What Are Shadow Prices in Linear Programming?

What Is a Shadow Price in Linear Programming?

Shadow prices measure the marginal value of relaxing a constraint. If a resource is scarce and binding, its shadow price tells you how much the objective (profit, cost, etc.) would improve if you had one more unit of that resource.

Key properties:

• Shadow prices come from the dual problem.
• They apply only to binding constraints.
• They are valid only within a range of feasibility.
• They represent marginal improvement, not total improvement.

Why Do Shadow Prices Matter?

1. They quantify the value of scarce resources

If a constraint is binding, the shadow price tells you exactly how much the objective would improve if you had one more unit of that resource.

2. They guide managerial decisions

Shadow prices help answer questions like:

• Should we buy more labor hours?
• Is it worth renting more machine time?
• Which resource is the true bottleneck?

3. They connect primal and dual problems

Shadow prices are the optimal dual variables. They reveal the economic meaning of the dual LP.

4. They support sensitivity analysis

Shadow prices are valid only within a certain range of the RHS. Outside that range, the basis changes and the shadow price no longer applies.

5. They identify redundant constraints

A constraint with a shadow price of zero is non‑binding and does not affect the optimal solution.

How to Interpret Shadow Prices (Step by Step)

Step 1: Identify binding vs. non‑binding constraints

A constraint is binding if its slack = 0 at the optimal solution. Only binding constraints can have non‑zero shadow prices.

Step 2: Read the shadow price from the final simplex tableau

In the final tableau:

• The shadow price equals the dual value associated with that constraint.
• It appears in the objective row under the slack/surplus variable.

Step 3: Interpret the sign

• Maximization: positive shadow price → more of the resource increases profit.
• Minimization: negative shadow price → more of the resource reduces cost.

Step 4: Apply the “one‑unit increase” rule

A shadow price of 7 means:

“If the RHS increases by 1 unit, the objective improves by 7 units.”

Step 5: Check the allowable range

Shadow prices are valid only within the RHS range where the current basis remains optimal. Outside that range, the shadow price changes.

Step 6: Connect to the dual problem

Each shadow price is the optimal value of a dual variable. This gives shadow prices a rigorous economic interpretation.

Step 7: Use shadow prices for real decisions

Shadow prices help determine:

• whether to acquire more of a resource
• whether a constraint is worth relaxing
• which constraints are bottlenecks
• the marginal value of capacity expansion

Numerical Example

Consider the LP:

\[ \text{Max } z = 5x_1 + 4x_2 \]

\[ \begin{aligned} 2x_1 + x_2 &\le 100 \\ x_1 + x_2 &\le 80 \\ x_1, x_2 &\ge 0 \end{aligned} \]

Suppose the optimal solution is:

• \(x_1 = 40\)
• \(x_2 = 40\)
• Both constraints are binding (slack = 0)

The final simplex tableau shows shadow prices:

• Constraint 1: 2
• Constraint 2: 1

Interpretation:

• One more unit of resource 1 increases profit by 2.
• One more unit of resource 2 increases profit by 1.

Resource 1 is the tighter bottleneck.