Discrete Applied Mathematics 13 (1986) 109-130

North-Holland

109

SPATIAL MARKET EQUILIBRIUM PROBLEMS AS NETWORK MODELS

Oscar BARROS and Andr6s WEINTRAUB Departamento de Ingenieria Industrial, Facuitad de Ciencias Fisicas y Matematicas, Universidad de Chile, Chile

Received September 1984

A series of equilibrium models for the agroindustrial sector is presented. The use of these models for governmental policy decisions is shown. In all cases, there is an inherent network structure, varying from a simple network flow problem to networks which are multiproduct, generalized and with additional constraints. We show solution techniques used to handle the

problems, and discuss in each case the advantages of specialized network algorithms.

1. Introduction

Since Samuelson [45], Takayama and Judge [49] introduced the notion of finding equilibrium solutions to spatially separated markets, through solving an optimiza- tion problem to maximize net social pay-off [51], a number of applications have come up, analyzing markets that range from energy to poultry.

The basic problem under study is to find sets of prices for spatially separated supply and demand points - represented by corresponding supply and demand functions - that satisfy the economic conditions of equilibrium [45]. A specific case of this problem is to find the equilibrium in a single-commodity transportation net- work with elastic (price dependent) supply and demand [5,20].

The original approach proposed by Takayama and Judge was to solve this prob- lem using quadratic programming. In such a case, supply and demand functions are restricted to be linear, i.e., price is a linear function of quantity. Several practical applications were made of this method [9, 10, 26, 31,48, 51]. In particular Judge and Takayama report applications to live-stock products in Japan, to broiler markets in the USA, to winter oranges market in Europe, to the world wheat market, to the banana market in Australia, to the north east milk market in the USA and to the rice and wheat market in India [31].

Lately these types of models have become increasingly popular, since they can be used to reproduce in the computer the workings and results of a competitive market under different assumptions about overall economic environment and policy options, and consequently support sectorial economic policy decisions [8, 14]. Thus, new solution algorithms for these problems have been proposed which represent im- provements, both in terms of computer efficiency and flexibility in modeling.

0166-218X/86/$3.50 © 1986, Elsevier Science Publishers B.V. (North-Holland)

110 O. Barros, A. Weintraub

Amongst others, the following methods have been proposed: linear programming [38], Dantzig-Wolfe decomposition [46~ fixed-point models and algorithms [28, 39, 34, 35], linear and nonlinear complementary programming [2, 36, 42, 46, 50] and Benders Decomposition [15, 44]. However there has been little empirical testing of these methods.

One method which has been applied in practice with good results is the one in the PIES model [16, 17]. This method uses a diagonalization approach and solves a sequence of convex programming problems [3, 18].

Despite the interest in these models and their intrinsic network structure, few pro- posals have been made to use network flow algorithms in solving this problem.

Glassey proposed a network algorithm for the single commodity problem based on principal pivoting [22].

Pang and Lee [41] presented a network specialization of parametric principal pivoting for the single commodity problem, extended by Pang [39] in an iterative scheme for the multicommodity case.

Asmuth, Eaves and Peterson [4] proposed solving an equivalent complementary problem using Lemke's algorithm for the multicommodity case.

For the single commodity case, Jones, Saigal and Schneider [30] proposed a variable dimension homotopy algorithm. This approach increases by one the mar- kets brought to equilibrium in each iteration, exploiting dominant market structures and is implemented using the underlying network structure.

There is small evidence to compare the relative advantages of these algorithms. Moreover, the algorithms are implemented on either small numerical examples or randomly generated networks and no application on a real life situation has been reported.

This paper departs from the theoretical approach to algorithm development pre- sented in the papers above in that we focus on solution techniques for solving real problems. Since 1976 we have been involved in the study of market equilibrium problems, through the analysis of the agriculture and agroindustry sector, specifi- cally, the sugar, wheat and corn, edible oil, dairy products and fruit markets [6, 7, 8]. These projects were developed with the purpose of aiding in government policy making in these areas, on issues such as price fixing, subsidies, tariffs and detection of monopolistic practices.

