlenkotter's algorithm [15]), followed by a m i n i m u m cost flow problem to compute the interdepot movements. Shortest path methods are then used to optimize the distribution of loaded containers. Our research represents the continuation of their study and its general objective is to integrate the location and transportation problems. More specifically, this paper aims to present the various formulations of the problem and to analyze the models and their properties. Such an analysis is a crucial step in the development of efficient solution procedures, since this important problem, from both a theoretical and a practical points of view, has never been treated before. As a matter of fact, we are able to identify in this paper the underlying structure and characteristics of the proposed formulations which have led us to consider several efficient ways to solve the problem. The detailed presentation of the specific algorithmic developments are, however, beyond the scope of this paper.
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Yet, a few computational results, using a real life case, are shown, in order to illustrate the worthiness of the approach. In classical location theory, the class of problems which more closely relates to our problem is usually called " t h e simple plant location p r o b l e m " [17] or " t h e uncapacitated facility location problem" [5]. We will refer to this m o d e l using the notation of Krarup and Pruzan [17] and will call it the (SPLP) problem. 2. N o t a t i o n , h y p o t h e s e s and definitions
The current formulation considers e m p t y m o v e m e n t s only. We then assume, for each client, known supplies (equal, in fact, to the n u m b e r of loaded vehicles received), and d e m a n d s (equal to the n u m b e r of loaded vehicles that need to be shipped) of vehicles of each type. We also assume that e m p t y vehicles have to transit through a depot and that, accordingly, no client to client m o v e m e n t s are allowed. In addition, since a strategic planning horizon is implicit in the problem, supply and d e m a n d are assumed nonsimultaneous. Consequently, e m p t y vehicles are sent to the depot as soon as they are available a n d are not kept by the client to satisfy future needs. Let C = {i Iclient}
n=lCl D = { j Icandidate depot} m = IOl
P = ( p I type of vehicle-commodity}
=lel T = ( t Itransport mode}. These elements make up a conceptual network G = (N, A), that generally parallels a physical network. Nodes are defined by clients and candidate depot sites: N = C U D. Arcs represent all possible movements, by all modes, between depots, from clients to depots and conversely: A c ( C x D X T)U(DxDX T) U (D X C x T) As client to client movements are not allowed, arcs in the C x C set are not defined. Furthermore, for any forbidden m o v e m e n t between a client a n d a depot, the corresponding arc is assumed not included in A. We now define O,p: supply at n o d e i of vehicles of type p, i ~ C, p ~ P; Dip: d e m a n d at node i of vehicles of type p, i ~ C, p ~ P; y = (y j): opening vector for depots 1 0
if depot i is open ~ D J YJ= otherwise x = (Xup,, Xjgpt): vector of type p vehicle flows, on m o d e t, between client i and depot j, i ~ C , j ~ D , p ~ P , t ~ T;
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T.G. Crainic et al., Models for multimode multicommodity LPs
w = (Wykpt): vector of type p vehicle flows, on mode t, between depot j and depot k, j, k ~ D , p ~ P , t ~ T ; fy(y, x, w): cost function to open and operate depot j, j ~ D; cijp,(x) and cjipt(x): transportation cost functions of vehicles of type p, on mode t between client i and depot j, i ~ C, j ~ D, p ~ P, t E T; sjkpt(w): transportation cost function of vehicles of type p, on mode t, between depots j a n d k, j, k ~ D , p ~ P , t ~ T . When linear costs and no capacities are assumed for the transportation links (as in the present formulation), traffic moves on the mode with the lowest unit cost. Consequently, we may simplify the notation by assuming that between each pair of nodes only one arc exists, in each direction, representing the mode (or mode combination) with the lowest unit transportation cost. The subscript t may then be omitted and we have:
Cijp(X ) = ci]pxiy p
i ~ C, j ~ D, p ~ P
Cjip(X ) = CjipXji p
j ~ D, i ~ C, p ~ P
sjkp(w )=sjkp~jk p
j, k ~ D ,
f j ( y , x, w I =fjy,
j ~ D.
p~P
We suppose that all unit costs are nonnegative. When this is not the case, simple transformations with no impact on the optimal solution may be used [17] to ensure this.
O; ~
7
kd
> F--l:-->
....
(cLients)
...... =>W-] ..... :-.>.
