Concept Architecture
Concept
Theoretically, Simplex Method is an optimisation algorithm for solving linear programming problems in which a linear objective function is maximised or minimised subject to a set of linear equality and inequality constraints. Developed by George Dantzig, the method is founded on the theory of convex polyhedra and demonstrates that, when an optimal solution exists, at least one optimum lies at a vertex of the feasible region. The algorithm exists to efficiently identify this optimal vertex without evaluating every feasible solution.
Mathematically, the Simplex Method represents a linear programming problem in standard form and iteratively moves between adjacent basic feasible solutions. At each iteration, a non-basic variable enters the basis while a basic variable leaves according to the pivot operation, producing a new feasible solution with an improved objective value. The process continues until no further improvement is possible, satisfying the optimality conditions.
In practice, the Simplex Method is implemented using simplex tableaux or revised simplex algorithms within optimisation software. In health economics it is applied to resource allocation, programme budgeting, production planning, workforce optimisation and service configuration where decision-makers seek to maximise health outcomes or minimise costs subject to budgetary, staffing or capacity constraints. Modern implementations are incorporated into optimisation packages rather than performed manually.
Purpose
Used to determine the optimal allocation of limited healthcare resources by maximising or minimising a linear objective function subject to linear constraints, supporting economic evaluation, resource planning and operational decision-making.
Mathematical Formulae
Primary Formula
Maximise (or minimise):
Z = c?x
Subject to:
Ax � b
x � 0
Supporting Formulae
Standard equality form:
Ax = b
x � 0
Objective function:
Z = ?c?x?
Reduced cost:
r? = c? ? c??B??A?
Basic solution:
x? = B??b
Optimality condition (maximisation):
r? � 0 for all non-basic variables
Related Mathematical Methods
- Linear Programming
- Revised Simplex Method
- Dual Simplex Method
- Interior-Point Method
- Branch-and-Bound Method
- Sensitivity Analysis
- Duality Theory
Example
A health authority wishes to allocate funding between two preventive programmes.
Maximise:
Z = 80x? + 60x?
Subject to:
5x? + 3x? � 300
2x? + 4x? � 180
x?, x? � 0
Applying the Simplex Method identifies the optimal combination of programme investments that maximises the total health benefit while remaining within available financial and operational constraints.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| SUMPRODUCT | =SUMPRODUCT(Costs,Activities) | Calculates the objective function value. |
| MMULT | =MMULT(A_Range,X_Range) | Evaluates linear constraints. |
| Solver Add-in | Objective cell with changing variable cells | Solves linear programming problems using the Simplex LP algorithm. |
| MIN | =MIN(Range) | Assists ratio testing during manual simplex calculations. |
| IF | =IF(Constraint<=Limit,"Feasible","Infeasible") | Checks whether candidate solutions satisfy constraints. |
VBA (Optional)
VBA can automate repeated optimisation runs using Excel Solver's Simplex LP engine to evaluate alternative healthcare resource allocation scenarios.
Sources
- Dantzig GB. Linear Programming and Extensions. Princeton University Press.
- Bertsimas D, Tsitsiklis JN. Introduction to Linear Optimization.
- Bazaraa MS, Jarvis JJ, Sherali HD. Linear Programming and Network Flows.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press.
- Drummond MF, Sculpher MJ, Claxton K, Stoddart GL, Torrance GW. Methods for the Economic Evaluation of Health Care Programmes.
- NICE. Health Technology Evaluation Manual.
Related Concepts (5)
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 the simplex method?
An optimisation algorithm that solves linear programming problems by moving systematically between feasible solutions until the optimum is found.
Source: Nocedal J, Wright SJ. Numerical Optimization. 2nd ed. Springer; 2006. doi:10.1007/978-0-387-40065-5.
How does the simplex method find an optimum?
The simplex method finds an optimum by moving systematically between feasible solutions of a linear programming problem until the best one is reached. It steps from one candidate solution to an adjacent, better one, repeating until no further improvement is possible. This systematic movement between feasible solutions to the optimum is how the simplex method works By walking along the edges of the feasible region from one corner to a better neighbour, the method reaches the optimum of a linear program, an approach distinct from cutting through the interior.
Source: Nocedal J, Wright SJ. Numerical Optimization. 2nd ed. Springer; 2006. doi:10.1007/978-0-387-40065-5.
What problems does the simplex method solve?
The simplex method solves linear programming problems, in which a linear objective is optimised subject to linear constraints, by moving systematically between feasible solutions until the optimum is found. It is built specifically for this linear setting. Solving linear programming problems in this way is the purpose of the simplex method By walking along the edges of the feasible region from one corner to a better neighbour, the method reaches the optimum of a linear program, an approach distinct from cutting through the interior.
Source: Nocedal J, Wright SJ. Numerical Optimization. 2nd ed. Springer; 2006. doi:10.1007/978-0-387-40065-5.
Between what does the simplex method move?
The simplex method moves between feasible solutions, stepping from one solution that satisfies the constraints to an adjacent one that improves the objective. It continues this progression until the optimum is found. Moving systematically between feasible solutions is the defining behaviour of the simplex method By walking along the edges of the feasible region from one corner to a better neighbour, the method reaches the optimum of a linear program, an approach distinct from cutting through the interior.
Source: Nocedal J, Wright SJ. Numerical Optimization. 2nd ed. Springer; 2006. doi:10.1007/978-0-387-40065-5.
When does the simplex method stop?
The simplex method stops when the optimum is found, that is, when no adjacent feasible solution improves the objective further. Because it moves systematically between feasible solutions seeking improvement, reaching a point with no better neighbour signals the optimum. Halting at this optimal feasible solution is how the simplex method terminates By walking along the edges of the feasible region from one corner to a better neighbour, the method reaches the optimum of a linear program, an approach distinct from cutting through the interior.
Source: Nocedal J, Wright SJ. Numerical Optimization. 2nd ed. Springer; 2006. doi:10.1007/978-0-387-40065-5.
How does the simplex method relate to an interior-point method?
The simplex method solves linear programming problems by moving between feasible solutions, which lie on the boundary of the feasible region, until the optimum is found, while an interior-point method approaches the optimum by traversing the interior of that region. Both solve linear programs but take different routes, along the boundary or through the inside. They are connected as contrasting algorithms for the same kind of problem By walking along the edges of the feasible region from one corner to a better neighbour, the method reaches the optimum of a linear program, an approach distinct from cutting through the interior.
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: 16 Jun 2026
Content version: 1.0.0
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- Persistent URI
- https://healtheconomics.wiki/concept/simplex-method
- Term code
- CS-OP-CO-003
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