The problems we solved share a common structure: raw inputs are produced in certain agricultural zones, usually by many small producers. These are aggregated geographically into representative major supply centers, each having a supply curve reflecting producers' alternatives and costs. From these, products are transported to plants for processing (with possible intermediate storing in warehouses). Final products are shipped to demand centers, where demand curves reflect consumer utility functions. Historic and present data is used to construct supply and demand functions, hauling and production costs.

In representing each market through models, with their peculiar characteristics, we concluded that there was an inherent network structure in every case. To capture

Spatial market equilibrium problems as network models 111

this characteristic is important for two reasons. Having a basic network structure aids in the characterization, modeling and analysis of the problem, given the insight it provides. This is particularly important when interacting with users with low mathematical sophistication, as was our case. The second, perhaps more obvious reason for taking advantage of network structures, is for computational purposes, given the extreme efficiency of network based codes. We proposed early on the potential of using the very efficient network flow algorithms for solving this type of problem [6]. We were guided in this idea by the good experience gathered from these methods for solving the similar traffic assignment problem, which have been extended to consider variable demands and interactions among different flows [1, 11,20, 27, 43].

There is a trade-off to consider here. Using a special purpose algorithm to exploit specific structures offers obvious computer cost advantages, highly desirable in large scale, often used systems. In one case we solved, the reduction of CPU time provided by a network code allowed us to work sets of problems on line. This gain however is obtained at the cost of possibly rigidizing somewhat the problem formu- lation and, in some cases, requiring the development of specific codes when the problem in network structure does not correspond to a straightforward case. We will refer to such options when discussing different markets.

The complexity of the models used depends on the type of market. In one case the market equilibrium could be represented as a single commodity network flow problem. Other markets led to models of multicommodities with cross elasticities at demand centers, non-network constraints and gains on arcs. The decision on whether to develop specific codes will often depend on its cost and the time available to produce a working tool.

We will discuss these issues, including the development of necessary software packages for users, and the type of policy analysis carried out for different markets. Section 2 covers the sugar market, Section 3 the dairy market and Section 4 the wheat-corn market. We treat the markets for which networks were actually used in more detail and give a general description of the other markets, discussing the potential of using network algorithms.

2. The sugar market

2.1. Description of the problem

The main participating elements or agents for this market are:

- Beet producers, a large number of small farmers distributed over an area of roughly one thousand by one hundred kilometers. They normally sell their produce to the nearest processing plant due to the transportation cost structure imposed by the linear geography of the region. For modeling purposes these farmers are grouped into four supply centers using geographical and accesability criteria. From

112 O. Barros, A. Weintraub

the econometrically estimated overall supply function, a characteristic supply func- tion is derived for each center h in a given year, which will be used in its inverse form:

Ph = fh(Uh)

where Ph is the price paid at center h and uh the beet it supplies 1.

- Ports, which act as supply centers of imported products that are competitive to sugar beet, namely crude cane sugar and refined sugar. There are four of these. Obviously, the supply functions for ports are constant, defined by the sugar import price plus tariff. Tariff is a government policy option.

- Plants, which process either sugar beet or crude cane sugar to produce refined sugar. There are six plants, four of which were state owned at the time of the study 2. This is a monopsonic situation due to the close attachment of beet pro- ducers to plants. Hence, beet purchase price is a policy decision, which will affect the level of beet supply according to the corresponding supply function for farmers. Clearly this decision is related to the price of imported sugar, including the tariff. Plants are characterized by their capacity and the variable costs of transforming beet and /o r crude cane into refined sugar. These costs have been determined to be roughly linear.

- Demand centers, which are groupings of many final consumers of sugar. Nine- teen of such centers were defined along geographical lines. Center i is characterized by an econometrically derived demand function of the following type for sugar (pre- sented in its inverse form):

Pi =gi(z i )

where Pi is the price charged for sugar at center i and zi the quantity of sugar it demands.

- Transportation network, which connects all the above elements. Transporta- tion costs have been determined to be linear.

Relationship among the elements defined are shown in Fig. 1. The natural cycle for this problem is one year, for there is no storage from one

year to the following, except for safety purposes. Within a year, production is mostly concentrated during the sugar beet harvesting season, so there is no need for defining multiple periods.

I The supply and demand functions for this market and others we will deal with contain many para-

meters which we will not state explicitly to avoid defining many different exogenous factors that were

determined to affect supply and demand. 2 Currently, all plants are run by the state.