~-...... L____At-...... LLA ( ] S - t_LL_A~O xjip>~O Wjkp>~O
forall i ~ C , j ~ D , p ~ P for all j, k ~ D , p ~ P
y j ~ (0, 1}
for all j ~ D .
(5)
Constraints (1) ensure that demand and supply requirements are met. Constraints (2) forbid exchanges between clients and closed depots ( M is a suitably large constant), while constraints (3) do the same for interdepot movements. Finally, constraints (4) correspond to the interdepot balancing requirements.
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Under somewhat mild hypotheses, the model may be considerably simplified. Supply and demand may thus advantageously replace the constant M in constraints (2), also tightening the feasible domain. Additionally, constraints (3) may be eliminated if interdepot transportation costs verify the triangle inequality. Formally: PROPOSITION 1 If the interdepot transportation cost functions verify:
s~kp / 0 f o r a l l ( r , s ) ~ A , p ~ P which is a minimum cost multiflow problem. [] Note that L(5;)) can be decomposed into I mutually exclusive minimum cost flow subproblems, one for each commodity. It is frequently necessary to compute a lower bound on the optimal solution of problem (L) by relaxing constraints (5). We define then the strong and weak linear relaxations of (L), denoted by (SRL) and (WRL) respectively, as the following problems: Minimize z = j~EDEfjYj + pEe y'~ [ ~.~ Y~.
(CijpXij
p Jr CjipXjip) -]- ~
~ SkjpWkjp j~D kED
i~C j~D
subject to E Xijp = Oip jED
for a l l i ~ C , p ~ P
E Xj.tp = Dip j~D
(SRL)
Xup i 0 yj>_.o
i~C
wjkp = 0
forall j ~ D
for all j ~ D ,
(9)
p~P
kED
for all i ~ C , j ~ D , for allk, j ~ D ,
p~P
p~P
for all j ~ D.
The equivalent formulations for the normalized model are called (SRLN) and (WRLN) respectively.
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Problems (SRL) and (L(fi)) are almost similar, except for constraints (8), which do not appear in the latter. These constraints may be seen as variable arc capacities on the xij p and Xjip flows. A problem such as (SRL) is usually more difficult to solve than standard minimum cost flow problems [18,19]. This is not, however, the case for (WRL); simply note that at the optimum we always have:
Y/= E
~ (Xup+Xj~p)/A
forall j ~ D
(10)
pEP i~C
where
A= E E (O,,+D,,). pEP iEC
This result is due to the fact that y/appears in only one type (9) constraint in the problem and that its cost, fj, is nonnegative. Thus, we can rewrite (WRL) by substituting the-expression defined in (10) to yj, to obtain an equivalent problem (WRL1): (WRL1) Minimize z= E [ Y'. Y'. ((C,yp + f J A ) x i j p + (cyip ' [ - f j / A ) X j i p ) pEP [ iECjED
+ E E skj w j l kED jED
J
subject to E Xijp = O~p jED
for a l l i ~ C , p ~ P
E Xjip = Dip j~D
Y'. xij p + E wkj - E
kED iEC Xijp, Xjip, Wkj p ~ 0
iEC
E wjk.=0
for a l l j ~ D ,
p~P
kED
for all i ~ C, k, j ~ D ,
p~P.
This problem has the following propriety: PROPOSITION 4
(WRL1) is an uncapacitated minimum cost multiflow problem.
Proof Simply note that (WRL1) and (L(.~)) have similar forms. The proof is then analogous to the one developed for proposition 3. [] (WRL1) can also be decomposed, by commodity, into ~ mutually exclusive minimum cost flow subproblems.
