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Increasing Practice Efficiency by Linear Programming
Increasing practice efficiency by linear programming
N eville D oherty, PhD, Hartford, Conn
L in e a r prog ram m ing is a m odern
m athem atical
te ch n iq u e useful fo f solving p ro b lem s that involve th e allo catio n of s c a rc e re so u rce s am ong alte rn a tive en d s. T h is pap er d is c u s s e s the te ch n iq u e and, with the use of a sim p lified e x a m p le , illu strates a potential ap p licatio n in dental p ractice .
Even a cursory glance at the flood o f current lit erature about the politics and econom ics of den tal health and dental health care could convince a skeptic that dentistry, although never a static science, is in the midst o f change and on the threshold of upheaval. Manifestations o f this dynamism are well known. They include ex pectations o f vast increases in the demand for dental care, the application o f new knowledge and technology, and the emerging prominence o f prepayment and prevention as rational alter natives to postpayment and dental disease. Along with these changes, traditional methods o f practice organization and planning are being modified. Because of the search for efficiency and productivity in meeting new demands and in delivering care, there are growing movements toward group practice, the division of tasks along more specialized lines, and the utilization of modern managerial techniques and aids.1 A s these changes occur, the dentist, long a traditional entrepreneur, will be faced with few options other than to improve his own manager ial capabilities, or to delegate his managerial responsibility externally to professional man
agement consultants or internally to a special ist. In dentistry, as in any other area that pro vides services, many decisions can and always will be made on the bases o f common sense, in stinct, and experience. H owever, new decisions require more specialized knowledge and under standing. The objective o f this paper is to illustrate a technical method of decision-making called lin ear programming. It is a product o f modern mathematics, and belongs in the family o f ana lytical techniques almost stigmatized by such terms as programming, operations research, and activity analysis. The technique was developed during World War II for scheduling of shipments o f food and war materials. The area of problem solving, how ever, has been o f interest for a longer period; literature on the subject in both mathematics and econom ics dates to the 1920s. In 1947, the seminal paper was published by the mathemati cian, Dantzig.2 Subsequently, the technique has been modified and refined and has found widespread use in industry, government, agri culture, and research organizations to solve complex personnel allocation, diet planning, transportation, business management, and en terprise planning problems.3 Because linear programming and other quan titative methods o f problem solving are based on mathematical concepts, some difficulty may be encountered in the interpretation o f the find ings into meaningful terms. Under such circum stances the dentist, despite concern about the efficiency o f his practice, may tend to equate his inability to understand with impracticality. H e may dismiss these new techniques, giving cre dence thereby to the thought that “ People would rather live with problems they cannot solve than with solutions they cannot understand.” JA D A , V o l. 85, N o ve m b e r, 1972 ■ 1099
Although much of the credit for demonstrat ing applications o f programming methods be longs to specialists known as operations re searchers, it is not necessary to be in operations research to gain an understanding of the prin ciples involved. Indeed, if these techniques are to benefit m odem dentistry, it is important that information about them be made available not only to researchers, but to teachers and practi tioners as well. Failure to do this could actually impede the development of new variations and refinements in technique and the application of new capabilities in solving dental problems. Thus, although this paper will not show the den tist how he can use linear programming in his office, I hope it will enhance his understanding o f technical approaches to decision-making and their potential for distinguishing seemingly con flicting goals.
P otential of linear program m ing The value of a linear programming solution to a problem lies in the fact that, under given con ditions, no other solution or plan of operation will give more efficient results. Mathematically, it is a technique to optimize a linear objective, subject to restraints that can be expressed or approximated in linear terms. Operationally, its value derives from the fact that the optimum solution, even for decisions involving four or five variables, is nearly always beyond one’s best judgment. Because the constraints on the practicing dentist are of a linear form, the appli cation of this technique is feasible as well as po tentially useful. The term linearity describes the fact that the equations used to express the relations among the variables in a problem are of the general form, y = a + b x (where a and b are any numbers). If one were to use such an equation to compute a table of values for x and y, and then plotted these values on a graph, he would find that he had drawn a set of points— all of which lie along a straight line; y = a + b x is, therefore, called a linear equation. When used in conjunction with a computer (such arrangements are available either through private or university facilities), linear program ming permits a comparison o f alternatives for planning on a scale unheard o f a few years ago. A fact of modern life is that solutions to every day, seemingly simple problems can involve 1100 ■ JADA, Vol. 85, November 1972
dozens o f restrictions and alternative courses of action. When such decisions are made by tra ditional methods, only a small subset of the al ternatives can be considered; as a result, there is a considerable risk that some of the better possibilities will be overlooked. With the use of linear programming, we can search among all feasible solutions to a particular class of such problems, that is, those with linear constraints and a linear objective. The technique has achieved a noteworthy record for finding the best solu tion or solutions in a matter of minutes, even seconds.4
