18
A HEURISTIC/SATISFICING APPROACH TO A PROBLEM OF FACILITY LOCATION
J.L. Rodgers* Department of Agricultural Economics and Marketing, Lincoln College N.Z.
Summary
Many operational research problems are not amenable to solution by generalized, optimizing solution procedures. The heuristic approach has been advanced as a means of obtaining satisfactory solutions to such problems. This paper first reviews the general nature of heuristics and then discusses a heuristic/satisficing approach which is capable of achieving acceptable sol utions to certain types of facility location problems. The method outlined builds upon earlier work on problems of locating agricultural commodity processing facilities and utilizes the iterative solution of a continually adjusted transhipment model. A simple example is given.
______Introduction
The determination of the number, size and location of processing facilities for a commodity which is available in unprocessed form at a given set of sources and is required in processed form at a given set of destinations is a problem which has been and is of considerable interest to researchers. If processing of the commodity was not required, the problem would involve the cost minimisation of a simple transportation model. But if the commodity has to be processed in some way at an intermediate location, the solution procedure is not so straight forward. Processing costs at any particular facility are usually characterised by economies
* Paper presented at the 1974 Conference of the Operational Research Society of New Zealand, August 23, 1974, in Christchurch 19
of scale (that is: the greater the throughput, the lower is the per unit processing cost). Consideration needs to be given to assembly costs (source to facility transport costs), processing costs, and distribution costs (facility to final destination transport costs). The problem might be further complicated by the requirement that the solution should not only be low cost, but also that it satisfy certain non-quantifiable criteria.
This paper discusses one approach which has been found useful in providing satisfactory solutions to certain problems of this type. The philosophy of the approach will be discussed followed by an outline of the solution method and finally a simple numerical example will be given.
2_.______Heuristics
Simon and Newell in an early contribution on heuristic problem solving, imply that the heuristic (as contrasted with the algorithmic) approach involves "intuition, insight and learning" (14, p.6) and that heuristic programming is the programming of computers to display these attributes in attempting problem solving tasks (14, p.7). On the other hand, Wiest, in a more recent paper, broadly describes heuristics so as to include any systematic method for solving problems from "systematic cut-and-try based on reasonable rules of thumb at one extreme (to) algorithms with their supporting theories and known properties at another extreme" (19, p.360), while Tonge defines heuristics as "principles or devices that contribute, on the average, to the reduction of search in problem solving activity. Heuristic programming is the construction of problem solving programs organised around 20
such principles or devices." (16, p.231). To add possibly more confusion, Keuhn and Hamburger offer a different view when they write: "We prefer to look at heuristic programming as an approach to problem sol ving where the emphasis is on working towards optimal solution procedures rather than optimal solutions." (7, p.644).
The above points of view are not exhaustive - for further discussion see (10) - but are sufficient to demonstrate that there exists significant differences of opinion on the important aspects of the heuristic approach.
Some clarification (or otherwise) might be obtained by considering the types of problems for which a heuristic approach might be appropriate. Simon and Newell (14, p.4), supported by Tonge (16, p.232) and Wiest (18, p.131), suggest that a heuristic approach is suited to those problems which may be called ill structured. Ill structure is a residual concept, i.e. a problem is said to be ill structured if it is not well structured. Simon (13, p.183) proposes that a problem is well structured to the extent that (a) any knowledge relevant to the problem can be represented in an acceptable model; (b) there exists definite criteria for judging the validity and acceptability of any solution; and (c) there exists a mechanizable procedure which is capable of solving an acceptable model.
Both Wiest (18,p.130) and Tonge (16, p.232) also comment that a heuristic approach is appropriate where the problem is too large to be solved by conventional analytical techniques or exhaustive search, 21
even with the aid of a computer (see (11, pp. 151-152))
Tonge (17, p. 25) lists some common character istics of existing heuristic procedures, including:- (a) decomposition of the original problem into a number of subproblems; (b) a high degree of dependence between the problem and the solution procedures used; (c) no guarantee that a satisfactory solution will be found.
It is concluded that the heuristic methods encompass a broad spectrum of specific techniques. The heuristic approach involves the development of a set of rules and procedures (heuristics) which hopefully will produce one or more satisfactory solution^ to a specific problem, but no guarantee of optimality of the solution can be given (12, p.39). The set of rules may be rigid or flexible but they must be sufficiently definitive to enable a convergent search for an approximate solution. Heuristic programming involves the writing of computer programs based on such sets of rules and procedures.
The last word is given to Keuhn and Hamburger: "The traditional operations research approach has been to search for optimum solutions. The heuristic approach differs in the following ways : (1) Explicit consideration is given to a number of factors (for example, computer storage capacity and solution time) in addition to the quality of the solution produced.
