einstein 2003; 1:105-9

105Linear programming applied to healthcare problems

ORIGINAL ARTICLE

Linear programming applied to healthcare problemsProgramação linear aplicada a problemas da área de saúde*

Frederico Rafael Moreira1

* Clinical Research Center of Hospital Israelita Albert Einstein.1 Statistician of the Clinical Research Center, Teaching and Research Institute.

Corresponding author: Frederico Rafael Moreira - Hospital Israelita Albert Einstein - Av. Albert Einstein, 627/701 - Morumbi - Prédio Joseh Feher (Bloco A) - Piso Chinuch - CEP 05651-901 - São Paulo(SP), Brazil. Tel.: (11) 3747.0456 - Fax: (11) 3747.0302 - e-mail: fredericorm@einstein.br

Received on July, 2003 – Accepted on November 20, 2003

ABSTRACTObjective and Method: To present a mathematical modelingtechnique by means of linear programming as an efficient tool tosolve problems related to optimization in healthcare. Twoapplications are approached: formulation of a balanced diet at aminimum cost and optimal allocation of resources for a set ofmedical interventions that comply with cost and medical visitrestrictions. Results: The balanced diet proposed would comprise1.4 glasses of skimmed milk/day and 100 g of salad/day (2/10 of a500 g portion) at a total minimum cost of R$ 2.55/day. The optimalsolution for the allocation model among the five types of medicalintervention programs maximizing quality-adjusted life year wasestablished as follows: use of 100% of intervention type 4 and 50%of intervention type 2, determining a maximum value of 20.5 QALY.Conclusion: In a world with increasingly scarce resources andevery day more competitive, linear programming could be used tosearch optimized solutions for healthcare problems.

Keywords: Resource allocation; Linear programming; Operationsresearch; Optimization

RESUMOObjetivo e Método: Apresentar a técnica de modelagemmatemática via programação linear como eficiente ferramentapara soluções de problemas que envolvam otimização para a áreade saúde. Duas aplicações foram tratadas: a formulação de umadieta balanceada a custo mínimo e a alocação ótima de recursospara um conjunto de intervenções médicas satisfazendo restriçõesde custo e visitas médicas. Resultados: A dieta balanceadaproposta seria composta por 1,4 copos de leite desnatado/dia e100 g de salada/dia (2/10 de porção de 500 g) a um custo mínimototal de R$ 2,55/dia. A solução ótima para o modelo de alocaçãoentre cinco tipos de programas de intervenção médica maximizandoos anos de vida ajustados para a qualidade foi assim determinada:utilização de 100% da intervenção tipo 4 e 50% da intervenção tipo2 determinando valor máximo de 20,5 AVAQ’s. Conclusão: Frentea um mundo competitivo e com recursos cada vez mais escassos,a programação linear pode ser utilizada na busca de soluçõesotimizadas para problemas da área de saúde.

Descritores: Alocação de recursos; Programação linear; Pesquisaoperacional; Otimização

INTRODUCTIONOperations Research is a science designed to providequantitative tools to decision-making processes. Itcomprises a set of mathematical optimization andsimulation methods and models, such as LinearProgramming, Non-linear Programming, CombinatoryOptimization, Theory of Queues, Dynamic Programming,Theory of Decisions, etc. Today, implementingoptimized solutions by linear programming has reducedcosts by hundreds or even thousands dollars in manymiddle to large-sized companies in severalindustrialized countries(1). Many researchers point outthe development of linear programming method as oneof the most important scientific advances of the secondhalf of the 20th century. Unfortunately, few Braziliancompanies use some optimization technique in theirprocesses, be them productive or not.

Linear programming has demonstrated to be analternative solution to plan brachytherapy, replacingthe traditional solutions based on trial and error(2).Linear programming has been used to formulatebalanced diets at minimum cost and complying with aset of nutritional restrictions(3-4).

The main objective of this article is to present anddisseminate the linear programming method, in orderto attain optimized solutions of healthcare problemsinvolving Economics and Nutrition. First, we presentthe formulation of a diet at minimum cost using onlytwo nutritional restrictions aiming at makinginvestigators in the field of healthcare more familiarwith the terms and potentialities of the methodreported.

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Later, we describe an optimized solution forproblems related to allocation of medical interventionsthat complies with a set of budget restrictions andmedical visits. Decision trees have been applied toeconomic analysis, such as cost-effectiveness, to choosethe best management strategies. The trees may indicatewhich interventions use less resource and representbetter quality of life. However, these techniques donot show how to effectively allocate resources in severalmedical intervention programs because they are notoptimization models. Thus, the second application hasthe purpose of presenting algebraic modeling by linearprogramming as a complementary tool to the traditionalmethods of economic analysis in health.

