Reduced Cost Calculator
This page uses a single-column page flow. The input area becomes three columns on large screens, two on medium screens, and one on mobile.
Example Data Table
| Variable | Objective type | cj | Shadow prices (y) | Column coefficients (a) | Zj = Σyiaij | Reduced cost | Meaning |
|---|---|---|---|---|---|---|---|
| x4 | Max | 9.000000 | [1.2, 0.8, 0.5, 0.3] | [2, 1, 3, 0] | 4.300000 | 4.700000 | Positive value suggests x4 can improve the objective. |
| x5 | Max | 6.000000 | [1.0, 0.5, 0.4, 0.2] | [2, 2, 1, 1] | 3.600000 | 2.400000 | Good entering candidate if feasibility stays valid. |
| x6 | Min | 4.000000 | [1.5, 0.6, 0.8, 0.4] | [1, 1, 2, 1] | 4.100000 | -0.100000 | Negative value can improve a minimization objective. |
Formula Used
rj = cj - yTaj = cj - Σ(yiaij)
rj = cj - Zj
ΔObjective ≈ rj × Δx
Here, cj is the objective coefficient of the tested variable, y is the vector of shadow prices, and aj is the variable column in the constraint matrix.
In a maximization model, a positive reduced cost usually signals improvement potential. In a minimization model, a negative reduced cost usually signals improvement potential.
How to Use This Calculator
- Enter the variable name and choose maximization or minimization.
- Select dual mode to compute Zj from shadow prices and column coefficients, or choose direct mode to enter Zj yourself.
- Provide the objective coefficient cj.
- Enter up to four shadow prices and four column coefficients when using dual mode.
- Set a trial step to estimate the objective effect of increasing the variable.
- Use the tolerance field to decide when a reduced cost should be treated as zero.
- Press the calculate button to view the result, interpretation, breakdown table, export buttons, and graph.
FAQs
1) What is reduced cost in optimization?
Reduced cost measures how far a variable’s objective coefficient is from the value implied by current dual prices. It helps judge whether a nonbasic variable can improve the current solution.
2) Why can reduced cost be positive, negative, or zero?
The sign depends on the objective direction and the relationship between cj and Zj. Zero means the variable is neutral at the current dual prices.
3) What does a positive reduced cost mean?
For a maximization model, a positive reduced cost usually means the variable may improve the objective if it enters the basis. Interpretation changes for minimization.
4) What does a negative reduced cost mean?
For a minimization model, a negative reduced cost usually means the variable may lower the objective if it enters the basis. In maximization, it is usually not attractive.
5) What is Zj in simplex terminology?
Zj is the value generated by multiplying the current shadow prices or basis information by the tested variable’s column. It represents the implied cost from the current basis.
6) Why does this calculator include a tolerance?
Optimization values often contain rounding noise. A tolerance lets you treat tiny reduced costs as zero, which helps avoid misleading interpretations around alternate optima.
7) What does the trial step output show?
It estimates the objective change from increasing the tested variable by the chosen amount. It is a quick sensitivity view, not a full re-optimization.
8) Can this calculator replace a full simplex solver?
No. It evaluates reduced cost for one variable using supplied data. A complete solver is still needed to update bases, feasibility, pivot rules, and full optimal solutions.