Concept Architecture
Concept
Theoretically, Linear Programming is a mathematical optimisation technique used to determine the best allocation of limited resources by maximising or minimising a linear objective function subject to a set of linear constraints. It is founded on optimisation theory, convex analysis and linear algebra and provides an exact solution for optimisation problems in which both the objective function and constraints are linear. Linear Programming forms one of the fundamental mathematical tools of operations research and health economics.
Mathematically, a Linear Programming problem consists of an objective function, decision variables and linear equality or inequality constraints defining a feasible region. The optimal solution occurs at an extreme point (vertex) of the feasible region, a result established by the fundamental theorem of linear programming. Solutions are typically obtained using the Simplex Method or Interior-Point Methods.
In practice, Linear Programming is extensively applied in health economics to optimise healthcare resource allocation, workforce planning, treatment selection, budget allocation and service delivery. It supports economic evaluation by identifying the allocation of constrained resources that maximises health outcomes or minimises total costs while satisfying operational, clinical and policy constraints.
Purpose
Used to determine optimal allocation of limited healthcare resources, maximise health outcomes or minimise costs, support operational decision-making and enable mathematical optimisation in health economics.
Mathematical Formulae
Primary Formula
Objective Function
Maximise (or Minimise)
Z = c?x
Subject to
Ax � b
x � 0
where:
- Z = objective value
- c = objective coefficient vector
- x = decision variable vector
- A = constraint coefficient matrix
- b = constraint vector
Supporting Formulae
Equality Constraint
Ax = b
Feasible Region
{x : Ax � b, x � 0}
Optimality Condition
Optimal solution occurs at a vertex of the feasible region.
Related Mathematical Methods
- Simplex Method
- Interior-Point Method
- Linear optimisation
- Integer programming
- Convex optimisation
- Duality theory
Example
A healthcare authority allocates funding between two preventive programmes.
Objective:
Maximise
Z = 12x? + 18x?
Subject to
4x? + 6x? � 240
x? + x? � 50
x? � 0
x? � 0
Linear Programming identifies the combination of programmes that produces the greatest total health benefit while remaining within available budget and operational constraints.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| SUMPRODUCT | =SUMPRODUCT(ObjectiveCoefficients,DecisionVariables) | Calculate the objective function value. |
| MMULT | =MMULT(ConstraintMatrix,DecisionVariables) | Evaluate linear constraints. |
| Solver | Maximise or minimise the objective subject to constraints | Solve Linear Programming problems. |
| IF | =IF(Constraint<=Limit,"Feasible","Constraint Violated") | Check constraint satisfaction. |
| SUM | =SUM(DecisionVariables) | Calculate total resource utilisation. |
VBA (Optional)
Automate Linear Programming optimisation using Excel Solver, evaluate healthcare resource allocation scenarios and generate optimal solutions for budgeting, workforce planning and health economic decision models.
Sources
- Dantzig GB. Linear Programming and Extensions.
- Bazaraa MS, Jarvis JJ, Sherali HD. Linear Programming and Network Flows.
- Hillier FS, Lieberman GJ. Introduction to Operations Research.
- Winston WL. Operations Research: Applications and Algorithms.
- Boyd S, Vandenberghe L. Convex Optimization.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation.
Related Concepts (3)
Library
Publications
1
Numerical Analysis of Parameterized Nonlinear Equations — Werner C. Rheinboldt, 1st Edition ed., 1986 (John Wiley & Sons)
A specialist numerical-analysis reference on iterative solution, continuation and computation for parameterized nonlinear systems, linked here to the ontology's numerical-method and optimization branch.
BookView source →
Frequently Asked Questions (6)
What is linear programming?
A method for optimising a linear objective function subject to linear equality and inequality constraints.
Source: Nocedal J, Wright SJ. Numerical Optimization. 2nd ed. Springer; 2006. doi:10.1007/978-0-387-40065-5.
What does linear programming optimise?
Linear programming optimises a linear objective function, seeking the values that make it as large or small as required, subject to linear equality and inequality constraints. Both the quantity being optimised and the restrictions on it are linear. Optimising a linear objective under linear constraints is what defines linear programming Because both the objective and the constraints are linear, the feasible region is a convex shape whose optimum lies at its edge, which is what specialised solvers are built to find.
Source: Nocedal J, Wright SJ. Numerical Optimization. 2nd ed. Springer; 2006. doi:10.1007/978-0-387-40065-5.
What constraints does linear programming allow?
Linear programming allows linear equality and inequality constraints, restrictions expressed as linear relations that the solution must satisfy. These constraints mark out the region of permissible solutions within which the linear objective is optimised. Permitting only linear equality and inequality constraints is intrinsic to what makes a problem linear programming Because both the objective and the constraints are linear, the feasible region is a convex shape whose optimum lies at its edge, which is what specialised solvers are built to find.
Source: Nocedal J, Wright SJ. Numerical Optimization. 2nd ed. Springer; 2006. doi:10.1007/978-0-387-40065-5.
What makes a problem suitable for linear programming?
A problem is suitable for linear programming when both its objective function and its constraints are linear, since linear programming is a method for optimising a linear objective subject to linear equality and inequality constraints. Linearity throughout is the requirement. This restriction to linear objectives and constraints distinguishes linear programming from more general optimisation Because both the objective and the constraints are linear, the feasible region is a convex shape whose optimum lies at its edge, which is what specialised solvers are built to find.
Source: Nocedal J, Wright SJ. Numerical Optimization. 2nd ed. Springer; 2006. doi:10.1007/978-0-387-40065-5.
What kind of objective function does linear programming use?
Linear programming uses a linear objective function, one that depends on its variables through a linear expression, and it optimises this subject to linear equality and inequality constraints. The linearity of the objective is what pairs with the linear constraints to make the problem a linear program. This linear objective is a defining element of linear programming Because both the objective and the constraints are linear, the feasible region is a convex shape whose optimum lies at its edge, which is what specialised solvers are built to find.
Source: Nocedal J, Wright SJ. Numerical Optimization. 2nd ed. Springer; 2006. doi:10.1007/978-0-387-40065-5.
How does linear programming relate to the simplex method?
Linear programming is a method for optimising a linear objective function subject to linear constraints, while the simplex method is an algorithm that solves linear programming problems by moving systematically between feasible solutions until the optimum is found. The simplex method is one way to solve a linear program. The two are connected, since linear programming is the problem and the simplex method is an algorithm for it Because both the objective and the constraints are linear, the feasible region is a convex shape whose optimum lies at its edge, which is what specialised solvers are built to find.
Source: Nocedal J, Wright SJ. Numerical Optimization. 2nd ed. Springer; 2006. doi:10.1007/978-0-387-40065-5.
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 3 Apr 2026
Content version: 1.0.0
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- Term code
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