How To Calculate Dual Variable Sensitivity Tables

Operations Research Calculator

Use this interactive calculator to estimate a dual variable, shadow price, allowable range status, and objective impact from changes in a linear programming constraint right hand side. Then review the expert guide below to understand how sensitivity tables are built and interpreted in practice.

Dual Variable Sensitivity Calculator

Calculator logic: if the constraint is nonbinding and slack is positive, the implied dual value is usually zero. If the basis remains unchanged within the allowable range, the shadow price equals the change in objective divided by the change in the right hand side.

Expert Guide: How to Calculate Dual Variable Sensitivity Tables

Dual variable sensitivity tables are one of the most useful outputs in linear programming. They tell you how much the objective function improves or worsens when the right hand side of a constraint changes by one unit, as long as the current optimal basis remains valid. In practical language, a dual variable often answers a very business focused question: what is one more unit of a resource worth right now? If you are solving production, transportation, staffing, blending, or budget allocation models, this number can be more valuable than the raw decision variable solution itself.

When people ask how to calculate dual variable sensitivity tables, they are usually trying to understand the shadow price column, the allowable increase and allowable decrease columns, and why some constraints have a dual value of zero. The short answer is that the table comes from the simplex basis and the dual relationship between the primal and dual models. The more operational answer is that you can calculate the core values from the current basis matrix, the objective coefficients of the basic variables, and the right hand side changes that preserve the same basis.

What a dual variable sensitivity table contains

A standard sensitivity report for constraints often includes the following pieces:

• Constraint name: the resource, demand, balance, or policy condition being tracked.
• Final value or activity: the left hand side value at the optimum.
• Right hand side: the available resource, required demand, or target limit.
• Slack or surplus: whether the constraint is binding.
• Shadow price: the marginal change in the objective for a one unit increase in the right hand side, assuming the current basis stays optimal.
• Allowable increase and allowable decrease: the interval over which the shadow price is valid without changing the basis.

If a constraint is not binding, its shadow price is often zero. That is because adding more of a resource that you are not fully using does not improve the objective at the margin. If a constraint is binding, the dual value is usually nonzero, and its sign depends on the problem formulation. For a maximization problem with less than or equal to resource constraints, the shadow price is commonly nonnegative. For a minimization problem with greater than or equal to requirements, the sign conventions can reverse.

Core formula used in sensitivity analysis

The most direct computational idea is this:

Shadow price = change in optimal objective / change in right hand side

This relationship is valid only when the right hand side change is small enough that the current basis remains optimal. In a full simplex setting, the exact dual vector is calculated by:

y^T = c_B^T B^-1

Here, B is the basis matrix formed by the columns of the basic variables, and c_B is the vector of objective coefficients for those same basic variables. Each component of y corresponds to the dual variable associated with one primal constraint. Once you have the dual vector, the estimated change in objective from a small right hand side change Δb is:

ΔZ = y^T Δb

If only one constraint changes, then the relevant dual variable is the shadow price for that row.

Step by step method to calculate a dual variable sensitivity table

1. Write the primal model clearly. Identify whether you have a maximization or minimization problem, the objective coefficients, and every constraint with sign restrictions.
2. Solve the model to optimality. In simplex terms, determine the final basis.
3. Build the basis matrix B. This contains the columns of the variables currently in the basis.
4. Collect the objective coefficients c_B. These are the objective coefficients corresponding to the basic variables.
5. Compute the dual vector. Use y^T = c_B^T B^-1.
6. Interpret each y value as a shadow price. Each value tells you the marginal objective impact of changing one right hand side by one unit, provided the basis does not change.
7. Find allowable increase and decrease. Apply right hand side perturbations and enforce primal feasibility for the current basis. This is done through x_B = B^-1 b and x_B(new) = B^-1(b + Δb).
8. Determine the largest interval preserving feasibility. The allowable range for each constraint is the largest positive and negative change that keeps all current basic variables feasible.
9. Publish the table. Include shadow prices, slack values, and allowable ranges.

Why allowable ranges matter

The most common mistake is to treat a shadow price as valid for any size of change. That is not correct. A shadow price is local to the current basis. Once a right hand side moves too far, a different constraint can become binding, a current basic variable can become infeasible, and the marginal value changes. The allowable increase and allowable decrease columns tell you when the reported shadow price is still reliable.

Suppose a factory has a labor constraint with a shadow price of 20. If the allowable increase is 30 hours, then adding 10 hours should improve a maximization objective by about 200. But adding 100 hours does not mean the objective improves by 2,000 using the same shadow price. The basis may change well before then, causing a new marginal value.

How to interpret binding and nonbinding constraints

• Binding constraint: slack equals zero. The resource or requirement is active at the optimum. These rows often carry nonzero dual values.
• Nonbinding constraint: slack is positive for less than or equal to constraints, or surplus is positive for greater than or equal to constraints. These rows often have a shadow price of zero.
• Equality constraint: always binding by definition, but its dual interpretation depends on the full model and sign restrictions.

Complementary slackness explains this pattern. In an optimal primal dual pair, a primal constraint with positive slack implies its corresponding dual variable is zero under standard sign conventions. That result is one reason sensitivity tables are so efficient: they summarize deep optimality conditions in a format managers can actually use.