(2) The evaluation of heuristic techniques is usually done by inductive rather than deductive procedures, that is, specific heuristics are justified not because they 22
attain an analytically derived solution (for example, an optimum) but rather because experimentation has proved that they are useful in practice." (7, p.229)
3_.______Sat is f icing
It has been stated that a heuristic approach to problem solving does not guarantee an optimal solution. This is not necessarily a limitation. Firstly, in many instances an optimal solution is not essential to management (8, p.179), so long as it is efficient or at least superior to guess-work (6, p.680). Secondly, an optimal solution, in the strict sense, may not even exist. This would be the case when no trade-offs can be given between a number of goals, but rather the decision maker requires that an acceptable solution should possess certain attributes (some of which may be unstated). For example the primary objective may be maximisation of profits while a set of secondary non-quantifiable objectives may need to be meteor when all objectives are of the satisficing type. As Eilon observes: "In fact, real problems are rarely attuned to single objectives, and the simple minded manipulations that are adopted through trade-offs to convert multi-objectives to a single objective are often so arbitrary as to cast serious doubts about the validity of optimisation models ”(2, p .36).
The satisficing approach argued by Eilon is supported by many other researchers. For example, in the area of farm management planning, Dent and Byrne write: "It has become clear, however, that the objectives of farmers, particularly their,long term objectives, cannot be reduced to a single criterion. ...The alternative is to offer the farmer a number of feasible plans” (1, p . 104). 23
The heuristic approach is well suited to problems involving multiple goal satisficing and where a number of alternative solutions are required (5, p.17).
4_.______The Facility Location Problem
The problem of determining the number, size and location of processing facilities is an ideal candidate for heuristic treatment. Except in special restricted cases, no optimal guaranteeing solution method is available (3, p.l); there often exists interest in non-numerical criteria for judging solution acceptability; and exhaustive search for acceptable solutions is impractical in most real problem situations (3 , p.2). In addition it is often desirable to provide a number of possible low cost solutions so that decision makers can consider jointly the comparative costs of alternative solutions and their comparative non-economic attributes (4, p.155).
5_.______Statement of the Problem
Given: m sources of some raw commodity supplying S.
(i = 1, 2, 3 ... m); t potential processing facilities, with potential capacity (k = 1, 2, ... t ) ; n destinations of the processed commodity demanding Dj (j = 1,2,3 ... n);
C-v = the per unit cost of transporting the commodity IK £ v ^ from the i source to the k facility, (assembly costs) ;
C, - = the per unit cost of transporting the commodity th th from the k facility to the j destination (distribution costs); = the per unit processing cost at the K facility and is some function of the throughout quantity (processing costs), i.e.,
m Pk = fk ( ■l1 x ik^’ for k = 1,2» t ’
where z x., = the throughput of the k1"^ i= 1 1K facility ; and
The problem is to determine x^, (the quantity shipped from the i**1 source to the k1"*1 facility) and x^j (the quantity shipped from the kt^1 facility to the j destination) for all i, j and k, so that total costs are as low as possible consistent with the solution satisfying an acceptable subset of
mt t|_ m m t n
E E C x + E f,( E x ) E x + E E n> i = l k=1 lk lk k=l L k i = 1 lk i = 1 lkJ k=1 j = l KJ
subiect to
E x.,= S., for i = 1,2 ... m (supply constraint) k=l lk 1
E x.,= K,, for k = 1,2 ... t (Capacity constraint) i=l lk k
m n E x ., = E x, . for k = 1,2 ... t (Input/Output constraint) i = l lk j = l k:i »
t E x, . = D . , for j = 1,2 ... n (Demand constraint) k=l kJ J
E S. = E D. = E K, , for feasibility i=l 1 j=l 3 k=l K
xij = 0, xk;. = 0, for all i, k, j
and such that an acceptable subset of 4> is satisfied. 26
Often (or more probably, usually) 4> is not completely known and so the approach adopted is to develop a set of low cost solutions with differing attributes so that the decision maker has a set of alternatives from which to choose. Alternatively if acceptable subsets of
If it were not for the complication of the per unit throughput cost of each facility being a function of the quantity processed, the problem would be a simple transhipment problem with the facility locations becoming intermediate transhipment points. Similarly if the throughput of each facility were known then the problem would also reduce to a transhipment problem.
6_.______A Monte Carlo Solution Approach
One approach under development uses Monte Carlo methods to obtain a number of low cost solutions. The problem is split into two subproblems both of which take on a transportation formulation. Subproblem 1 involves the minimisation of transportation costs from sources to facilities, while subproblem 2 involves minimisation of transportation costs from facilities to destinations. The size, number and location of facilities are chosen randomly so that total 'actual' capacity is equal to total 'required' capacity. The two separate transportation problems are optimised and the total cost of the solution calculated. If this total cost is below some preset level, the solution is reported. The procedure is relatively fast and a large numbeT of'solutions can be investigated, resulting in a reasonable number of low cost solutions. 27
The selection of any particular potential facility location and its size is governed by predetermined probabilities. By varying these probabilities the researcher is able to direct the search to any desired extent. Tt is in the selection of these probabilities that heuristics are involved. From the researcher's knowledge of the problem and by observation of the nature of the low cost solutions that are reported, the probabilities can be adjusted to investigate feasible solutions which may prove interesting. It is possible that some such heuristics can be incorporated into the program to allow automatic resetting :>f the probabilities.