Spatial market equifibrium problems as network models 113

SUPPLY FUNCTION

H Ph

t/ Uh= ~Ej Xhj

SUPPLY FUNCTION

t . Uh' H'TH CRUDECANE SUGAR

SUPPLY CENTER (PORTS)

FROM OTHER SUPPLY CENTERS

TO OTHER TO OTHER PROCESSING DEMAND

P L A N T S , C E N T E R S

TH SUGAR BE j TH PROCESSING I TH DEMAND SUPPLY CENTER,/ PLANT / CENTER

FUNCTION

Fig. 1. Sugar market

Pi

DEMAND FUNCTION

z i = ~ x j i

In establishing a model for this problem, it is clear that the most relevant decision maker is the government. The model should help authorities to set, in a consistent way, the price that state plants should pay farmers, and the tariff for imported pro- ducts. In making these decisions, there are several objectives that can be pursued:

(i) If government follows a free market philosophy, even if it owns the processing plants, it can try to determine the prices that a competitive market would generate and set them accordingly.

(ii) If government wishes to maximize its revenues, it can use its monopsonic power and determine prices to do so.

(iii) Government may wish to subsidize either producers or consumers and deter- mine prices accordingly, evaluating the economic impact of the decision.

2.2. The model

We present a simplified version of the model, which considers only the essential technical characteristics. For a more detailed version see [7]. Since technical trans- formation coefficients are similar for all plants, all flows will be measured in refined sugar equivalent in the model. Note that this definition is fundamental in allowing to define a single commodity to model the problem.

1 1 4 O. Barros, A. Weintraub

Variables

Xhj: amount of sugar equivalent going from beet supply center h to plant j .

xh,j: amount of sugar equivalent going from port h' to plant j .

xji: amount of sugar equivalent going from plant j to demand center i .

Parameters

CAPj: capacity of plant j .

Chj : unit transportation cost from center h to plant j .

Cji : unit transportation cost from plant j to center i.

kj : unit production cost at plant j .

Constraints

Typical flow conservation equations at plants are:

E Xhj + E Xh,j= ~. Xji , Vj. h h"

Capaticy constraints at plants are:

xj= E xji_<cAPj, vj. I

Total amount produced at supply center h and total amount demanded at center i are respectively:

Uh = Xhj, Y h . J

Zi "= E Xji, Vi. J

These are unknown variables which should satisfy supply and demand functions respectively.

Objective function

To obtain values for the variables that correspond to the pure competition equi- librium, it is well known that the following net social pay-off function should be maximized [43,49]:

i Z = ~i gi(~i)d~i- ~ J~(~h) d~h " 0 h

- -E k jx j - - E ChjXhj-- ~.. Cjixji j h,j J.I

Spatial market equilibrium problems as network models 115

where the first two terms are respectively the consumers and producers surplus; the third, production costs; and the fourth and fifth correspond to transportation costs.

From Fig. 1, the inherent network structure of the problem becomes apparent. By defining all variables in pure sugar equivalent, the model results in a single com- modity pure network flow problem with convex costs in the arcs.

2.3. Solution techniques and implementation

It is obviously advantageous to use a special purpose network code to handle this problem. In addition, in order to avoid solving a problem with nonlinear objective, supply and demand functions were step linearized in a conventional way (see Fig. 1). For supply and demand functions a linearization with fourteen steps was considered an adequate approximation. Clearly one variable needs to be defined for each step. Convexity of the cost function assures consistent optimal solutions.

A primal simplex specialized code was developed for this case along the lines of [24]. In order to facilitate the use of the model, an on-line Decision Support System was implemented.

