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 <j> is a set of desirable solution attributes. 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 <j>. This can be formulated as a cost minimisation problem as follows:- Minimise 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 ..