W hen is linea r program m ing possible? In addition to the objectives a dentist has for his practice, such as treating more patients, reduc ing costs, increasing net income, and making better use of his assisting staff and equipment, three conditions must exist before a problem can be formulated and solved by linear program ming. The absence of any one of these either simplifies the problem to such an extent that the solution is easy or introduces such a complex problem that linear programming cannot be used to obtain a solution. The first condition is that either more than one limited resource or more than one restric tion on the solution to the problem must exist. Suppose, for example, that adequate equipment is available but that the number o f assistants is the only limiting resource to produce more dental services. In such a situation, the entire practice should be managed to produce the alter native with the greatest benefit, depending on practice goals. This benefit can be thought of in terms o f services or incomes or both, per assist ant. O f course, most managers are faced with limited resources or restrictions, or both, that limit their activities. Alternative courses of action, the second con dition, must be available to the decision-maker. If a practice is structured to provide only one service in a given way, the question may arise: Is it better to provide the maximum number of these services or to let the practice idle? H ow ever, even the dentist whose practice is suited only for the provision of one kind of service must make decisions as to the allocation of time, use o f assistants, method of treatment, and so forth. Thus, many courses of action usually ex
ist even in the most simple operation. The third condition necessary in forming a lin ear programming problem involves input-output relationships between resources (labor, capital, equipment, and so forth) and products (alter native dental services that can be provided). The combination in which various resources can be used to provide different amounts o f services must be specified. A lso, if income effects are of prime concern, the prices or expected prices of the resources used and services provided must be known, or at least estimated. If these three conditions—two or more limit ed resources or restrictions, alternative courses o f action, and input-output relationships— are present, the approximate conditions under which a management decision must be made usually can be expressed mathematically. The use of linear programming in the solution o f a hypo thetical practice problem containing each of these three conditions can be illustrated in a situation. First, a caution: real life, with all its vagaries, is too complicated to program, even on the most sophisticated electronic equipment. For this reason w e use mathematical models to simplify reality and, thereby, to express it in manageable terms. The level of mathematical sophistication to which such methods can be advanced is quite remarkable; here, however, the use of mathematics is limited to the level necessary to illustrate a technique with poten tially useful applications in dentistry.
A linea r pro g ra m m in g exam ple ■ The problem : Suppose a practice offers gen eral dentistry and pedodontics. The practice uses two operatories, three assistants, and one dentist. The dentist and his assistants each work an eight-hour day to provide the services; each operatory, therefore, is also available for eight hours. The dentist who owns and manages the practice is, let us say, curious to examine the potential o f operations research for means to improve the efficiency o f his practice— not in terms of improving techniques that his experi ence has enabled him to develop to a sophisticat ed level, but in terms o f organizing the use of his scarce resources (himself, his assistants, and his equipment) in the most productive manner possible. Examination of the practice records indicates that an operatory is used for 45 minutes, on the
average, for each pedodontic and general dental care appointment. A lso, delivery of a pedodon tic service requires IV2 hours o f assistants’ time and 15 minutes of the practitioner’s time. G en eral dentistry, on the other hand, takes one hour of assistants’ time and a half hour of the den tist’s time. On the basis o f expected charges for pedodontic and general care (a schedule may have been worked out with a union or insurance company, for example), a net income o f $8 for a pedodontic service and $6 for a general den tal service can be estimated. The problem is to find the combination o f services that would yield the best returns on the scarce or limited resourc es. ■ Linear program ming form at: Several tech niques can be used to obtain a solution to this problem. The linear programming technique, however, can be shown simply. P equals the number o f pedodontic services to be provided; G equals the number of general dental services to be provided; T equals hours of operatory time unused; A equals hours o f as sistants’ time unused; and D equals hours of dentist’s time unused. The problem can be represented by these re lationships. Maximize returns (the objective function): 8P+6G . Subject to the constraints: (a) 0.7 5 P + 0 .7 5 G + T = 16 hours of operatory time; (b) 1.50P+ 1 .00G + A = 24 hours of assist ants’ time; (c) 0 .2 5 P + 0 .5 0 G + D = 8 hours o f dentist’s time. So that Ps=0, Gs=0, AsM), Ds=0, Ts=0 (non negativity conditions). This is the standard form o f a programming problem. It consists o f three parts: the function, called the objective function, whose value is to be maximized or minimized. In this instance it is dollars per service: $8 for a pedodontic ser vice and $6 for a general dental service. The constraints are the resource limitations. The nonnegativity conditions are statements that preclude use of negative amounts of resources; for example, one cannot use less than zero hours of one’s own time. What do these parts mean? And how can they be used? A start is made by examining the con straints. The first equation (a) says that the amount of operatory time used in pedodontic services, plus the amount used in general den Doherty: LINEAR PROGRAMMING ■ 1101