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An upper bound on the optimal solution of (L) may be obtained by defining a feasible solution (y*, x*, w*) for (L) as follows: YP -- ._{113 otherwiseif there exists at least i E C and p ~ P , such that x~.p>Oorxfip>O for all j ~ D, where (x*, w*) is an optimal solution for problem (WRL1). Note that this bound may not be very tight. The (WRL1) formulation is actually used to compute bounds in a branch-andbound algorithm [7,8]. The problem is decomposed according to commodities, each resulting in a minimum cost flow subproblem being solved by using RNET [16]. A FORTRAN implementation of this algorithm has been used to solve several test problems, including a real case application with 90 candidate depots, 300 clients, 12 commodities and a network of 5,130 multicommodity arcs. This problem has been solved in 12 hours on an PC/AT personal computer equipped with a DSI/32 coprocessor card. Constraints (7) in problem (L) are "complicating". To dispose of them, two approaches may be used: the penalty method [20] and the Lagrangian relaxation of the problem [2, chap. 4]. The first approach gives rise to the following formulation, denoted by (PRL): (PRL) Minimize
Z = E fJYJ'q'- p~P E [ iECj~D E E (CiJpxijp'q- CptjXpJi') + E E E [Kijp(Min{O, Oip)J- Xijp p~P i~C j~D
k,j~OE SkjpWkjp]
})2
+ Kj,p(Min{O, D,pyj-Xjip })2] subject to
E Xijp = O,p jED for a l l i ~ C , p ~ P E Xjip ~- Dip j~D E Xijp-}- E Wkjp-- E Xjip-- ~. wjkp=O i~C k~D i~C k~D Xijp, Xjip, Wkjp ~ 0
(0,1}
for all j ~ D ,
p~P
f o r a l l i ~ C , j, k ~ D , p ~ P for all j ~ D
where Kijp and Kji p are strictly positive penalty constants. This problem has the following propriety when dropping the zero-one constraint on the y variables:
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PROPOSITION 5 The strong relaxation of (PRL) is an uncapacitated minimum cost multiflow problem with convex are costs.
Proof Note first that the constraint sets of the relaxed problem and of problem (L(y)) are similar. Thus, by proposition 3, (PRL) has a minimum cost multiftow network structure. In addition, the last part of the objective function, related to the penalty functions, is clearly convex; the objective function of (PRL) is then convex, which completes the proof. [] An attractive approach to solve (PRL) is to adapt the relaxation techniques developed by Bertsekas and Tseng [3,4] for minimum cost network problems. We are currently pursuing this avenue and the preliminary results are very encouraging. The Lagrangian relaxation of problem (L) with respect to constraints (7) has also the structure of a minimum cost multiflow problem. Details may be found in [6]. This approach also offers interesting algorithmic perspectives that we plan to explore in the near future.
5. Dualization
To characterize the dual properties of the initial problem (L), let us consider the strong linear relaxation of the normalized problem: (SRLN) Minimize z e ( y , x, w) = j~D
p~P
+ ~
i
j
~ SkjpWkjp
kED j E D
subject to
xij p = 1
for all i ~ C, p ~ P
(uip)
E Xjip = I jED
for all i ~ C, p ~ e
(Oip)
x~jp10
=
E i~C
-
for all j E D ,
pep
for all k, j ~ D ,
peP.
(LN( ~, ~)), as (L(y)) and (WRL1), can be decomposed into ¢ mutually exclusive minimum cost flow subproblems, one for each commodity. Let ~ be a n optimal solution for this problem. We have enough information to compute bounds on the optimal solution of (SRLN), z e ( y *, x*, w*):
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T.G. Crainic et al., Models for multimode multicommodity LPs
PROPOSITION 6 zP(fi, ~ l X ) ~ < z e ( y *, x*, w*)~ 0 =, y~ = x~jp and
.r)ip > 0 ~ yj = xj~p y/> x~j.p ~ y,.j.p= 0
for a l l i ~ C ,
j~D,
p~P
Yj > Xjip :=~ ~[jip = 0
~+jp and ~y~pcan then be seen as prices to be paid in order to increase the value of yy, i.e. to "open" depot j. Let (Y, ~) be an optimal solution for (SRLN(~)), and let y be defined as follows: = } for a l l j ~ D . y=j = Max {= xiip, xj~p i~C,p~P
Note that .~y will always be lesser or equal to 1, because Xiyp and xyit, are such that: E xiip = 1 and E x i i p = l f o r a l l i e C a n d p ~ P . j~D
jED
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T.G. Crainic et al., Modelsfor multimode multicommodity LPs
We can again compute bounds on the optimal solution of (SRLN),
ZP(y *, x*, w*) using the above information: PROPOSITION 8
Proof (i) First, recall that zP(Y, ~ 17) is the optimal value of the objective function of (SRLN(~)); thus it is also equal to the optimal value of the objective function of its dual (DLN(Ft)). It follows that ze(~, w ly) is a lower bound on the optimal solution of (DLN), and hence on zP(y *, x*, w*). (ii) On the other hand, (.P, ~, ~) is clearly a feasible solution for problem (SRLN); consequently, ze(.P, ~, ~) is an upper bound on the optimal solution of (SRLN), i.e. ze(y *, x*, w*). 0 Let us close this section by mentioning that problems (SRLN(~)) and (SRLN(~)) may be used in common to compute bounds on the optimal solution of (LN). As a matter of fact, we have developed an algorithm [13,14] where they are solved cyclically (SRLN(~) by DUALOC [15] and SRLN(~) by R N E T [16]), using as input for one problem the output of the other, previously solved. This algorithm is very efficient. The same real case application problem, under the same computing conditions, was solved in 4 hours. It also attains very good solutions (less than 0.5% from the lower bound) even if its convergence properties are not well defined yet.
