Linear Programming Explained with Example: Constraints, Shadow Prices, and Real-World Applications
1. What are the three essential components of a linear programming model?
Answer:
A linear programming model consists of decision variables, an objective function, and a set of constraints.
- Decision variables represent quantities a manager can control (e.g., production levels).
- The objective function defines the goal (e.g., maximize profit or minimize cost).
- Constraints represent the limits on resources such as capacity, labor, or materials.
2. In the insulation production example, what are the decision variables and objective function?
Answer:
The decision variables are:
- b = truckloads of type B insulation produced per day
-
r = truckloads of type R insulation produced per day
The objective function is to maximize total contribution: Maximize 950b+ 1200r
subject to production, loading, and chemical resource constraints.
3. What are the main constraints in the insulation example?
Answer:
- Machine capacity: ( 1.4b + 2.8r ≤ 70 )
- Loading dock limit: ( b + r ≤ 30 )
- Chemical (flame retardant) limit: ( 3b + r ≤ 65 )
- Non-negativity: ( b ≥ 0, r ≥ 0 )
4. What is the role of “shadow prices” in linear programming?
Answer:
Shadow prices represent the marginal value of relaxing a constraint by one unit.
They show how much the objective function (profit or cost) would change if more of a constrained resource were available.
For example, in the insulation problem, the shadow price of the loading dock constraint ($700) means that one extra truck per day could increase contribution by $700.
5. What does it mean if a constraint has “slack”?
Answer:
Slack is the unused portion of a resource.
If a constraint has slack, it means that the resource is not fully utilized and is non-binding.
For instance, in the insulation example, the chemical constraint had a slack of 15 units, meaning 15 canisters of the flame retardant were unused.
6. What is the interpretation of a “reduced cost” in LP results?
Answer:
Reduced cost is the difference between an activity’s direct contribution and the opportunity cost of the resources it uses.
- If reduced cost = 0 → the activity is part of the optimal plan.
-
If reduced cost > 0 → the activity is not profitable under current conditions.
In the case, the reduced cost for “Type X” insulation was $214.28, indicating it should not be produced unless its contribution increases by more than $214.28.
7. How does linear programming help in “pricing out” new activities?
Answer:
“Pricing out” means comparing a new product’s unit contribution with the opportunity cost of resources it consumes.
If opportunity cost exceeds contribution, the new product is not profitable.
This method ensures that new initiatives do not reduce overall profit by misallocating scarce resources.
8. What is the managerial interpretation of equilibrium in LP models?
Answer:
At equilibrium, shadow prices act like internal resource prices, balancing supply and demand within the organization.
Each product manager produces at levels where the unit contribution equals the transfer cost of resources.
No manager has an incentive to change behavior — this mirrors market stability.
9. What are the main steps in applying linear programming in practice?
Answer:
- Formulate the model (define decision variables, constraints, objective).
- Collect data (resource limits, costs, and returns).
- Run the model using software (e.g., Excel Solver, GAMS, or CPLEX).
- Analyze results (study shadow prices, reduced costs, and sensitivity).
- Implement the solution while validating assumptions with stakeholders.
10. What are the limitations of linear programming in real-world decision-making?
Answer:
- It assumes linearity of relationships (which may not hold in practice).
- It assumes divisibility of variables (fractional outputs may not be realistic).
-
It assumes certainty of data (real-world data may vary).
To handle uncertainty, methods like stochastic programming and sensitivity analysis are used.
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