METHODSDescribing a linear programming method: Optimizationmodels are defined by an objective function composedof a set of decision-making variables, subject to a set ofrestrictions, and presented as mathematical equations.The objective of optimization is to find a set of decision-making variables that generates an optimal value forthe objective function, a maximum or minimum valuedepending on the problem, and complies with a set ofrestrictions imposed by the model. Such restrictionsare conditions that limit the decision-making variablesand their relations to assume feasible values. In linearprogramming models, the objective function is linear,that is, it is defined as a linear combination of decision-making variables and a set of constants, restricted to aset of linear equality or inequality equations. Therefore,we have a model composed of an objective function,restrictions, decision-making variables and parameters.

The chart represented in figure 1 shows definitions andinteractions among these components. A solution of aproblem is called optimal when the decision-makingvariables assume values that correspond to the maximumor minimum value of the objective function andcomplies with all restrictions of the model.

An algebraic representation of a generic formulation oflinear programming model could be presented asfollows:

To maximize or minimize the objective function:

Z = c1 x1 + c2 x2 +............ cn xn (a)

It is subject to restrictions:

a11 x1 + a12 x2 +.............. a1n x n ≤ r1 (b)a21 x1 + a22 x2 +.............. a2n xn ≤ r2 (c)

.... .......am1 x1 + am2 x2 +............. a m n x n ≤ r m (d)

x j ≥ 0 (j = 1,2,............,n) (e)

where:

(a) represents the mathematical function encoding theobjective of the problem and is called objective function(Z) in linear programming, this function must be linear.(b)-(e) represents the linear mathematical functionencoding the main restrictions identified.(e) non-negativity restriction, that is, the decision-making variables may assume any positive value or zero.“xj” corresponds to the decision-making variables thatrepresent the quantities one wants to determine tooptimize the global result.“ci” represents gain or cost coefficients that eachvariable is able to generate.“rj “ represents the quantity available in each resource.“ai j” represents the quantity of resources each decision-making variable consumes.

The diet problem: Description: To illustrate anapplication of linear programming in diet formulation,let us assume that a diet, for justifiable reasons, isrestricted to skimmed milk and a salad using well-knowningredients. We know that the nutritional requirementswill be expressed as vitamin A and calcium controlledby their minimum quantities (in milligrams). Table 1summarizes the quantity of each nutrient available infoods and their daily requirement for good healthconditions of an individual, as well as the unitary costof these foods. The objective is to minimize the totaldiet cost and comply with nutritional restrictions.

Objective functionMeasure of effectiveness as a mathematicalfunction of decision-making variables. It maximizesor minimizes a measure of performance

RestrictionsSet of linear equality or inequality equations thatthe decision-making variables must comply with.

Decision-making variablesUnknown variables that will be determinedby the solution of the model. They arecontinuous and non-negative variables.

ParametersThey are previouslyknown constants orcoefficients

It is subject to

Components

Figure 1. Linear programming model components

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107Linear programming applied to healthcare problems

Table 1. Costs, nutrients and predetermined nutritional restrictions

Nutrient Milk (glass) Salad (500 mg) Minimum Nutritional

Requirement

Vit. A 2 mg 50 mg 11 mg

Calcium 50 mg 10 mg 70 mg

Cost/unit R$1.50 R$3.00

Decision-making variables:X1 = quantity of milk (in glasses)/dayX2 = quantity of salad (in 500 g portions)/day

Objective function (Z): The function to be minimized– total diet cost – is the objective function of thisproblem. It is defined by the combination of foods X1

(milk) and X2 (salad) and their unit costs, R$ 1.50 (onereal and fifty cents) and R$ 3,00 (three reals),respectively. The cost function is a linear function ofX1 and X2, that is: Z = 1.5X1 + 3.0X2

Restrictions: The total quantity of vitamin A in thisdiet should be equal or greater than 11 mg. The foodformulation should provide at least 70 mg of calcium.We could not take into account negative quantities offood; thus, X1 and X2 should be non-negative quantities(X1 ≥ 0; X2 ≥ 0)

Mathematical model for the diet problem

To minimize Z = 1.5X1 + 3X2

x1, x2

it is subject to

2X1 + 50X2 ≥ 11 (vitamin A restriction)50X1 + 10X2 ≥ 70 (calcium restriction) X1 ≥ 0; X2 ≥ 0 (non-negativity restriction)

Description of the problem of healthcare resourceallocation: Researchers want to allocate resourcesamong five medical intervention programs for a certainpopulation aiming at maximizing the quality-adjustedlife year (QALY). QALY is a measure that considersquantity and quality of life related to the medicalintervention applied. It is estimated that a maximumof 300,000 monetary units be spent and the number ofmedical visits is expected to remain as 40,000, at most.