6. Application The models presented in this paper were initially developed as part of a strategic and tactical planning system for a large European maritime container transportation and distribution company. The company operates worldwide maritime lines to and from 23 major European ports. Arriving ships carry containers of two sizes and about ten major types each, loaded with imported goods, as well as empty containers returning after exports. The loaded containers must be further transported to their final destination, usually an industrial firm located somewhere in the Western European continent. Transportation modes are of various types, including railroad, trucking, barges, and mixed modes (railroad from the port to an inland terminal and then trucking, for example). The empty containers may be dispatched wherever they are needed for exports (see below).
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After imports, the unloaded empty containers may either return to the port of origin or be transported to any depot (port or inland terminal). Similarly, exporting industrial firms need empty containers to be delivered to them, from the export port or any of the aforementioned locations. After loading, the container is transported to the port either directly or through a mixed mode. Export containers are loaded on ships together with empty containers sent abroad to cope with the worldwide trade unbalance of certain products and container types. Regional trade unbalances in Europe require the ground transportation of empty containers directly between depots, in addition to transportation between depots and clients. In the year 1986, among 130 possible locations over five European countries, 87 ports or inland terminals have been actually used by the company. A total of 240 000 ground container shipments have been made, of which 95 000 (40%) were empty movements. In order to propose a strategic/tactical plan for this problem that would go beyond the initial sequential approach of Dejax et al. [9,10], we wanted to design a global model for the empty container transportation problem. This is what led us to identify the class of location models with interdepot balancing requirements, the mathematical characteristics of which have been presented in this paper. In our application, clients have been aggregated into 300 client zones, and 12 container types and 9 transportation modes or tarification types have been considered. The goal of the study was to determine the depots to use and the corresponding flows between depots and clients, as well as between depots, in order to meet the demand for empty containers while minimizing the total cost of transportation and depot operations. The depots (either ports or inland terminals), serve for the collection, storage and distribution of the empty containers, as well as break points for mixed mode transportation [12]. Both, the branch-and-bound approach and the dual-based optimization method, have been used to propose solutions to the problem, that would be implemented by the European firm in preparation to their short term planning and real time dispatching decisions. The conclusions call for a significant decrease in the number of depots to be permanently used (48 instead of 87), as well as important changes in the client to depots associations for empty container transportation after imports or before exports and specific recommendations for the inter-depot balancing flows. This strategic/tactical plan results in an important (47%) saving in annual cost. These savings have to be considered with care since the strategic/tactical plan is based upon a number of aggregation hypotheses and is only intended as a framework for improving the actual operations in terms of what inland terminals to operate and what logistics (client-depots) zones to design, both for imports and exports. It is also intended as a tool to simulate future policies in terms of transportation modes, tarifs, trade-offs between maritime and ground transportation policies and costs. However, we believe that these results are significant enough per se to demonstrate the usefulness of building the model developed in this paper.
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7. Conclusions We have presented and analyzed the properties of a formulation of a multimode multicommodity location-distribution problem with interdepot balancing requirements. The model displays a number of theoretically and practically very interesting characteristics. A network structure emerges continually throughout the analysis, not only in the model itself, but also, and most importantly, in all subproblems, relaxations, dualizations, etc. Given the richness and the well-known general efficiency of network models, this greatly facilitates the development of efficient algorithms for our problem. In fact, we believe that it is more appropriate to develop specific algorithms for this problem than to use traditional algorithms for location models with general additional constraints. As for the algorithmic approach, several methods are possible and may advantageously build u p o n the structure and properties displayed by our model: enumeration, decomposition, dualization, relaxation, etc. We are presently exploring some of these avenues and others will be undertaken soon. Preliminary results were presented in this paper and more will be communicated as soon as they become available. Two extensions of the linear model present considerable interest and will be studied further: the integration of both loaded and empty movements into a single formulation and the consideration of capacities at all, or some, depots. And, of course, we also plan to look, in the near future, into the stochastic and nonlinear aspects of the problem.
