# Linear programming applied to healthcare problems

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2 106 Moreira FR Later, we describe an optimized solution for problems related to allocation of medical interventions that complies with a set of budget restrictions and medical visits. Decision trees have been applied to economic analysis, such as cost-effectiveness, to choose the best management strategies. The trees may indicate which interventions use less resource and represent better quality of life. However, these techniques do not show how to effectively allocate resources in several medical intervention programs because they are not optimization models. Thus, the second application has the purpose of presenting algebraic modeling by linear programming as a complementary tool to the traditional methods of economic analysis in health. METHODS Describing a linear programming method: Optimization models are defined by an objective function composed of a set of decision-making variables, subject to a set of restrictions, and presented as mathematical equations. The objective of optimization is to find a set of decisionmaking variables that generates an optimal value for the objective function, a maximum or minimum value depending on the problem, and complies with a set of restrictions imposed by the model. Such restrictions are conditions that limit the decision-making variables and their relations to assume feasible values. In linear programming models, the objective function is linear, that is, it is defined as a linear combination of decisionmaking variables and a set of constants, restricted to a set of linear equality or inequality equations. Therefore, we have a model composed of an objective function, restrictions, decision-making variables and parameters. Figure 1. Linear programming model components Components Decision-making variables Unknown variables that will be determined by the solution of the model. They are continuous and non-negative variables. Objective function Measure of effectiveness as a mathematical function of decision-making variables. It maximizes or minimizes a measure of performance It is subject to Restrictions Set of linear equality or inequality equations that the decision-making variables must comply with. Parameters They are previously known constants or coefficients The chart represented in figure 1 shows definitions and interactions among these components. A solution of a problem is called optimal when the decision-making variables assume values that correspond to the maximum or minimum value of the objective function and complies with all restrictions of the model. An algebraic representation of a generic formulation of linear programming model could be presented as follows: To maximize or minimize the objective function: Z = c 1 + c +... c n (a) It is subject to restrictions: a 11 + a a 1n r 1 (b) a 1 + a +... a n r (c) a m1 + a m +... a m n r m (d) x j 0 (j = 1,,...,n) (e) where: (a) represents the mathematical function encoding the objective of the problem and is called objective function (Z) in linear programming, this function must be linear. (b)-(e) represents the linear mathematical function encoding the main restrictions identified. (e) non-negativity restriction, that is, the decisionmaking variables may assume any positive value or zero. x j corresponds to the decision-making variables that represent the quantities one wants to determine to optimize the global result. c i represents gain or cost coefficients that each variable is able to generate. r j represents the quantity available in each resource. a i j represents the quantity of resources each decisionmaking variable consumes. The diet problem: Description: To illustrate an application of linear programming in diet formulation, let us assume that a diet, for justifiable reasons, is restricted to skimmed milk and a salad using well-known ingredients. We know that the nutritional requirements will be expressed as vitamin A and calcium controlled by their minimum quantities (in milligrams). Table 1 summarizes the quantity of each nutrient available in foods and their daily requirement for good health conditions of an individual, as well as the unitary cost of these foods. The objective is to minimize the total diet cost and comply with nutritional restrictions.