It is also assumed that each patient receives onlyone intervention. Moreover, fraction values of QALY,intervention costs and number of medical visits areallowed. The objective it to maximize total QALYaccumulated by interventions in order to comply withthe budget restrictions and the number of medical visitsdescribed in table 2.

Table 2. QALY, intervention cost and medical visits values

Identification of

intervention Intervention cost N. of medical

programs QALY (x 103 monetary units) visits (x 103)

1 10 100 40

2 15 50 50

3 15 50 50

4 13 40 15

5 9 120 30

Maximum value 300 40

Formulation of the mathematical model• Decision-making variables

Xi = fraction of each intervention i to be designatedby the model.

• Objective functionTo maximize the function Z defined as total quality-adjusted life year due to interventions:10X1 + 15X2 + 15X3 + 13X4 + 9X5The parameters of this equation represent thenumber of QALY obtained as a result of theintervention i.

• RestrictionsBudget restrictions:100X1 + 50X2 + 50X3 + 40X4 + 120X5 ≤ 300. Thecost, including all interventions, should not exceed300,000 monetary units.Visits: 40X1 + 50X2 + 50X3 + 15X4 + 30X5 ≤ 40Xi ≥ 0; Xi ≤ 1 i = 1, 2, 3, 4, 5. The decision-makingvariables may assume any value between 0 and 1.

Mathematical Model

To maximize Z = 10X1 + 15X2 + 15X3 + 13X4 + 9X5

x1, x2, x3, x4, x5

It is subject to a set of restrictions:

100X1 + 50X2 + 50X3 + 40X4 + 120X5 ≤ 30040X1 + 50X2 + 50X3 + 15X4 + 30X5 ≤ 400 ≤ X1 ≤ 1; 0 ≤ X2 ≤ 1; 0 ≤ X3 ≤ 1; 0 ≤ X4 ≤ 1; 0 ≤ X5 ≤ 1

RESULTSGraphic solution for the diet problemWhen we deal with decision-making variables (twofoods), a geometric representation is possible andconvenient for didactic purposes. In order to explorethe geometry of the problem, the restrictions were firstrepresented in a Cartesian plan (figure 2), identifyingthe feasible region, that is, the region of the Cartesianplan that complies with the set of restrictions of themodel. Second, we have to minimize cost, which is a

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constant for each combination of X1 and X2. Hence,different costs generate parallel lines where cost is aconstant in each line. Then, the cost lines were drawnin the graphic to obtain the minimum cost that complieswith nutritional restrictions. Observe that cost (z)decreases as we move towards the intersection of linesthat identify calcium and vitamin A restrictions. Theexact point in which cost is minimized corresponds tothe intersection of these lines. This point is easily foundby simultaneously determining the solution of theequations: 2X1 + 50X2 = 11 and 50X1 + 10X2 = 70.This solution, X1 = 1.4 and X2 = 0.2, corresponds tothe total minimum cost of Z = R$ 2.55. In other words,the optimal solution corresponds to a diet of 1.4 glassesof skimmed milk/day and 100 grams of salad/day (2/10of a 500 g-portion), at a minimum cost of R$ 2.55.

Figure 2. The feasible region is limited by the lines 2X1 + 50X2 = 11(vitamin A) and 50X1 + 10X2 =70 (calcium) and is identified by the arrows.The lines in red (lines z) represent the ‘objective function’ to be minimizedin order to have a minimum cost diet. The point in black (letter a) determinesthe optimal solution X1 = 1.4 X2 = 0.2, corresponding to a minimum cost ofR$ 2.55 (line in bold red).

Analytical solution for the diet problem: The SimplexMethod is an algorithm created to algebraically obtaina solution. An algorithm is a set of rules that must befollowed step by step, so that, in the end, the desiredresult is attained. The Simplex Method was created byGeorge Dantzig and other scientists of the AmericanAir Force Department, in 1947(5). Microsoft ExcelSolver uses a basic implementation of Simplex Methodto solve linear programming problems. The optimalsolution obtained for this problem by Microsoft ExcelSolver indicates a diet composed of 1.4 glasses of skimmedmilk/day and 100 grams of salad/day (X1 = 1.4; X2 = 0.2

of a 500 g-portion), corresponding to total minimumcost of R$ 2.55.

