Kansas Geological Survey, Bulletin 194, pt. 1, originally published in 1969
Originally published in 1969 as part of "Short Papers on Research in 1968," Kansas Geological Survey Bulletin 194, part 1, p. 25-27. This is, in general, the original text as published. The information has not been updated.
The goal of optimal utilization of a region's mineral resources has been widely discussed and has been approached from several points of view. Heretofore, no known application of linear programming to this problem is in existence. This study is an attempt to devise such a model.
Osage County, rich in unexploited mineral reserves, was chosen as an ideal basis for a closed-region linear programming model. The model contains 43 activities which are operated subject to 18 constraints. Optimal solutions can be determined analytically for any desirable objective that is data-possible, such as maximizing the corporate profits within the county.
One of the most important elements of the economic process is the conversion of resources from one form to another. The task of determining which products should be produced with a given set of mineral resources can be quite formidable when one is searching for the "best" solution. For instance, a region producing only five products and using any or all of five raw materials finds that there is a minimum of 25 decision alternatives available to it, assuming that it will produce only one of the products. Should it be decided to produce more than one product in this region, the number of decision alternatives could conceivably become very large. Under certain assumptions, primarily linearity, such problems can be solved by the method of linear programming. A suitable definition of linear programming for our purposes is the maximization or minimization of a linear objective function subject to linear constraints. An example will assist in clarifying the terminology.
Assume that the economic development commission of a region knows that the personal income (P.I.) realized from various activities is $2.00 per ton for coal mined, $100.00 per million kilowatt-hours power produced, and $5.00 per ton of Portland cement produced, these being the only feasible activities of this region. The objective is to maximize personal income and thus a mathematical statement of the objective function is
P.I. = $2A + $100B + $5C
where A, B, and C are the units of coal, electric power, and Portland cement, respectively.
This function as it stands will of course be maximized by producing an infinite number of units of each, but since the raw materials labor, coal, clay, and limestone are limited, there must be some finite solution that satisfies the constraints imposed upon the resources, such that no other solution gives a greater value for the objective function.
Let us examine what is meant by constraints. Each of the activities above competes with the others for the available raw materials coal and labor. The substitutability of these resources in the activities is one of the salient features of linear programming, since it provides us with alternative uses concerning which we must make our decisions. Thus we must make judicious use of our resources by applying them where they will return the greatest personal income.
An additional constraint common to all linear programming problems is that the variables themselves must not be negative. This constraint should appear to be neither arbitrary nor confining since it is difficult to conceive of, say, negative coal.
Finally we should understand the meaning of linearity as it applies to the linear programming problem. We assume that the relationships existing among all of the variables in our problem are linear. It has already been pointed out that the objective function is linear; we must also set up the constraints on the problem as a linear system. This will become evident as the model begins to develop.
Very large reserves of water, coal, clay and shale, limestone, sand and gravel, and sandstone are known to exist in Osage County, as well as probable reserves of marble, yellow ocher, and salt. These 11 commodities with the addition of labor and six intermediate commodities form a model constrained by 18 limited resources in the operation of 43 activities thought to be suitable uses for these resources.
(The numbers in parentheses indicate the resources used in the activities as numbered earlier.)
Immediately evident is that all resources and activities are measured in physical units and thus our model is dependent only on current technology in the industries and not on inflatable values.
Let X1, . . . . . ., X43 represent the operating levels of the activities described above and add X44,. . . . . . , X61 to represent disposal activities of the limited resources that do not become fully utilized. (Note: The addition of X44,. . . . . , X61 [called slack variables] is necessary to provide an equality condition in each of the 18 constraints on the model.) Let R1, . . . . . . , R61 represent the amount of personal income (or tax revenue, corporate profit, etc.) resulting from the operation of X1, . . . . . , X61 at the level of one unit. (Note: Obviously, R44, . . . . . , R61, are all zero since a disposal activity adds nothing to the region.) We now can write our objective function:
Objective = R1X1 + . . . . + R61X61
The goal is now to maximize this function subject to constraints as shown:
a11X1 + . . . + a1,61X61 = C1
. . .
. . .
. . .
. . .
. . .
A18,1X1 + . . . + a18,61X61 = C18
C1, . . . . . . , C18 represent the negatives of the limitations on each of the resources and are expressed in terms of man-hours, tons, etc., per year. (Note: The reason for C1, . . . . . , C18 being negative is purely computational and not of real importance to the study. It should also be noted that the restraint on each intermediate commodity is zero since the amount produced is equal to the amount consumed.) Physically, the limitations are the maximum availability of these resources annually. The primary limitations here are capital investment and the availability of markets for the products. Thus the limitations must be forecast or estimated by such methods as are available.
The matrix aij represents the production coefficients, which are of three types:
An awareness of the assumptions and drawbacks of a linear programming formulation must be maintained if its results are to be meaningful and valid.
Discussed above is the assumption of linearity. The primary factors included in linearity are the absence of fixed costs and the neglect of efficiency or inefficiency due to size. However, within a limited range of production, the errors involved are slight.
Transportation effects have been eliminated in one of two ways--either the commodity is used entirely within the region or it is marketed to outsiders f.o.b. mine or factory. Obviously the market must be forecasted correctly for our model to produce meaningful results.
The usefulness of this linear programming model has been touched on earlier with respect to optimizing the use of limited mineral resources. With a simple change of the coefficients in the objective function, it is possible to maximize personal income, tax revenue, corporate profits, gross revenue, or any other desirable function that is data-feasible. The production coefficient matrix is not affected by changing the objective function, thus the model is quite versatile. Other manipulations can be made by varying the limiting constraints to reflect market considerations or capital investment limitations.
It is hoped that eventually a similar model will be constructed for the entire State of Kansas, resulting in the development of the relatively untouched resources of Kansas by private industry.
Kansas Geological Survey, Short Papers on Research in 1968
Placed on web July 26, 2011; originally published in Feb. 1969.
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