3 107 Table 1. Costs, nutrients and predetermined nutritional restrictions Nutrient Milk (glass) Salad (500 mg) Minimum Nutritional Requirement Vit. A mg 50 mg 11 mg Calcium 50 mg 10 mg 70 mg Cost/unit R\$1.50 R\$3.00 Decision-making variables: = quantity of milk (in glasses)/day X = quantity of salad (in 500 g portions)/day Objective function (Z): The function to be minimized total diet cost is the objective function of this problem. It is defined by the combination of foods (milk) and X (salad) and their unit costs, R\$ 1.50 (one real and fifty cents) and R\$ 3,00 (three reals), respectively. The cost function is a linear function of and X, that is: Z = X Restrictions: The total quantity of vitamin A in this diet should be equal or greater than 11 mg. The food formulation should provide at least 70 mg of calcium. We could not take into account negative quantities of food; thus, and X should be non-negative quantities ( 0; X 0) Mathematical model for the diet problem To minimize Z = X, it is subject to + 50X 11 (vitamin A restriction) X 70 (calcium restriction) 0; X 0 (non-negativity restriction) Description of the problem of healthcare resource allocation: Researchers want to allocate resources among five medical intervention programs for a certain population aiming at maximizing the quality-adjusted life year (QALY). QALY is a measure that considers quantity and quality of life related to the medical intervention applied. It is estimated that a maximum of 300,000 monetary units be spent and the number of medical visits is expected to remain as 40,000, at most. It is also assumed that each patient receives only one intervention. Moreover, fraction values of QALY, intervention costs and number of medical visits are allowed. The objective it to maximize total QALY accumulated by interventions in order to comply with the budget restrictions and the number of medical visits described in table. Table. QALY, intervention cost and medical visits values Identification of intervention Intervention cost N. of medical programs QALY (0 3 monetary units) visits (0 3 ) Maximum value Formulation of the mathematical model Decision-making variables X i = fraction of each intervention i to be designated by the model. Objective function To maximize the function Z defined as total qualityadjusted life year due to interventions: X + 15X + 13X + 9X The parameters of this equation represent the number of QALY obtained as a result of the intervention i. Restrictions Budget restrictions: X + 50X + 40X + 10X 300. The cost, including all interventions, should not exceed 300,000 monetary units. Visits: X + 50X + 15X + 30X 40 X i 0; X i 1 i = 1,, 3, 4, 5. The decision-making variables may assume any value between 0 and 1. Mathematical Model To maximize Z = X + 15X + 13X + 9X,, x 3, x 4, x 5 It is subject to a set of restrictions: X + 50X + 40X + 10X X + 50X + 15X + 30X ; 0 X 1; 0 X 3 1; 0 X 4 1; 0 X 5 1 RESULTS Graphic solution for the diet problem When we deal with decision-making variables (two foods), a geometric representation is possible and convenient for didactic purposes. In order to explore the geometry of the problem, the restrictions were first represented in a Cartesian plan (figure ), identifying the feasible region, that is, the region of the Cartesian plan that complies with the set of restrictions of the model. Second, we have to minimize cost, which is a