The basic idea of the Decision Support System is to give proper facilities not only for running the model but for generating, updating and processing the information needed for setting it up [8]. Typical functions performed by the computerized sys- tems are: displaying current values of and accepting changes in data used by the model, e.g. a parameter of a demand function; setting up relationships, e.g. taking an updated value for a parameter and recalculating the new form of a demand func- tion; calculating model coefficients, e.g. taking the new formula for a demand function, performing a step-linearization on it and calculating the integral of the linearized function to determine the coeficients of the objective function; running the model with recalculated coefficients; and displaying results in an easy to in- terpret fashion. The system is processed at a large government central computer center and accessed and operated by means of a remote minicomputer terminal located at the user's place of work. Typically, a user would like to test the con- sequences of a change in one of the parameters (policy decision), say the tariff on imported raw sugar. This is done interactively. The user revises in the display the status of the input data and parameters. He then initiates the set-up of relationships and calculation of coefficients of the linearized network flow model, which are displayed on completion of the calculations. If satisfied, he initiates the solution process of the model.

For each case, the tables of results include the following principal variables: quan- tity supplied and producers' gross and net revenues by location, rate of capacity utilization by processing plant, gross and net revenue by plant, volumes imported by type of sugar and port, price to consumers by location and profits of importers. An example of the results is shown in Table 1.

Typical problems run were 100 nodes and 1000 arcs. Solution CPU times were in the order of a few seconds at an IBM 370-145. The fast response obtained by

116 O. Barros, A. Weintraub

Table I. Sample outputs for sugar sector model

Solution definition b Beet price Domestic Consumer Producer Revenue less

(US$/ton) production and producer income variable costs,

(000 tons) surplus all plants

1. Max. net social pay-off 37.0 251 +6.3 n.c. a n.c. a

2. Max. gross revenue less

var. costs, all plants 28.7 137 +1.3 +0.4 +0.9

3. Beet price fixed at 38.5 38.5 208 +2.6 +9.4 -6 .8

4. Beet price fixed at 37.0 37.0 237 +5.0 +10.2 -5 .2

5. Beet price fixed at 34.0 34.0 213 +4.8 +6.2 -1 .4

6~ Beet price fixed at 31.0 31.0 159 +2.7 +2.7 0.0

7. Beet price fixed at 28.0 28.0 117 0.0 0.0 0.0

a n.c. = not calculated.

b Quantity of refined sugar fixed at 401,000 tons in solutions 2-7 but allowed to vary in solution 1.

using a network algorithm allowed the users to work on line. The model can be solved to simulate various pricing regimes: competitive, mono-

polistic-monopsonistic (maximizing net plant revenues), various combinations of the above and fixed price ranges. Different objectives can be easily considered by modifying the objective function. For example to maximize the plants' net benefit (monopsonistic/monopolistic point of view) supply functions are replaced by the marginal cost curves of beet production and demand functions by the marginal return curves. Other changes in the objective function allow to consider minimum support prices, subsides and other policy actions without changing the structure of the problem and hence the solution procedure.

A common use of the model is in deciding on the price to be paid to the beet farmers. Typically, hundreds of different runs are made to consider different scenarios of international market sugar price, tariff and pricing policies. Hence, the importance of providing the fast response guaranteed by a network flow algorithm. Of course this would have not been possible with a general purpose L.P.

3. The dairy market

3.1. Description o f the problem

The main elements to consider in this market are:

- Fluid milk supply centers, representing aggregated farm producers with econo- metrically derived associated (inverse) supply functions:

phi t t = f~t(Uht), Vh, l,t

Spatial market equilibrium problems as network models 117

where ptht is the price of input 1 in milk 3 at center h in period t and utm, the total amount supplied by center h.

- P l a n t s , where fluid milk is processed into 9 products (pasteurized milk, powder milk, yoghurt, cheese, etc.). Cost functions for plants are marginally increasing, i.e. total cost for plant j is:

¢(rJ) where rJ is production volume of plant j in period t and ¢( . ) is convex.

- P o r t s , through which powder milk is imported, paying a given tariff.

- D e m a n d cen ters , where the nine products interact, leading to demand functions with cross elasticities:

P[k t t = g i k ( Z n , . . Z[k , . . Z[m), Vi , k, t

where ptik is the price of product i at center k in period t and z[k is the quantity demanded of product k at center i in period t.

- S t o r a g e for some products (cheese, butter).

- T r a n s p o r t a t i o n n e t w o r k .

The relationship among these elements is shown in Fig. 2. This problem has mul- tiple periods in a one year cycle to account for seasonality in milk supply.