tal services, plus the amount unused must equal 16 hours, and that the amount of operatory time needed is three quarters of an hour for both a pedodontic service and a general dental service. This equation represents the operatory or capi tal equipment restriction in this problem. In Figure 1, it is shown graphically by the line AB. Each point on AB represents a combination of appointments for pedodontic and general care that is exactly equal to 16 hours. Line AB also indicates that, if operatories were the only lim ited resource and if all available operatory time were used for pedodontic or general dentistry, or both, a dentist could handle an additional pa tient for pedodontic care by giving up one pa tient for general care. Thus, operatories sub stitute at a 1:1 ratio between general and pedo dontic services. The second constraint (b) says that 1Vi hours of assistants’ time per pedodontic service, plus one hour of assistants’ time per general dental service, plus the time unused must equal 24 hours a day. Therefore, this equation represents the labor restriction. It is shown graphically by the line CE in Figure 1. Each point on CE repre sents a combination of assistants’ time, for pedo dontic and general care that requires the use of exactly 24 hours. Line CE also indicates that, if labor were the only limited resource and if all available labor were used, the dentist could schedule W2 additional appointments in general dentistry by taking one less pedodontics ap pointment. Thus, labor substitutes at a 1.5:1 ratio between general and pedodontic services. The third constraint (c) says that 0.25 hours (15 minutes) of the dentist’s time per pedodontic service, plus 0.50 hours (30 minutes) of the den tist’s time per general service, plus the unused dentist’s time must equal eight hours per day. Consequently, this equation represents the pro fessional or managerial constraint in this prob lem. It is shown graphically by the line FH in Figure 1. Each point on FH represents a com bination of pedodontic and general dental ser vices that requires the use of exactly eight hours o f the dentist’s time. The line also indicates that if the dentist’s time was the only limited resource and if all his available time were used, he could treat one additional general dental patient by taking two less children. Thus, management substitutes at a 0.5:1 ratio between general and pedodontic services. The heavy line FY E in Figure 1 represents the production possibility curve. This means 1102 • JADA, Vol. 85, November 1972
Fig 1 ■ Resource restrictions.
that under the conditions specified, it is possible to produce only those combinations of general or pedodontic services, or both, that appear on on or below and to the left of the production pos sibility curve. N o combination o f services to the right and above the line FY E can be produced because of the lack of one or more of the limited resources. The area bounded by FY E is, in fact, the area of technical feasibility; the line AB, de fining the operatory restraint, lies totally above this area. The dentist has more than enough operatory time to provide any of the combina tion of services permitted by the scarce resourc es— his assistants and himself. For this reason, the operatory restriction can be ignored for the time in this analysis; it is not a strategic factor in this particular problem. (Such a situation is probably more common than we like to think. It may arise because of the indivisibility of equip ment, for example, a dentist cannot buy 1Vi oper atories. It may arise because of the common sense dictum: if it is there, use it, even if the pa tient waits unattended for 15 minutes!) The objective function says that $8 per ped odontic service provided plus $6 per general ser vice provided will equal the return. This return must be the maximum return under the condi tions specified in the constraints (a, b, and c). Thus, the solution must fall on the production possibility curve presented in Figure 1. The production possibility curve seen in Fig ure 2 bounds the region of feasible output solu tions. The object of a programming calculation, however, is to pick the optimal combination among the feasible output combinations. For the example, this is defined as the level of out put yielding the highest return to the practice.
Fig 2 ■ Optimum solution.
This point is located through the use of isoreve nue curves. A ny two points on such a curve rep resent combinations o f general and pedodontic services that yield the same incomes, for exam ple, income at combination M is the same as that at combination N . In a linear program these iso revenue curves are always straight lines and they are parallel. The equation for a typical isoreve nue curve in the example is: income equals 8P+ 6G. The curve representing a $100 income level, for instance, has the equation: 100=8P +6G , or G = 100—4/3P, so that the isorevenue curve is a straight line, the slope of which is -4 /3 . Similar ly, all lines representing greater income levels will be higher than, but with the same slope as the $100 line; those lines representing lesser in come levels will be lower than the $100 line. Sev eral of these lines, representing increasing in come levels, are shown in Figure 2. The solution to the problem is geometrically represented by the point in the feasible region that is tangent to (lies beneath and touches) the highest attainable isorevenue curve. This point is found by moving to higher and higher isoreve nue curves until one reaches the production pos sibility curve. In this example the result or solu tion occurs at point Y (Fig 2); that is, at a point where P equals 8 and G equals 12. Therefore, the provision o f 8 pedodontic services and 12 general services would enable the practice to make the most efficient use of its scarce re sources.