Simple numerical example

Imagine a maximization model with one labor constraint. The original right hand side is 100 labor hours, and the optimal profit is 5,000. You test a new right hand side of 110 labor hours and re optimize. The new profit is 5,200. The change in objective is 200, and the change in right hand side is 10. Therefore, the shadow price is:

200 / 10 = 20

That means each additional labor hour is currently worth 20 in profit, if the basis stays the same. If slack at the original optimum is zero, the constraint is binding, which supports a nonzero shadow price. If the allowable increase is at least 10, then this estimate is consistent with the sensitivity range.

Metric	Original case	Updated case	Interpretation
Right hand side	100	110	Resource availability increased by 10 units
Optimal objective	5,000	5,200	Profit increased by 200
Slack	0	Usually still 0 or near 0 within basis range	Constraint is binding
Implied shadow price	20 objective units per additional resource unit

Relation between primal and dual models

Every linear program has an associated dual. In a standard primal maximization problem with less than or equal to constraints and nonnegative variables, the dual is a minimization problem with greater than or equal to constraints. The dual variables represent implicit values placed on the primal constraints. In economics this is often called a marginal value. In management science it is often called a resource price. In software reports it is usually labeled shadow price or dual value.

The sensitivity table exists because optimal dual values can be recovered directly from the final simplex tableau or from the basis inverse. Once you understand that, the table becomes less mysterious. It is not an extra report layered on top of optimization. It is a direct expression of the optimality conditions already present in the solved model.

Typical signs and meanings of dual values

Primal problem structure	Typical dual sign	Managerial interpretation	Practical example
Maximization with less than or equal to resource constraints	Nonnegative	More resource usually cannot hurt the objective	Extra machine time or labor hours
Minimization with greater than or equal to requirement constraints	Often nonpositive under standard sign conventions	Increasing a requirement usually raises cost	Minimum service level or demand satisfaction
Equality constraints	Unrestricted in sign	Marginal value can be positive or negative depending on the model	Balance equations in network flow

Useful statistics from operations research practice

Linear programming and sensitivity analysis are not niche tools. They are used widely in engineering, logistics, energy planning, agriculture, public policy, and revenue management. University operations research programs consistently teach post optimal analysis as a core topic because a single optimal solution rarely answers the full decision problem. In applied settings, planners often need to know whether to buy one more unit of labor, increase a capacity cap, or relax a demand requirement. The dual variable table answers exactly those questions at marginal scale.

• Many standard optimization textbooks and university course notes treat sensitivity analysis as a required component of linear programming interpretation, not an optional appendix.
• Commercial solvers such as CPLEX, Gurobi, and spreadsheet based simplex engines routinely report dual prices and reduced costs because these are among the most actionable outputs for analysts.
• In resource allocation models, constraints with zero slack frequently dominate managerial attention because they indicate bottlenecks and carry the highest marginal value.

How to compute allowable increase and allowable decrease

Suppose the basis remains fixed and only the right hand side vector changes. The basic solution becomes:

x_B(new) = B^-1(b + Δb) = B^-1b + B^-1Δb

To preserve feasibility, every component of x_B(new) must satisfy the variable sign restrictions, usually nonnegativity. For a single row change, you can isolate one perturbation parameter and derive a set of inequalities. Solving those inequalities gives the allowable decrease and allowable increase. This calculation is basis specific. If the basis changes, the old range no longer applies.

Common mistakes analysts make

1. Ignoring the allowable range. This causes incorrect extrapolation of the shadow price.
2. Confusing reduced costs with dual values. Reduced costs belong to variables, while shadow prices belong to constraints.
3. Forgetting sign conventions. The meaning of a positive or negative dual value depends on the primal form.
4. Using dual values from integer programming models as if they were ordinary LP shadow prices. For mixed integer models, post optimal interpretation is more nuanced.
5. Assuming zero slack always means huge value. A binding constraint can still have a small or even zero marginal value in degenerate cases.

When the calculator on this page is most useful

The calculator above is ideal when you already know the original and re optimized objective values for a right hand side change and you want a fast estimate of the implied shadow price, basis range status, and interpretation. It is especially useful for teaching, quick reporting, and sanity checking solver output. For a complete industrial sensitivity report across all constraints, however, you would usually rely on a solver that exposes the final basis or directly prints a sensitivity table.

Recommended authoritative references

If you want to study the theory behind dual variables, simplex sensitivity analysis, and post optimal interpretation in more depth, these university and government resources are excellent starting points:

• MIT OpenCourseWare, Optimization Methods in Management Science
• Cornell University notes on sensitivity analysis in linear programming
• National Institute of Standards and Technology, quantitative methods resources

Final takeaway

To calculate a dual variable sensitivity table, solve the linear program, identify the optimal basis, compute the dual vector from y^T = c_B^T B^-1, and then determine the allowable right hand side ranges that keep the basis feasible. In day to day business terms, the reported shadow price is the marginal value of relaxing or tightening a constraint, and the allowable range tells you when that marginal value remains trustworthy. If you remember those two ideas, you will interpret sensitivity reports far more accurately and make better optimization decisions.