Since plants are privately owned, this situation is basically a competitive one. Hence, the objective in setting a model is to predict the competitive equilibrium this market will attain under given conditions. Policy decisions that will affect the equi- librium are:

(a) Tariff on imported milk. (b) Minimum support prices established by the government for farmers. (c) New infra-structure; i.e. new roads and/or plants and/or technologies (e.g.

long life milk) planned by government or private agents. The government is also interested in determining market imperfections. This can

be done by comparing model outputs with real life situations; thus finding out, for example, that there is a monopoly pricing in a given submarket, such as yoghurt sales in a given region.

3 Although milk is apparently just one product, in reality it contains two basic elements besides water, fat and non-fat matter. These are separated in the process and used on their own in different products

(e.g. fat to produce butter).

I 18 O. Bar ros , A . Weintraub

PRODUCTS STORED FROM PREVIOUS PEr . . . . PERIOD t - 1

PERIOD t

SUPPLY FUNCTION

TO TO OTHER PROCESSING PLANTS

P~I H TH MILK / J THpRocESSIN ITHDEMAN D PRODUCER OR/ PLANT CENTER /

'MpORter I / , I h ~/ XhJ I .I j

+

J t _-~£.x t

Ul~l J hjl

PLAT r COST FUNCTIOI

FROM OTHER MILK PRODUCERS

//

t =9ik (z l l .... Ztlm )

t ' t =~Ej .t' <_tXjik

PERIOD t . l

PRODUCTS STORED TO PERIOD t'

Fig. 2. Dairy market.

3 . 2 . ~ The model

To develop a model representative of this sector, we define:

Variab les 4

t . X'hjl. quantity of input l sent from supply center or port h to plant j in period t.

4 Besides the ones def ined in the problem description.

Spatial market equilibrium problems as network models 119

xJ~ : production of product k in period t which is stored and sent from plant j to demand center i for consumption in period t' (>t).

YJk : inventory of product k at plant j at the end of period t.

Parameters

akt: quantity of input l contained in product k.

Cthjt" unit transportation cost for production input 1 from supply center h to plant j in period t.

cJik: unit transportation cost for product k from plant j to demand center i in period t.

dJk: inventory carrying cost for one unit passing from period t to t + 1.

Sk: number of periods that product k can be in storage.

Constraints

(a) Material balance at supply centers:

Uthl ~ t = xhjl, Vh, t, 1=1,2 (1) J

(1 = 1 for fat material, l = 2 for non-fat material).

(b) Material balance at plants:

~ Xthjt ~ tt' = attxj ik, V j, t, 1 = 1, 2, h i ,k , t '>t

rJ = xaj , v j , t. h,I

(c) Material balance at demand centers:

(2)

(3)

XJ~tk = ztk, Vj, k , t . (4) j,t'<--t

(d) Inventory definition:

yjtk= ~ rt' Xjik, V j, k, t. (5) i , l '>l , r<t

(e) Storage limitation:

xjtiCk = O, t '-- t > s k, Vj, k, t, t'. (6)

Object ive f unc t i on

(i) Consumers' plus producers' surplus is given by the following line integral [20, 49]:

120 0. Barros, A. Weintraub

Oo E E ' ' ' ' = Gin) d~ik gik(~i l , " " , Lt dO k

where zi' = (z[l, ... ,Z[m), minus the term:

t ' "b t l 01 = h.,., ~" J I o f/t(~m) d~ , t .

(ii) Plant production and inventory costs are:

02 2 t t @)(r)) + , , = djkY)k. j,t j,t

(iii) Transportation costs are:

= E cb, xb, + E h,j,t j,i,k,t<t'

(7)

(iv) Net social pay-off is:

0----- 0 0 - 0 1 - 0 2 - 0 3 .

(8)

(9)

CJik XJ~'k. (10)

(11)

3.3. Solution techniques and implementation

In examining the problem, there is a clear multicommodity network structure. We were thus motivated to take advantage of this and develop a specific code. However, deadline pressures for supplying a working tool compelled us to use a general pur- pose LP code, as will be discussed next. Nevertheless, the network structure of the problem was used in the analysis of the problem formulation and presentation of results with users, which was of great aid in clarifying and gaining insight into the problem. In a parallel track, we developed a network based code, which we discuss at the end of this section.