The result also could be obtained by solution o f the simultaneous equations defined by the constraints (a, b, and c). Because one o f the equations is redundant, that is, a linear function of another one, the solution is reduced to two equations and two unknowns. In complex and realistic problems, more powerful and compli cated computational techniques would be nec essary. These results can be used geometrically by inspecting Figures 1 and 2, or arithmetically by replacing the P’s and G ’s in the constraints with 8 ’s and 12’s, to obtain the additional informa tion: —0.75(8)+0.75( 12)+ T = 16; T equals one hour of operatory time that can go unused. — 1.50(8)+1.00(12)+A=24; A equals zero hours of the assistants’ time that should go un used. — 0.25(8)+0.50(12)+D =8; D equals zero hours of the dentist’s time that should go unused. A lso by solving for the objective function (in com e equals 8P+6G ) for the values P equals 8 and G equals 12, the maximum possible net re turn is $136 per day. N o other combination of services will return $136 or more, and all other combinations will return less than $136. The importance of this last result cannot be underestimated. It shows that net income per service, alone, can be a poor indicator of the op timum delivery pattern. Pedodontics, which in this situation produces an average net income of $8 per service, optimally should be provided in a ratio to general services o f 2:3, even though general dental services produce only $6 per ser vice. Other information is not evident in the so lution but is available with use of the linear pro gramming procedure. For example, it will in dicate the value o f an additional hour of the as sistants’ and dentist’s time, the so-called mar ginal value or marginal contribution to the prac tice. A lso the optimal pattern can be found of shifts between pedodontic and general services that should be adopted if their relative fees vary or if other changes occur (such as different time schedules for the assistants or the dentist, or both).
Advantages of solving problem s by linear program m ing Many advantages in the use o f linear program ming aid in the solution of problems about which Doherty: LINEAR PROGRAMMING ■ 1103
decisions must be made. Several o f these ad vantages have been illustrated or discussed and may be summarized. The manager, or decision-maker, is certain that no other feasible solution— given his fixed resources or restrictions, his alternatives, and his input and output relationships— will permit him to satisfy his practice objectives, whether they are increased income, reduced costs, or any reorganization o f his practice to improve his delivery o f dental care. The manager, given the linear programming solution, knows exactly what an additional unit of his limited resources is worth. H e knows the range over which income may vary before a change in the practice plans should be considered. In addition, he knows the types of changes that are necessary if his income varies outside this known range. The manager knows exactly those conditions under which the solution to the problem has been obtained. These conditions are stated in the equations that represent the mathematical for mulation o f the problem.
Disadvantages of solving problem s by lin e a r program m ing One should not become too optimistic, how ever, and consider the linear programming tech nique the only innovative procedure that can be used in the analysis o f complex managerial prob lems. Certain disadvantages to linear program ming probably will not be overcome in the near future. Some of the more serious of problems that would be encountered in application of pro gramming to practice planning are given. Managers or operators of practices, because they may be unfamiliar with the procedure, would need assistance in interpreting solutions, especially if these solutions are obtained by computer. Even professional management services do not tend to offer such radical departures from traditional budgeting methods. More refined data are needed to formulate a detailed programming problem than are nor mally maintained in the records kept in the aver age practice. Large problems require the use of computers operated by experienced personnel. The solu tion o f a single large problem could, therefore, becom e expensive. 1104 ■ JADA, Vol. 85, November 1972
Sum m ary and conclusions A s practices grow in size, as the lines o f spe cialization becom e more closely drawn, and as the search for greater efficiency becomes para mount, more practice problems are likely to be solved by mathematical programming tech niques. I have illustrated only one such technique here: linear programming. Because of its relative simplicity and understandability as well as its valuable results, the technique has been widely used in many fields. The full advantages of lin ear and other programming techniques in the dental field cannot be obtained until more ef ficient data collecting and processing methods are established; the use o f the technique is in creased sufficiently to offset the cost of machine time and analysts’ salaries; and dentists or their managerial consultants becom e interested in the concept of programming and are willing and able to learn to interpret the solutions obtained from programmed problems. The universities, especially those with den tal schools, may be in the best position for ex perimenting with these techniques and dissem inating the findings to the dental world. In many ways, the dental student, the dentist, and all per sons in the dental delivery system are being prepared for change and for a period of unpre cedented increase in demand for care. This in crease will be accompanied by greater empha sis on practice efficiency. Mathematical pro gramming has been developed specifically to aid the search for greater efficiency. Adoption of such an approach to dental practice will not be easy; however, it can be done and it can be re warding. M ost of all, programming offers too important an advance in knowledge to be dis missed as either excessively complex or ab stract and not to be considered in the search for better ways to deliver more dental care. Dr. Doherty is with the department of behavioral sciences and community health, the University of Connecticut Health Cen ter, McCook Hospital, 2 Holcomb St, Hartford, 06112. 1. Brandhorst, W.S. Dental economics. J Am Col Dent 37:119 April 1970. 2. Dantzig, G.B. In Koopmans, T.C., ed. Activity analysis of pro duction and allocation. New York, John Wiley & Sons, Inc., 1951, p 339. 3. Vajda, S. Readings in linear programming. New York, John Wiley & Sons, Inc., 1967. 4. Baumol, W.J. Economic theory and operations analysis, ed 2. Englewood Cliffs, NJ, Prentice-Hall, 1965, p 84.
