Optimum Allocation of Resources: Linear Programming for CMA Inter

What this chapter covers

Optimum Allocation of Resources - LPP: topics in the order to study them

1. 1Introduction to Linear Programming and FormulationEvery later method starts from a correct model, so learn variables, objective, constraints and assumptions first.
2. 2Graphical Method of Solving LPPIt shows visually what a feasible region and an optimal corner point are, which makes simplex easier to follow.
3. 3Special Cases in Graphical SolutionOnce you can draw a normal solution, you can spot exceptions like multiple optima, unbounded and infeasible problems.
4. 4Simplex MethodIt handles problems with more than two variables and builds on the corner point idea, using slack variables and tables.
5. 5Duality and Sensitivity AnalysisIt reads meaning out of the final simplex table, so it comes last and needs the simplex steps to be solid.

How to prepare Optimum Allocation of Resources - LPP

1. Learn the formulation routine: define the decision variables with units, write the objective, list each constraint with its resource, and add non-negativity.
2. Practise formulating at least a handful of word problems without solving them. Check every constraint sign (≤ for limited resources, ≥ for minimum requirements).
3. Solve graphically: convert each constraint to an equation, plot the line, shade the feasible region, list the corner points and compute the objective at each one in a small table.
4. Study each special case on a drawn example so you recognise the pattern: parallel objective line for multiple optima, open region for unbounded, no common region for infeasible.
5. Practise simplex in a fixed layout: add slack variables, build the initial table, choose the entering column and leaving row by the ratio test, and stop when the optimality condition is met. Keep arithmetic neat.
6. Read the final table for duality and sensitivity: shadow prices, unused resources and ranges. Write one line of interpretation for each result.
7. Finish with timed mixed questions and MCQs on definitions, so you can switch between methods in the exam.

Common mistakes in Optimum Allocation of Resources - LPP

• Wrong constraint signs or mixing up resources in formulation

  Fix: Build a small table of resources against products first, then write each constraint straight from a row.

• Shading the wrong side of a line in the graph

  Fix: Test the origin or another point in each inequality before shading and mark the feasible region clearly.

• Taking the optimum from a non-corner point or skipping corner points

  Fix: Solve intersecting constraints algebraically and list all corner points with the objective value in a table.

• Missing a special case and forcing a single answer

  Fix: Check for ties in corner point values, an open region or an empty region before writing the conclusion.

• Arithmetic slips in simplex iterations

  Fix: Divide the pivot row by the pivot element first, then update each row, and check that the basic columns stay as unit columns.

• Stopping at the number and giving no interpretation

  Fix: State the product mix, the maximum profit or minimum cost, and the unused resources in a closing line.

Last-day revision: Optimum Allocation of Resources - LPP

• ✓An LPP has decision variables, a linear objective function, linear constraints and non-negativity restrictions.
• ✓Define variables clearly with units before writing any equation.
• ✓Limited resources give ≤ constraints; minimum requirements give ≥ constraints.
• ✓The feasible region is the area satisfying all constraints together.
• ✓If an optimal solution exists for a bounded feasible region, it lies at a corner point.
• ✓Test every corner point in the objective function and pick the best for max or min as asked.
• ✓Multiple optimal solutions occur when the objective function line is parallel to a binding constraint that forms an edge of the feasible region at the optimum, so two adjacent corner points give the same optimal value.
• ✓Infeasible means no common feasible region; unbounded means the objective can improve without limit.
• ✓In simplex, add a slack variable for each ≤ constraint to turn it into an equation.
• ✓Pick the entering variable by the optimality rule and the leaving variable by the minimum ratio.
• ✓Every LPP has a dual; the dual of a maximisation problem is a minimisation problem.
• ✓A shadow price shows how much the objective changes for one extra unit of a binding resource, within a valid range.

Optimum Allocation of Resources - LPP practice questions

• In a maximisation linear programming problem with three structural constraints (all of the ≤ type) and two decision variables, the dual prob…
• By the complementary slackness property of duality, if a primal constraint has positive slack (unused resource) at the optimal solution, the…
• In the graphical solution of a maximisation LPP, the feasible region is found to be unbounded in the direction of increasing objective value…
• Which of the following is a basic assumption of linear programming?
• In a graphical LPP, a redundant constraint is best described as one that
• In a graphical LPP, the objective function is maximised and its iso-profit line is moved parallel to itself away from the origin. The optimu…
• In a simplex iteration, x1 is the entering variable. Its column coefficients in three constraint rows are 4, 5 and −3, with corresponding RH…
• A graphical LPP has constraints x + y <= 6 and x + y >= 10 with x, y >= 0. The correct conclusion is that the problem:

Optimum Allocation of Resources - LPP in other exams

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

• CMA FinalLinear Programming
• ACCA Applied SkillsLimiting factors