The general purpose L P approach

In order to apply a standard LP code to the model above, the following lineariza- tion procedure was applied.

The exact dependence of the demand of a given product on the demands of other products in a given center at a given time (omitting indexes to avoid unnecessary complexity in notation), was determined to be:

Pl = gl (Zl, Z2) : al - bn zl - b12z2 (12)

where, without loss of generality, Pl is the price charged to product 1, z~ the amount demanded for product 1, z2 the amount demanded for product 2, and a~, b l l and bi2 a r e parameters. That is, the demand for a given product has only cross elasticity with one other product (its substitute). Then, we step-linearized the above demand function around a given point (Zl, z2) in the following way.

We first fixed z2 = z2 and let z~ vary around zl. Doing this, the demand function is only dependent on zl, i.e.:

Spatial market equilibrium problems as network models 121

Pl = a ' - -b l l z l . (13)

Thus we stelS-linearized this function in the same manner as in the sugar market. But in general z2:~'2, which meant a parallel displacement of function (13) and hence of its step-linearized version, or what is the same, a variation in the length of each step. It is easy to calculate a formula to determine the change in length of the step as a function of the displacement of z2 around z'2.

In this manner we had a step-linearized version of (12) as a function of the dis- placements of (zl, z2) around (~q, z2)- Then the line integral of this function in (7), corresponding to the consumers' surplus, led to a linear function of (z~, z2).

The same procedure was applied for all products. Now, in determining the producers' surplus, a straightforward step-linearization

in the sugar market fashion was applied to the supply functions, since they depended on just one variable.

Finally, plant productions costs were piecewise linearized in the usual way, thus completing the linearization of the model.

The method above has been used on a routine basis to determine the equilibium for the dairy market for given conditions specified by the parameters of the prob- lem. Specific applications have been, among others, to establish a policy on the tariff for imported milk; to evaluate the feasibility of building a plant to process a today discarded by-product of the milk processing; and to establish monopolistic behavior for given products at certain demand centers.

However, the relatively large size of the model (almost a thousand constraints and a few thousand variables) has precluded its use in an interactive and day to day basis as in the sugar market. Hence the motivation to develop a more efficient network based algorithm.

The network approach

In approaching this problem by network based algorithm methods, we note that a transformation factor is needed at the plants to account for the processing of inputs contained in fluid milk into products, thus destroying the network structure. However, this problem may be overcome with the following remark. Milk and dairy products are composed of fat and non-fat elements in fixed proportions. Hence, all flows in the network may be represented as some combinations of these two elements. Thus, material balance equations at plants (2) can be transformed by con- sidering the levels of fat and non-fat component for milk and dairy products as follows.

Let:

a~,: fat contained in product k per unit of non-fat matter, in period t.

B~: fat contained in milk per unit of non-fat matter, in period t.

Hence we have the following balance equations, which replace constraints (2):

122 O. Barros, A . Weintraub

~ x t h j = ~, ,,tr .~jik, V j, t, (2a) h i,k,t'>t

o~tk v t t, Xthj = ~ l~t "~jik, V j , t . (2b)

h i,k,t'>-t pI¢

The first equation is the conservation of non-fat matter and the second one corres- ponds to fat.

We can view this problem as a multiproduct network flow problem with addi- tional (common) constraints (say (2b)), or add the fat conservation constraints to the objective function above, multiplied by Lagrange multipliers. This latter idea means the value of these multipliers should be iterated until the fat conservation constraints are satisfied, i.e., using a Lagrangean relaxation approach. A good starting value can be taken as the current price for fat which closely corresponds to the butter price. This value being close to the optimal Lagrange multiplier makes the latter approach attractive. For a given value of the multiplier, the problem is stated in the same general form already described.

Now, in solving this problem with network algorithmic approaches, several ideas are available, proposed originally for solving traffic assignment problems [6, 13, 19]. For example, it is possible to use a feasible direction algorithm on the whole problem (e.g. Frank-Wolfe [33]). We developed the following decomposition by products algorithm taking as a reference the network representation in Fig. 3.

The approach is based on working sequencially on each product at a time, while keeping the flows of the remaining products, fixed. Note that flows from supply nodes to plants correspond to milk, and thus are common to all products. The algor- ithm for an N-product problem has the following steps:

(i) Determine a feasible solution for the problem:

max z (given by (11) plus the Lagrangean term corresponding to (2b)). subject to (1), (2a) and (3).