Acknowledgements We wish to thank the following institutions which have financially contributed to this project; Programmes de cooprration France-Qurbec, Natural Sciences and Engineering Research Council of Canada, te Fonds F.C.A.R. du ministrre de rEducation du Qurbec pour l'aide et le soutien h la recherche. We also want to thank two anonymous referees whose comments have contributed to the improvement of the presentation of this paper.
References [1] C.H. Aikens, Facility location models for distribution planning. Eur. J. Oper. Res. 22 (1985) 263. [2] D.P. Bertsekas, Constrained Optimization and Lagrange Multiplier Methods (Academic Press, 1982). [3] D.P. Bertsekas, A unified framework for primal-dual methods in minimum cost network flow problems, Mathematical Programming 32 (1985) 125.
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[4] D.P. Bertsekas, P.A. Hossein and P. Tseng, Relaxation methods for network flow problems with convex arc costs, SIAM J. Control and Optimization 25 (1987) 1219. [5] G. Cornuejols, G.L. Nemhauser and L.A. Wolsey, The uncapacitated facility location problem, in: Discrete Location Theory, eds. R.L. Francis and P. Mirchandini (Wiley-Interscience) forthcoming. [6] T.G. Crainic, P. Dejax and L. Delorme, Probl6mes de localisation-distribution avec 6quilibrage: 616merits de mod~lisation et propri6t6s, Publication #490, Centre de recherche sur les transports, Universit6 de Montr6al, 1986. [7] T.G. Crainic, L. Delorme and P. Dejax, A branch-and-bound approach for location/ distribution problems with balancing requirements, Paper presented at the TIMS/ORSA National Meeting, New Orleans, 1987. [8] T.G. Crainic, L. Delorme and P. Dejax, A branch-and-bound procedure for location/ distribution problems with balancing requirements, Centre de recherche sur tes transports, Universit6 de Montr6al, 1988. [9] P.J. Dejax, Thy. Durand and F. Servant, A planning model for loaded and empty container ground transportation, Paper presented at the TIMS/ORSA National Meeting, Los Angeles, 1986. [10] P.J. Dejax and F. Servant, Trois application significatives d'un mod61e de localisation de d6p6ts, Annals of the First International Meeting in Industrial Engineering and Management in France, AFCET, Paris, (1986) 587. [11] P.J. Dejax and T.G. Crainic, A review of empty flows and fleet management models in freight transportation, Transportation Science 21 (1987) 227. [12] P.J. Dejax, T.G. Crainic and L. Delorme, Strategic-tactical planning of container land transportation systems, Paper presented at the EURO IX-TIMS XXVIII Meeting, Paris, 1988. [13] L. Detorme, T.G. Crainic and P. Dejax, A dual based algorithm for location/distribution problems with balancing requirements, Centre de recherche sur les transports, Universit6 de Montreal, 1988. [14] L. Delorme, T.G. Crainic and P. Dejax, Dual methods for location/distribution problems with balancing requirements, Paper presented at the TIMS/ORSA National Meeting, Washington, 1988. [15] D. Erlenkotter, A dual-based procedure for uncapacitated facility location, Oper. Res. 26 (1978) 992. [16] M.D. Grigoriadis and T. Hsu, The Rutgers minimum cost network flow subroutines (RNET documentation), Department of Computer Science, Rutgers University, 1980. [17] J. Krarup and P.M. Pruzan, The simple plant location problem: survey and synthesis, Eur. J. Oper. Res. 12 (1983) 36. [18] L. Schrage, Implicit representation of variable upper bound in linear programming, Mathematical Programming Study 4 (1975) 119. [19] L. Schrage, Implicit representation of generalized variable upper bounds in linear programming, Mathematical Programming 14 (1978) 11. [20] W.I. Zangwill, Nonlinear programming via penalty functions, Management Science 13 (1967) 344.
Annex H e r e follows the p r o o f of p r o p o s i t i o n 1 of section 3, which states t h a t w h e n the i n t e r d e p o t t r a n s p o r t a t i o n cost f u n c t i o n s verify
Sikp