4 108 Moreira FR constant for each combination of and X. Hence, different costs generate parallel lines where cost is a constant in each line. Then, the cost lines were drawn in the graphic to obtain the minimum cost that complies with nutritional restrictions. Observe that cost (z) decreases as we move towards the intersection of lines that identify calcium and vitamin A restrictions. The exact point in which cost is minimized corresponds to the intersection of these lines. This point is easily found by simultaneously determining the solution of the equations: + 50X = 11 and X = 70. This solution, = 1.4 and X = 0., corresponds to the total minimum cost of Z = R\$.55. In other words, the optimal solution corresponds to a diet of 1.4 glasses of skimmed milk/day and 100 grams of salad/day (/10 of a 500 g-portion), at a minimum cost of R\$.55. Salad Calcium Vitamin A Milk Figure. The feasible region is limited by the lines + 50X = 11 (vitamin A) and X =70 (calcium) and is identified by the arrows. The lines in red (lines z) represent the objective function to be minimized in order to have a minimum cost diet. The point in black (letter a) determines the optimal solution = 1.4 X = 0., corresponding to a minimum cost of R\$.55 (line in bold red). Analytical solution for the diet problem: The Simplex Method is an algorithm created to algebraically obtain a solution. An algorithm is a set of rules that must be followed step by step, so that, in the end, the desired result is attained. The Simplex Method was created by George Dantzig and other scientists of the American Air Force Department, in 1947 (5). Microsoft Excel Solver uses a basic implementation of Simplex Method to solve linear programming problems. The optimal solution obtained for this problem by Microsoft Excel Solver indicates a diet composed of 1.4 glasses of skimmed milk/day and 100 grams of salad/day ( = 1.4; X = 0. a Z x =.55 of a 500 g-portion), corresponding to total minimum cost of R\$.55. Solution for the resource allocation problem: The optimized solution for the model of medical intervention program allocation calculated by Microsoft Excel Solver corresponds to the complete use (100%) of the intervention program number 4 and a fraction of 50% for the intervention program number, not using the interventions number 1, 3 and 5 ( = 1; X = 0.5; X 3 = 0; X 4 = 1; X 5 = 0). This numerical solution provides a maximum value of 0.5 QALY for the objective function of this model. The graphic solution is easily obtained when there are two decision-making variables (as shown in the example of diet). When there are three decisionmaking variables it is still possible to have a graphic solution despite the difficulty in identifying the intersections between the planes defined in a threedimensional space. In cases of four or more decisionmaking variables, graphic solution is impossible and the only alternative is an analytical solution by Simplex Method. DISCUSSION The demand for efficient decisions in healthcare delivery gives opportunity to application of optimization techniques in problems related to resource allocation, which could be a complementary tool to economic evaluation models. The numerical results presented in the modeling of medical intervention program allocation show this potential applicability ( = 0; X = 50%; X 3 = 0; X 4 = 100%; X 5 = 0). The diet proposed in the second model provides a reduced number of restrictions, decision-making variables (milk and salad) and the type of restriction used since it has the didactic purpose of demonstrating that this type of algebraic modeling and its reporting to physicians are efficient. Low-income populations could benefit from formulation of balanced diets at a minimum cost, prepared based on foods available and/or affordable, thus complying with dozens or hundreds of restrictions. Potential applications of optimization methods may include assessment of economic impact of several therapies by means of diseases and evaluation of product prices. For instance, optimized prices of product components could be assessed considering some restrictions, such as expenses with research and development, cost of alternative treatments, marketing strategies and estimates for sales projection. Mathematical modeling using linear programming may be applied to problems related to optimized resource allocation in healthcare.

5 109 The objective could be, for example, to maximize QALY and life years, or to minimize the number of individuals who develop complications, the cost of treatment, as well as to define the components for diet formulation or medications. Technical information about construction of these models is found in Bazarra (6) and Hillier & Lieberman (7). Large-sized models involving hundreds of restrictions and variables may be solved by means of algebraic modeling software, such as AIMMS (Advanced Interactive Mathematical Modeling System), GAMS (General Algebraic Modeling System), AMPL (Algebraic Mathematical Programming Language), LINDO (Linear Interactive and Discrete Optimizer). Technical information comparing this software is found in the publication OR/MS Today, August 001 (8). CONCLUSION The type of algebraic modeling presented in this article known as linear programming could be considered a useful tool to support decision-making processes in healthcare. In a world with increasingly scarce resources and every day more competitive, the search of optimized solutions to replace traditional methods based on common sense and trial and error may become an issue of survival for many organizations. REFERENCES 1. Lieberman G, Hillier F. Introduction to mathematical programming. New York: McGraw-Hill; D Souza WD, Meyer RR, Thomadsen BR, Ferris MC. An iterative sequential mixed-integer approach to automated prostate brachytherapy treatment plan optimization. Phys Med Biol 001;46: Colavita C & D Orsi R. Linear programming and pediatric dietetics. Br J Nutr 1990;64: Henson S. Linear programming analysis of constraints upon human diets. J Agric Econ 1991;4: Dantzig GB. Origins of the simplex method In: Nash SG, editor. A history of scientific computing. New York: ACM Press; Bazaraa MS, Jarvis J, Sherali HD. Linear programming and network flows. nd ed. New York: Wiley, Hiller FS, Lieberman GJ. Introduction to operations research. New York: Mc- Graw Hill, Fourer R. 001 Software Survey: linear programming. OR/MS Today 001;8:58-68.

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