7_.______A Deterministic Approach
A second and more promising approach based on earlier work by Logan and King (9) and more recently Stammer (15) involves a procedure of iteratively solving a continually adjusted transhipment model. The basic cost matrix structure utilized in this approach is presented in Figure 1.
Submatrix S-,, contains the per unit transport costs from the i +■ source to k 1* Vi facility plus the per unit through put cost of the kt^1 facility. Submatrix S ^ contains pro hibitively high cost elements to prevent direct shipment from
source to destination. Submatrix S 2 ^ is a square submatrix with zero cost elements down the leading diagonal and usually prohibitively high cost elements elsewhere. If shipment between intermediate points (i.e., the facilities) is allowed, the non-diagonal elements of this submatrix are set to reflect
the cost of such shipment. Submatrix S 2 2 contains the per - 28 -
Figure 1
Cost/Requirements Table
t h unit transport costs from the k facility to the . th , j destination.
Since the throughput of each facility is unknown an iterative solution procedure is suggested which employs a number of heuristics as shown in Figure 2. 29
Figure 2
Stammer suggested that the procedure be modified to the extent that Heuristic 3 should become:
Any facilities with zero throughput should retain their previous level of unit processing costs. 30
Provided that the processing cost functions are well behaved, both procedures converge to a stable solution. In most instances, the Stammer technique yields a solution with a cost as low as that obtained by the Logan and King method. The solution achieved by both methods are local optima; that is, a small change involving the re-routing of one unit of the commodity would result in a higher cost solution. However, lower cost solutions might exist involving significant facility size changes, the dropping of a facility currently in the solution, or the introduction of a facility not present. In addition both procedures yield only one solution each (often the same) and to achieve different solutions it is necessary to redefine potential facility locations and/or their capacity constraints, or alternatively modify the heuristics.
To this end a technique known as forcing has been developed. Simply stated., forcing entails the artificial manipulation of the unit processing costs so as to encourage a facility into the solution or increase its throughput if it is already there, or alternatively force it from a solution or decrease its present throughput. Forcing may be applied at any stage during the iterative search for a stable solution, (that is before, during or after).
Some general forcing rules found useful are:- (1) Any location which has been dropped from consideration early in the iterative process is a good candidate for 'forcing in'.
(2) Any location which is present in a stable solution, but whose throughput is small is a good candidate for 'forcing out'. 31
(3) If there are two locations present in a stable solution which are'geographically close', one of them is a good candidate for 'forcing out'.
Heuristicis which are specific to a particular problem depend on the researchers knowledge of the system and his observation of the “solution characteristics during the search procedure. For example, in determining forcing strategies, observations of the following type may prove useful:
(1) Good solutions tend to contain facilities which are source (or conversely destination) b iased.
(2) Certain pairs of locations tend to enter the same good solution (or conversely are never present in the same good solution).
Finally a knowledge of at least some of the 8_._____ The Computer Program A computer program,"An Interactive Facility Location System (AIFLS)", has been developed on the IBM 1130 which incorporates both the Logan and King and the Stammer iterative search procedures and which permits 32 the forcing technique to be applied. As the name implies the system allows user interaction with the program at execution time. While all input may be through the card reader it has been found more convenient to allow direct data manipulation through the console key board. The usual mode of operation is to input the initial problem data through the card reader and initiate initial iterative searches using the standard Logan and King, and Stammer methods. After this, the search for alternative low cost solutions is directed by the user from the keyboard using forcing and other data manipulation procedures. The user is thus able to consider each successive solution before deciding upon the next search strategy. 9_.