N eville D oherty, PhD, Hartford, Conn
L in e a r prog ram m ing is a m odern
m athem atical
te ch n iq u e useful fo f solving p ro b lem s that involve th e allo catio n of s c a rc e re so u rce s am ong alte rn a tive en d s. T h is pap er d is c u s s e s the te ch n iq u e and, with the use of a sim p lified e x a m p le , illu strates a potential ap p licatio n in dental p ractice .
Even a cursory glance at the flood o f current lit erature about the politics and econom ics of den tal health and dental health care could convince a skeptic that dentistry, although never a static science, is in the midst o f change and on the threshold of upheaval. Manifestations o f this dynamism are well known. They include ex pectations o f vast increases in the demand for dental care, the application o f new knowledge and technology, and the emerging prominence o f prepayment and prevention as rational alter natives to postpayment and dental disease. Along with these changes, traditional methods o f practice organization and planning are being modified. Because of the search for efficiency and productivity in meeting new demands and in delivering care, there are growing movements toward group practice, the division of tasks along more specialized lines, and the utilization of modern managerial techniques and aids.1 A s these changes occur, the dentist, long a traditional entrepreneur, will be faced with few options other than to improve his own manager ial capabilities, or to delegate his managerial responsibility externally to professional man
agement consultants or internally to a special ist. In dentistry, as in any other area that pro vides services, many decisions can and always will be made on the bases o f common sense, in stinct, and experience. H owever, new decisions require more specialized knowledge and under standing. The objective o f this paper is to illustrate a technical method of decision-making called lin ear programming. It is a product o f modern mathematics, and belongs in the family o f ana lytical techniques almost stigmatized by such terms as programming, operations research, and activity analysis. The technique was developed during World War II for scheduling of shipments o f food and war materials. The area of problem solving, how ever, has been o f interest for a longer period; literature on the subject in both mathematics and econom ics dates to the 1920s. In 1947, the seminal paper was published by the mathemati cian, Dantzig.2 Subsequently, the technique has been modified and refined and has found widespread use in industry, government, agri culture, and research organizations to solve complex personnel allocation, diet planning, transportation, business management, and en terprise planning problems.3 Because linear programming and other quan titative methods o f problem solving are based on mathematical concepts, some difficulty may be encountered in the interpretation o f the find ings into meaningful terms. Under such circum stances the dentist, despite concern about the efficiency o f his practice, may tend to equate his inability to understand with impracticality. H e may dismiss these new techniques, giving cre dence thereby to the thought that “ People would rather live with problems they cannot solve than with solutions they cannot understand.” JA D A , V o l. 85, N o ve m b e r, 1972 ■ 1099
Although much of the credit for demonstrat ing applications o f programming methods be longs to specialists known as operations re searchers, it is not necessary to be in operations research to gain an understanding of the prin ciples involved. Indeed, if these techniques are to benefit m odem dentistry, it is important that information about them be made available not only to researchers, but to teachers and practi tioners as well. Failure to do this could actually impede the development of new variations and refinements in technique and the application of new capabilities in solving dental problems. Thus, although this paper will not show the den tist how he can use linear programming in his office, I hope it will enhance his understanding o f technical approaches to decision-making and their potential for distinguishing seemingly con flicting goals.
P otential of linear program m ing The value of a linear programming solution to a problem lies in the fact that, under given con ditions, no other solution or plan of operation will give more efficient results. Mathematically, it is a technique to optimize a linear objective, subject to restraints that can be expressed or approximated in linear terms. Operationally, its value derives from the fact that the optimum solution, even for decisions involving four or five variables, is nearly always beyond one’s best judgment. Because the constraints on the practicing dentist are of a linear form, the appli cation of this technique is feasible as well as po tentially useful. The term linearity describes the fact that the equations used to express the relations among the variables in a problem are of the general form, y = a + b x (where a and b are any numbers). If one were to use such an equation to compute a table of values for x and y, and then plotted these values on a graph, he would find that he had drawn a set of points— all of which lie along a straight line; y = a + b x is, therefore, called a linear equation. When used in conjunction with a computer (such arrangements are available either through private or university facilities), linear program ming permits a comparison o f alternatives for planning on a scale unheard o f a few years ago. A fact of modern life is that solutions to every day, seemingly simple problems can involve 1100 ■ JADA, Vol. 85, November 1972
dozens o f restrictions and alternative courses of action. When such decisions are made by tra ditional methods, only a small subset of the al ternatives can be considered; as a result, there is a considerable risk that some of the better possibilities will be overlooked. With the use of linear programming, we can search among all feasible solutions to a particular class of such problems, that is, those with linear constraints and a linear objective. The technique has achieved a noteworthy record for finding the best solu tion or solutions in a matter of minutes, even seconds.4
