Skip to content

CMA Final · Strategic Cost Management

Linear Programming for CMA Final Strategic Cost Management

Linear programming is a method to find the best value of a linear objective, such as maximum profit or minimum cost, subject to linear limits on resources. You solve it by formulating the model, then using the graphical method for two variables or the simplex method for larger ones, and reading the result.

What this chapter covers

Linear programming (LP) helps a firm decide how to use scarce resources such as machine hours, labour and materials. You pick the quantities of products to make so that profit is highest or cost is lowest, without breaking any limit. The chapter starts with formulation, moves to solving, and ends with interpretation.

You first learn to turn a business story into an objective function and constraints. Then you solve small problems with the graphical method and larger ones with the simplex method. Next come duality and shadow prices, which tell you what one more unit of a resource is worth. Finally, sensitivity analysis shows how far the data can change before the answer changes.

In Strategic Cost Management this chapter links to product mix decisions, limiting factors, make-or-buy and pricing. Contribution per unit is usually the objective coefficient, so your marginal costing basics feed straight into it. A shadow price is also a decision tool: it tells you how much extra you would pay to relax a constraint. Expect both numerical questions and short MCQs on interpretation.

This chapter is worth the effort because it is rule-based and predictable. Once you can formulate cleanly and run the steps, marks are easy to collect, and partial marks are available even if the final figure slips. Formulation, shadow price meaning and sensitivity ideas also suit MCQs, so the chapter can earn marks in Section A as well as in the written questions. It also builds the decision-oriented thinking the paper rewards: a clear recommendation backed by the working.

Linear Programming: topics in the order to study them

  1. 1Introduction to Linear Programming and FormulationEvery later method starts from a correct model, so learn objective functions, constraints and non-negativity first.
  2. 2Graphical Method of Solving LPPIt shows visually what a feasible region and optimal corner point are, which makes simplex easier to understand.
  3. 3Simplex MethodIt extends the same logic to more than two variables using slack variables and tableaus, so you need the graphical idea first.
  4. 4Duality and Shadow PricesYou read dual values from the final simplex tableau, so this topic comes after you can solve one.
  5. 5Sensitivity Analysis and Special CasesIt builds on the optimal solution and shadow prices, and special cases like infeasibility and unboundedness make sense only once you know normal solving.

How to prepare Linear Programming

Treat this chapter as a skill, not a reading task. You learn it by solving problems on paper until the steps run on their own.

  1. Practise formulation alone first. Take ten word problems and write only the variables, objective and constraints. Check units and the direction of each inequality.
  2. Solve graphical problems by plotting constraint lines, marking the feasible region and testing every corner point in the objective function.
  3. Learn the simplex steps as a fixed routine: add slack variables, set up the first tableau, choose the entering column, find the leaving row by the minimum ratio, and repeat until the optimality test is met.
  4. Write each tableau neatly with row and column labels. Check each iteration before moving on, because one slip carries forward.
  5. Read the final tableau for the answer, the unused resources and the shadow prices, and state their meaning in a sentence.
  6. Practise sensitivity and special cases as short interpretation questions, such as the range where a shadow price holds.
  7. Finish with timed mixed problems and end each answer with a clear recommendation in business terms.

Common mistakes in Linear Programming

  • Using selling price instead of contribution per unit in the objective function

    Fix: Compute contribution as price minus variable cost per unit and use only that, unless the question clearly says otherwise.

  • Writing the wrong inequality direction or mixing units in constraints

    Fix: Read each limit once more after writing it and convert all data into the same unit before forming the constraint.

  • Testing only some corner points in the graphical method

    Fix: Solve for every corner of the feasible region and put each into the objective function, then compare.

  • Arithmetic errors in simplex iterations

    Fix: Divide the pivot row first, check that the pivot column becomes a unit column, and verify the right-hand side stays non-negative.

  • Stating a shadow price without its meaning or limits

    Fix: Write one sentence: each extra unit of this resource raises profit by this amount, within the range where the current solution stays valid.

  • Ending with numbers and no recommendation

    Fix: Close with the product mix, total contribution, unused capacity and a short business suggestion.

Last-day revision: Linear Programming

  • Formulation order: define decision variables, write the objective function, list constraints, add non-negativity.
  • The objective coefficient is usually contribution per unit, not selling price.
  • The optimal solution of a feasible, bounded LP lies at a corner point of the feasible region.
  • Graphical method handles two decision variables.
  • Simplex adds a slack variable to each ≤ constraint to make it an equation.
  • Entering variable in maximisation is the column with the most positive value in the Cj − Zj row.
  • Leaving variable comes from the minimum non-negative ratio of the right-hand side to the pivot column value.
  • Stop when no Cj − Zj value in a maximisation problem is positive.
  • Shadow price is the change in the optimal objective value for one extra unit of a scarce resource.
  • A resource with spare capacity (positive slack) has a shadow price of zero.
  • The dual of a maximisation problem is a minimisation problem, and dual variables correspond to primal constraints.
  • Infeasible means no feasible region exists, and unbounded means the objective can grow without limit.

Linear Programming practice questions

Linear Programming in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Linear Programming: frequently asked questions

How should I start Linear Programming for CMA Final?

Start with formulation, since every method depends on it. Practise turning word problems into an objective function and constraints before you try to solve anything. Then move to the graphical method and simplex.

Do I need to learn both the graphical and simplex methods?

Yes. The graphical method is limited to two variables but builds your understanding of the feasible region and corner points. Simplex is needed for larger problems and for reading shadow prices from the final tableau.

What is a shadow price in simple terms?

It is the extra profit you gain, or cost you save, from one more unit of a scarce resource. It is zero for a resource that is not fully used. Use it to judge whether paying more for extra capacity is worthwhile.

Can Linear Programming be asked in the MCQ section?

Yes, it can. Questions on formulation, the meaning of slack, shadow prices, or special cases like unboundedness suit objective format. Revise the definitions and interpretation points along with the numerical steps.