(Note that constraints (4) and (5) need not be considered explicitly.) Make n = 1.

. tt' (ii) Fix all variables Xhjl, xjik, Z~k, Vi, j, k, t, t', except those with index k = n, at the values given by the feasible solution and solve the following problem:

m a x z

subject to (1), (2a) and (3) and given values Xhjl, it' X)ik, Z~k, Vi, j , l , t , t ' , k ~ n .

(iii) The feasible solution is now the one obtained in (ii). Take n --n + 1. If n > N make n--1 and go to (iv). Otherwise go to (ii).

(iv) Termination rule: if after a complete cycle of solving N one commodity problems, no (significant) improvement is obtained, stop.

Spatial market equilibrium problems as network models 123

SUPPLY CENTERS PLANTS DEMAND CENTER,PRODUCT

Period

t -1 ,t x l i k

Period t

t Xhj

t r j

I.t-1, t~l i~jik I I

Fig. 3. Network flow representation for the dairy market.

124 0. Barros, A. Weintraub

There are several forms in which step (ii) can be implemented. Following the algorithms proposed for the traffic assignment problem, we can mention [37], where the convex simplex method specialized to networks is used to determine an im- proving feasible direction, followed by an unidimensional search or [52, 53] where the improving feasible direction step consists in finding a set of disjoint circuits with minimum total (negative) marginal cost. Such an approach was proposed in [6].

We did an experimental implementation of this algorithm for the dairy market problem. It came out to about 400 nodes and 600 links. We obtained a solution which was similar to that obtained with the linearization approach. Since results were encouraging we have been working on a more refined version of the algorithm.

When cross elasticities appear in supply or demand functions, as in some versions of our dairy problem, conditions of symmetry on the Jacobian of cost functions (integrability conditions) are required. When they are not satisfied, no equivalent optimization problem exists for the equilibrium conditions.

The problem can however be transformed into a variational inequality, which characterizes the equilibrium solution. Conditions under which an algorithm can converge to an equilibrium condition are not trivial. Particularly intuitive is the approach of diagonalization proposed by Florian [ 18], where a sequential decompo- sition by products is used and the asymmetric part of the costs are fixed using the flow from the previous iteration. One form in which the conditions necessary to have local convergence for nonlinear problems can be stated, is that the diagonal elements in the matrix representing the Jacobian be 'larger' than the non-diagonal elements. We note that the diagonalization method for solving asymmetric cost variational inequalities is identical to the Jacobi method for solving systems of linear and nonlinear equations. Sufficient conditions for the local convergence of the Jacobi method have been derived by Florian and Spiess [21], Pang and Chan [40] and sufficient conditions for global convergence of the Jacobi method have been derived by Dafermos [12].

In [47], computational tests done on small problems showed that when the asym- metric term in the matrix were of relatively small importance, the deviation between the solution to an equivalent optimization problem and the real equilibrium condi- tions was small. In particular we found in one of our dairy market problems, which had non-symmetric Jacobian, that the application of the optimization approach - using the linearizing method - led to a true equilibrium.

4 . T h e w h e a t - c o r n m a r k e t

4.1. Description of the problem

The basic characteristics of the problem are the following:

- Wheat and corn have independent supply and demand functions.

Spatial market equilibrium problems as network models 125

- Wheat and corn share warehousing, with limited capacity.

- In hauling, storing and processing, there exists a significant level of losses.

- Wheat and corn can be stored, so the model is a multiperiod one, in one year cycles.

The main elements or agents involved are:

- Supply centers, through aggregated corn and wheat production areas, and ports for imports.

- Warehouses, where wheat and corn are stored and dryed. Associated to this aggregate transhipment are convex (marginally increasing) handling costs, storage capacities and losses.

- Mills, where wheat is ground. Again, costs, capacities and production losses occur. In general, the percentage of losses will depend on the quality of production facilities. Wheat processing produces a marketable by-product which is incor- porated to the objective function.

- Demand centers. Demand for final products is mostly derivative, e.g. bread using flour, foodstock for corn. Separate demand functions are defined for wheat and corn.