W hen is linea r program m ing possible? In addition to the objectives a dentist has for his practice, such as treating more patients, reduc ing costs, increasing net income, and making better use of his assisting staff and equipment, three conditions must exist before a problem can be formulated and solved by linear program ming. The absence of any one of these either simplifies the problem to such an extent that the solution is easy or introduces such a complex problem that linear programming cannot be used to obtain a solution. The first condition is that either more than one limited resource or more than one restric tion on the solution to the problem must exist. Suppose, for example, that adequate equipment is available but that the number o f assistants is the only limiting resource to produce more dental services. In such a situation, the entire practice should be managed to produce the alter native with the greatest benefit, depending on practice goals. This benefit can be thought of in terms o f services or incomes or both, per assist ant. O f course, most managers are faced with limited resources or restrictions, or both, that limit their activities. Alternative courses of action, the second con dition, must be available to the decision-maker. If a practice is structured to provide only one service in a given way, the question may arise: Is it better to provide the maximum number of these services or to let the practice idle? H ow ever, even the dentist whose practice is suited only for the provision of one kind of service must make decisions as to the allocation of time, use o f assistants, method of treatment, and so forth. Thus, many courses of action usually ex
ist even in the most simple operation. The third condition necessary in forming a lin ear programming problem involves input-output relationships between resources (labor, capital, equipment, and so forth) and products (alter native dental services that can be provided). The combination in which various resources can be used to provide different amounts o f services must be specified. A lso, if income effects are of prime concern, the prices or expected prices of the resources used and services provided must be known, or at least estimated. If these three conditions—two or more limit ed resources or restrictions, alternative courses o f action, and input-output relationships— are present, the approximate conditions under which a management decision must be made usually can be expressed mathematically. The use of linear programming in the solution o f a hypo thetical practice problem containing each of these three conditions can be illustrated in a situation. First, a caution: real life, with all its vagaries, is too complicated to program, even on the most sophisticated electronic equipment. For this reason w e use mathematical models to simplify reality and, thereby, to express it in manageable terms. The level of mathematical sophistication to which such methods can be advanced is quite remarkable; here, however, the use of mathematics is limited to the level necessary to illustrate a technique with poten tially useful applications in dentistry.
A linea r pro g ra m m in g exam ple ■ The problem : Suppose a practice offers gen eral dentistry and pedodontics. The practice uses two operatories, three assistants, and one dentist. The dentist and his assistants each work an eight-hour day to provide the services; each operatory, therefore, is also available for eight hours. The dentist who owns and manages the practice is, let us say, curious to examine the potential o f operations research for means to improve the efficiency o f his practice— not in terms of improving techniques that his experi ence has enabled him to develop to a sophisticat ed level, but in terms o f organizing the use of his scarce resources (himself, his assistants, and his equipment) in the most productive manner possible. Examination of the practice records indicates that an operatory is used for 45 minutes, on the
average, for each pedodontic and general dental care appointment. A lso, delivery of a pedodon tic service requires IV2 hours o f assistants’ time and 15 minutes of the practitioner’s time. G en eral dentistry, on the other hand, takes one hour of assistants’ time and a half hour of the den tist’s time. On the basis o f expected charges for pedodontic and general care (a schedule may have been worked out with a union or insurance company, for example), a net income o f $8 for a pedodontic service and $6 for a general den tal service can be estimated. The problem is to find the combination o f services that would yield the best returns on the scarce or limited resourc es. ■ Linear program ming form at: Several tech niques can be used to obtain a solution to this problem. The linear programming technique, however, can be shown simply. P equals the number o f pedodontic services to be provided; G equals the number of general dental services to be provided; T equals hours of operatory time unused; A equals hours o f as sistants’ time unused; and D equals hours of dentist’s time unused. The problem can be represented by these re lationships. Maximize returns (the objective function): 8P+6G . Subject to the constraints: (a) 0.7 5 P + 0 .7 5 G + T = 16 hours of operatory time; (b) 1.50P+ 1 .00G + A = 24 hours of assist ants’ time; (c) 0 .2 5 P + 0 .5 0 G + D = 8 hours o f dentist’s time. So that Ps=0, Gs=0, AsM), Ds=0, Ts=0 (non negativity conditions). This is the standard form o f a programming problem. It consists o f three parts: the function, called the objective function, whose value is to be maximized or minimized. In this instance it is dollars per service: $8 for a pedodontic ser vice and $6 for a general dental service. The constraints are the resource limitations. The nonnegativity conditions are statements that preclude use of negative amounts of resources; for example, one cannot use less than zero hours of one’s own time. What do these parts mean? And how can they be used? A start is made by examining the con straints. The first equation (a) says that the amount of operatory time used in pedodontic services, plus the amount used in general den Doherty: LINEAR PROGRAMMING ■ 1101