Solution for the resource allocation problem: Theoptimized solution for the model of medical interventionprogram allocation calculated by Microsoft Excel Solvercorresponds to the complete use (100%) of theintervention program number 4 and a fraction of 50%for the intervention program number 2, not using theinterventions number 1, 3 and 5 (X1 = 1; X2 = 0.5; X3 = 0;X4 = 1; X5 = 0). This numerical solution provides amaximum value of 20.5 QALY for the objective functionof this model.

The graphic solution is easily obtained when thereare two decision-making variables (as shown in theexample of diet). When there are three decision-making variables it is still possible to have a graphicsolution despite the difficulty in identifying theintersections between the planes defined in a three-dimensional space. In cases of four or more decision-making variables, graphic solution is impossible andthe only alternative is an analytical solution by SimplexMethod.

DISCUSSIONThe demand for efficient decisions in healthcaredelivery gives opportunity to application of optimizationtechniques in problems related to resource allocation,which could be a complementary tool to economicevaluation models. The numerical results presented inthe modeling of medical intervention programallocation show this potential applicability (X1 = 0; X2 =50%; X3 = 0; X4 = 100%; X5 = 0). The diet proposed inthe second model provides a reduced number ofrestrictions, decision-making variables (milk and salad)and the type of restriction used since it has the didacticpurpose of demonstrating that this type of algebraicmodeling and its reporting to physicians are efficient.Low-income populations could benefit from formulationof balanced diets at a minimum cost, prepared basedon foods available and/or affordable, thus complyingwith dozens or hundreds of restrictions. Potentialapplications of optimization methods may includeassessment of economic impact of several therapies bymeans of diseases and evaluation of product prices.For instance, optimized prices of product componentscould be assessed considering some restrictions, suchas expenses with research and development, cost ofalternative treatments, marketing strategies andestimates for sales projection. Mathematical modelingusing linear programming may be applied to problemsrelated to optimized resource allocation in healthcare.

Vitamin A

Milk

Calcium

Salad

2

1.5

1

0.5

0.2

0.5 1 1.5 2

a

Zx =2.55

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The objective could be, for example, to maximize QALYand life years, or to minimize the number of individualswho develop complications, the cost of treatment, as wellas to define the components for diet formulation ormedications. Technical information about constructionof these models is found in Bazarra(6) and Hillier &Lieberman(7). Large-sized models involving hundreds ofrestrictions and variables may be solved by means ofalgebraic modeling software, such as AIMMS (AdvancedInteractive Mathematical Modeling System), GAMS(General Algebraic Modeling System), AMPL (AlgebraicMathematical Programming Language), LINDO (LinearInteractive and Discrete Optimizer). Technical informationcomparing this software is found in the publicationOR/MS Today, August 2001(8).

CONCLUSIONThe type of algebraic modeling presented in this articleknown as linear programming could be considered auseful tool to support decision-making processes inhealthcare. In a world with increasingly scarce resources

and every day more competitive, the search of optimizedsolutions to replace traditional methods based oncommon sense and trial and error may become an issueof survival for many organizations.

REFERENCES1. Lieberman G, Hillier F. Introduction to mathematical programming. New York:

McGraw-Hill; 1991.

2. D’Souza WD, Meyer RR, Thomadsen BR, Ferris MC. An iterative sequentialmixed-integer approach to automated prostate brachytherapy treatment planoptimization. Phys Med Biol 2001;46:297-322.

3. Colavita C & D’Orsi R. Linear programming and pediatric dietetics. Br J Nutr1990;64:307-17.

4. Henson S. Linear programming analysis of constraints upon human diets. JAgric Econ 1991;42:380–93.

5. Dantzig GB. Origins of the simplex method In: Nash SG, editor. A history ofscientific computing. New York: ACM Press; 1990.

6. Bazaraa MS, Jarvis J, Sherali HD. Linear programming and network flows. 2nd

ed. New York: Wiley, 1990.

7. Hiller FS, Lieberman GJ. Introduction to operations research. New York: Mc-Graw Hill, 2000.

8. Fourer R. 2001 Software Survey: linear programming. OR/MS Today2001;28:58-68.