- Transportation network, with linear costs and product losses in the hauling process.

Relationships among these elements are shown in Fig. 4. There are seven supply centers, 27 warehouse groupings, five importing ports,

seven mill groupings and seven demand centers.

4.2. The model

The model represents the above stated characteristics in a similar form as the sugar and dairy market and is omitted [7].

4.3. Solution techniques and implementation

Again, for the purpose of satisfying client's needs, the problem was solved with an LP package. In doing this, a linearization similar to the sugar market was per- formed. The resulting LP has been solved routinely. One interesting application was to determine if wheat buyers were acting in agreement to lower the prices of the pro- duct.

We can again analyze how a more efficient approach might be used by taking advantage of its network form. Through linearization of the objective function, each product can be modelled as a generalized network flow problem, and solved

126 O. Barros, A. Weintraub

PRODUCTS STORED FROM PREVIOUS PERIODS PERIOD t-1

PERIOD t

SUPPLY FUNCTION

I = I : CORN

WHEAT TO 1 OTHER MILLS

COST l IFUNCTION

k / ~ j TH STO =ANDr DRYINGj (

H TH WHEAT-CORN ~ .\ . , , . jr.tt

I : 2 : WHEAT WHEAT TO OTHER / I

h / [J ' LL

WHEAT FROM FLOUR TO OTHER OTHER

H TH WHEAT-CORN STORAGE DEMAND iMPORTER CENTERS

j" TH FLOUR IMPORTERS

j , ,

//~,'HF LO U R TO ER DEMAND

CENTERS

t t t j"i2

] TH DEMAND 1 ,~CENTER

DEMAND t FUNCTION

P i k

\ = t

Z i k

BY -PRODUCT

it ' Xjil

PERIOD t°l

PRODUCTS STORED TO OTHER PERIODS

Fig. 4. Wheat-corn market.

using the efficient proposed primal algorithms for these problems [23]. There exists however the common capacity constraints in the warehouses and mills. In order to solve the two product problem as a generalized network we can view the problem [32]:

- As a block diagonal LP with common constraints, by applying typical decom- position techniques such as price directive decomposition (Dantzig-Wolfe) or primal partitioning.

- By relaxing the capacity constraints and putting them in the objective function,

Spatial market equilibrium problems as network models 127

with Lagrangean multipliers representing shadow prices of these constraints as penalty multipliers.

In this case, the proposed approaches may be less efficient due to the relatively large number of common constraints. However, its potential improves by con- sidering that usually a small percentage of the warehouses and mills are at full capacity.

Of particular interest could be the relaxation/restriction approach proposed by Glover and Klingman [25]. In this setting, in one model, the additional constraints are put in the objective function with corresponding Lagrange multipliers, which leads to an upper bound (for the maximization problem) on the optimal solution. Appropriate values of the multipliers are searched to obtain a tight bound. In the other model, through variation of RHS values a feasible solution and a lower bound on the optimal solution are obtained. The procedure iterates until the difference between the lower and upper bound is within a given tolerance.

The advantage of this approach is that each model would essentially require solving just a generalized network flow problem, via use of subgradient optimiza- tion.

However, as we discussed above, it is not clear whether it is worthwhile to use a specialized code for this type of problem. Development of an appropriate code is not trivial, and its efficiency is doubtful, as it is well known that the relative advantage of using network based codes decreases substantially when gains and multicommodities are introduced.

5. Conclusion

We have presented several market equilibrium problems, discussing their model- ling, solution techniques and implementation. We have not shown other markets, such as fruits - apple, pear, grape, apricot and nectarine markets - given their rela- tive similarity in modelling to cases already shown.

There is a clear underlying network structure for all these problems, which is use- ful in their analysis and modelling. In terms of solution techniques, we have seen that each case must be analyzed according to its characteristics. For some problems, a conventional network based code can be applied directly. Other cases will require the development of specific codes if we wish to take advantage of network struc- tures, as in the particular multiproduct structure of the da]ry problem. It is not always clear, as in the wheat-corn market where we have multiproducts with gains and additional constraints, whether developing network based algorithms is worth- while. This leads to an interesting area of algorithmic research.

128 O. Barros, A. Weintraub

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