tal services, plus the amount unused must equal 16 hours, and that the amount of operatory time needed is three quarters of an hour for both a pedodontic service and a general dental service. This equation represents the operatory or capi tal equipment restriction in this problem. In Figure 1, it is shown graphically by the line AB. Each point on AB represents a combination of appointments for pedodontic and general care that is exactly equal to 16 hours. Line AB also indicates that, if operatories were the only lim ited resource and if all available operatory time were used for pedodontic or general dentistry, or both, a dentist could handle an additional pa tient for pedodontic care by giving up one pa tient for general care. Thus, operatories sub stitute at a 1:1 ratio between general and pedo dontic services. The second constraint (b) says that 1Vi hours of assistants’ time per pedodontic service, plus one hour of assistants’ time per general dental service, plus the time unused must equal 24 hours a day. Therefore, this equation represents the labor restriction. It is shown graphically by the line CE in Figure 1. Each point on CE repre sents a combination of assistants’ time, for pedo dontic and general care that requires the use of exactly 24 hours. Line CE also indicates that, if labor were the only limited resource and if all available labor were used, the dentist could schedule W2 additional appointments in general dentistry by taking one less pedodontics ap pointment. Thus, labor substitutes at a 1.5:1 ratio between general and pedodontic services. The third constraint (c) says that 0.25 hours (15 minutes) of the dentist’s time per pedodontic service, plus 0.50 hours (30 minutes) of the den tist’s time per general service, plus the unused dentist’s time must equal eight hours per day. Consequently, this equation represents the pro fessional or managerial constraint in this prob lem. It is shown graphically by the line FH in Figure 1. Each point on FH represents a com bination of pedodontic and general dental ser vices that requires the use of exactly eight hours o f the dentist’s time. The line also indicates that if the dentist’s time was the only limited resource and if all his available time were used, he could treat one additional general dental patient by taking two less children. Thus, management substitutes at a 0.5:1 ratio between general and pedodontic services. The heavy line FY E in Figure 1 represents the production possibility curve. This means 1102 • JADA, Vol. 85, November 1972
Fig 1 ■ Resource restrictions.
that under the conditions specified, it is possible to produce only those combinations of general or pedodontic services, or both, that appear on on or below and to the left of the production pos sibility curve. N o combination o f services to the right and above the line FY E can be produced because of the lack of one or more of the limited resources. The area bounded by FY E is, in fact, the area of technical feasibility; the line AB, de fining the operatory restraint, lies totally above this area. The dentist has more than enough operatory time to provide any of the combina tion of services permitted by the scarce resourc es— his assistants and himself. For this reason, the operatory restriction can be ignored for the time in this analysis; it is not a strategic factor in this particular problem. (Such a situation is probably more common than we like to think. It may arise because of the indivisibility of equip ment, for example, a dentist cannot buy 1Vi oper atories. It may arise because of the common sense dictum: if it is there, use it, even if the pa tient waits unattended for 15 minutes!) The objective function says that $8 per ped odontic service provided plus $6 per general ser vice provided will equal the return. This return must be the maximum return under the condi tions specified in the constraints (a, b, and c). Thus, the solution must fall on the production possibility curve presented in Figure 1. The production possibility curve seen in Fig ure 2 bounds the region of feasible output solu tions. The object of a programming calculation, however, is to pick the optimal combination among the feasible output combinations. For the example, this is defined as the level of out put yielding the highest return to the practice.
Fig 2 ■ Optimum solution.
This point is located through the use of isoreve nue curves. A ny two points on such a curve rep resent combinations o f general and pedodontic services that yield the same incomes, for exam ple, income at combination M is the same as that at combination N . In a linear program these iso revenue curves are always straight lines and they are parallel. The equation for a typical isoreve nue curve in the example is: income equals 8P+ 6G. The curve representing a $100 income level, for instance, has the equation: 100=8P +6G , or G = 100—4/3P, so that the isorevenue curve is a straight line, the slope of which is -4 /3 . Similar ly, all lines representing greater income levels will be higher than, but with the same slope as the $100 line; those lines representing lesser in come levels will be lower than the $100 line. Sev eral of these lines, representing increasing in come levels, are shown in Figure 2. The solution to the problem is geometrically represented by the point in the feasible region that is tangent to (lies beneath and touches) the highest attainable isorevenue curve. This point is found by moving to higher and higher isoreve nue curves until one reaches the production pos sibility curve. In this example the result or solu tion occurs at point Y (Fig 2); that is, at a point where P equals 8 and G equals 12. Therefore, the provision o f 8 pedodontic services and 12 general services would enable the practice to make the most efficient use of its scarce re sources.