______A Simple Example The following example is presented to illustrate the approach outlined above. Any similarity between the example and familiar real world situations is purely intentional. The story begins (with apology to Tolkien). The successful cultivation of the legendary weed of Middle Earth requires very specific soil and climatic conditions. After the War of the Rings when life settled back to normal, it was found that the weed could be grown in twelve localities in add-ition to the Shire. Except for the inhabitants of Minas Tirith, the locals prefer to process their own tobacco ("it looses all its flavour when processed in bulk"), and so by Royal decree the Weed Marketing Corporation (W.M.C.) was established and made responsible for the processing and marketing of weed for the Minas Tirith market and for export. Apart from locally processed and consumed weed, all production must be sold to the W.M.C. - 33 Export demand is at an all time high, owing to the discovery that Shire weed (as it had become known) had no injurious effects on health and in fact was beneficial (witness the longevity of hobbits). The W.M.C., not wishing to exploit their monopolistic position, were prepared to sell (for a "reasonable profit") all weed that was not consumed locally. They skilfully negotiated long term trade agreements with the countries to the North, South, East and West so that markets were ensured for all surplus production. They then contracted researchers at the University of Middle Earth (U.M.E.) to determine the number, size and location for the planned new tobacco processing facilities. The terms of reference for the research study were that the whole marketing operation should not involve unnecessary costs but at the same time a view should be kept of certain political considerations. For example, the people of Esgoroth were crying out for some industry in their region, while the inhabitants of Hobbiton were not at all sure that they wanted a large factory in their village ("it will foul-up the air and pollute the streams"). The research group(operations researchers, econometricians, engineers - the usual type) gathered the following data: (a) The expected excess production per annum for each source region (Table 1). (b) The expected demand per annum of Minas Tirith and the expected sales to each export region (Table 1). (c) Twelve potential plant locations and the scale cost curves applicable for each location (Table 2). (d) Per unit transport costs, 36 (i) from source regions to potential plant locations (Table 3a); (ii) from potential facility locations to ports (Table 3b); (iii)from ports to final destinations (Table 3c) Having decided to use the heuristic facility location procedures (outlined earlier) , the researchers constructed a three stage transhipment model as shown in figure 3. Figure 3 Cost/Requirements Table for Middle Earth W.M.C. model Non Port Facility Ports Dest inations Supply Locat ion 1 2 . . 8 ) . . 12 13 .... 17 1 ' 500 2 ( Sources Source to Facility Inactive transport :osts + (all cost processing cos ts elements =°° 13 700 14 0 Non-port K Non- \ 00 :acilities Inactive Ports 'o port (all cost Locations . Lransport elements = °° \ :osts 21 0 K 22 )K Inactive Port to final Ports (all cost \\ " destination elements 00 \ transport 25 = 00) 0 costs K Demand K K ____K fC...... K 500. . . 600 1 37 This model is limited in that it does not allow facility- to-facility and port-to-port transhipment. In addition, as it is presently constructed, it is not possible to separately alter the capacity contraints on port-based facilities and the corresponding ports. Some simple modifications to the model would correct these limitations. The heuristic procedures where applied to the model. Table 4 presents the size and location of facilities for some of the solutions obtained by the U.M.E. team. Table 4 Some selected solutions to the W.M.C. facility location problem THROUGHPUT (in bundles) PLANT LOCATION 1 2 3 4 5 6 7 1. Hobbiton 2400 2400 2. Rivendell 800 3. Esgaroth 600 600 4. Tharbad 2100 2100 2100 5. Lorien 1250 1250 650 6. Isengard 7. Edoras 4050 2600 4050 8. Doom 9. Havens 2400 M O O 2400 2400 2400 10.Lond Dae r 2100 11. Belfalas 1400 2050 1400 12.Minas 4600 4600 4600 1000 Tirith TOTAL COST p.a. 66241 66736 66996 >7533 68461 58833 69635 (in gold pieces) 38 Solutions 2 and 3 were obtained using the standard Logan and King, and Stammer procedures respectively. Forcing was required to achieve solution 1 and the remainder. Discussion of Results It is noted that in going from a centralised to a decentralised location pattern the total cost increases. However, if the government has serious feelings concerning decentralisation it might prefer a solution such as 7 to the others, (some people are complaining that Minas Tirith is already too large and industrialised); the additional cost of this solution is of the order of 51. If only lip service is paid to regional development, solution 4 which involves an additional cost of about 2%, might be attractive. Postscript The U.M.E., team selected 20 or so solutions which they considered covered all possible desirable attributes. These were fully documented in the final report which was presented to the W.M.C. A decision is still pending. 39 REFERENCES 1. Dent, J.B. and P.F. Byrne, "Investment Planning by Monte Carlo Simulation", Review of Marketing and Agricultural Economics, 37(2), June 1969. 2 . li i 1 o n , S . , "Mathematical Modelling for Management", Tnterfaces, 4 (2), February 1974 3. Elshat'e i , A. N . Optimal Location of Facilities, Unpublished Ph.D. thesis., University of Birmingham, 1972. 4. Ferguson, D.C., W.O. McCarthy and J.L. Rodgers, "Determining Alternative Locations for Processing Facilities", Review of Marketing and Agricultural Economics, 3T) (’4) , December 1972 5. Hillier, F.S. and G.J. An Introduction to Operations Liberman , Research, Holden Day, 1967 6 . Joseph, J .A ., "Heuristic Approach to Non standard Form Assignment Problems", Operat ions Research,15(4), July - August T957 7. Keuhn A.A. and M.J. 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