The result also could be obtained by solution o f the simultaneous equations defined by the constraints (a, b, and c). Because one o f the equations is redundant, that is, a linear function of another one, the solution is reduced to two equations and two unknowns. In complex and realistic problems, more powerful and compli cated computational techniques would be nec essary. These results can be used geometrically by inspecting Figures 1 and 2, or arithmetically by replacing the P’s and G ’s in the constraints with 8 ’s and 12’s, to obtain the additional informa tion: —0.75(8)+0.75( 12)+ T = 16; T equals one hour of operatory time that can go unused. — 1.50(8)+1.00(12)+A=24; A equals zero hours of the assistants’ time that should go un used. — 0.25(8)+0.50(12)+D =8; D equals zero hours of the dentist’s time that should go unused. A lso by solving for the objective function (in com e equals 8P+6G ) for the values P equals 8 and G equals 12, the maximum possible net re turn is $136 per day. N o other combination of services will return $136 or more, and all other combinations will return less than $136. The importance of this last result cannot be underestimated. It shows that net income per service, alone, can be a poor indicator of the op timum delivery pattern. Pedodontics, which in this situation produces an average net income of $8 per service, optimally should be provided in a ratio to general services o f 2:3, even though general dental services produce only $6 per ser vice. Other information is not evident in the so lution but is available with use of the linear pro gramming procedure. For example, it will in dicate the value o f an additional hour of the as sistants’ and dentist’s time, the so-called mar ginal value or marginal contribution to the prac tice. A lso the optimal pattern can be found of shifts between pedodontic and general services that should be adopted if their relative fees vary or if other changes occur (such as different time schedules for the assistants or the dentist, or both).
Advantages of solving problem s by linear program m ing Many advantages in the use o f linear program ming aid in the solution of problems about which Doherty: LINEAR PROGRAMMING ■ 1103
decisions must be made. Several o f these ad vantages have been illustrated or discussed and may be summarized. The manager, or decision-maker, is certain that no other feasible solution— given his fixed resources or restrictions, his alternatives, and his input and output relationships— will permit him to satisfy his practice objectives, whether they are increased income, reduced costs, or any reorganization o f his practice to improve his delivery o f dental care. The manager, given the linear programming solution, knows exactly what an additional unit of his limited resources is worth. H e knows the range over which income may vary before a change in the practice plans should be considered. In addition, he knows the types of changes that are necessary if his income varies outside this known range. The manager knows exactly those conditions under which the solution to the problem has been obtained. These conditions are stated in the equations that represent the mathematical for mulation o f the problem.
Disadvantages of solving problem s by lin e a r program m ing One should not become too optimistic, how ever, and consider the linear programming tech nique the only innovative procedure that can be used in the analysis o f complex managerial prob lems. Certain disadvantages to linear program ming probably will not be overcome in the near future. Some of the more serious of problems that would be encountered in application of pro gramming to practice planning are given. Managers or operators of practices, because they may be unfamiliar with the procedure, would need assistance in interpreting solutions, especially if these solutions are obtained by computer. Even professional management services do not tend to offer such radical departures from traditional budgeting methods. More refined data are needed to formulate a detailed programming problem than are nor mally maintained in the records kept in the aver age practice. Large problems require the use of computers operated by experienced personnel. The solu tion o f a single large problem could, therefore, becom e expensive. 1104 ■ JADA, Vol. 85, November 1972
Sum m ary and conclusions A s practices grow in size, as the lines o f spe cialization becom e more closely drawn, and as the search for greater efficiency becomes para mount, more practice problems are likely to be solved by mathematical programming tech niques. I have illustrated only one such technique here: linear programming. Because of its relative simplicity and understandability as well as its valuable results, the technique has been widely used in many fields. The full advantages of lin ear and other programming techniques in the dental field cannot be obtained until more ef ficient data collecting and processing methods are established; the use o f the technique is in creased sufficiently to offset the cost of machine time and analysts’ salaries; and dentists or their managerial consultants becom e interested in the concept of programming and are willing and able to learn to interpret the solutions obtained from programmed problems. The universities, especially those with den tal schools, may be in the best position for ex perimenting with these techniques and dissem inating the findings to the dental world. In many ways, the dental student, the dentist, and all per sons in the dental delivery system are being prepared for change and for a period of unpre cedented increase in demand for care. This in crease will be accompanied by greater empha sis on practice efficiency. Mathematical pro gramming has been developed specifically to aid the search for greater efficiency. Adoption of such an approach to dental practice will not be easy; however, it can be done and it can be re warding. M ost of all, programming offers too important an advance in knowledge to be dis missed as either excessively complex or ab stract and not to be considered in the search for better ways to deliver more dental care. Dr. Doherty is with the department of behavioral sciences and community health, the University of Connecticut Health Cen ter, McCook Hospital, 2 Holcomb St, Hartford, 06112. 1. Brandhorst, W.S. Dental economics. J Am Col Dent 37:119 April 1970. 2. Dantzig, G.B. In Koopmans, T.C., ed. Activity analysis of pro duction and allocation. New York, John Wiley & Sons, Inc., 1951, p 339. 3. Vajda, S. Readings in linear programming. New York, John Wiley & Sons, Inc., 1967. 4. Baumol, W.J. Economic theory and operations analysis, ed 2. Englewood Cliffs, NJ, Prentice-Hall, 1965, p 84.