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CMA Final · Strategic Cost Management

Transportation Problem for CMA Final Strategic Cost Management

The transportation problem finds the cheapest way to ship goods from several sources to several destinations, given supply, demand and unit costs. You formulate a cost table, get a starting solution by NWCR, least cost or VAM, then test it with MODI and improve it until no cell has a negative opportunity cost.

What this chapter covers

Transportation is a special case of linear programming. You have sources with fixed supply (plants, warehouses) and destinations with fixed demand (markets, depots). Each route has a unit cost. The aim is to decide how many units to send on each route so that total cost is lowest, with all supply used and all demand met.

The chapter has a fixed routine. First you formulate the problem as a cost matrix. Then you build an initial basic feasible solution (IBFS). Then you check it for optimality with the MODI (u-v) method and revise it if needed. Finally you handle special cases: degeneracy, unbalanced problems and maximization.

In Paper 16 it sits with the other quantitative decision tools, such as linear programming and other optimization topics. It is also a real cost-management decision: where to produce and ship from. Questions often end with a recommendation or a comment on the total cost, so the working must lead to a clear decision.

Transportation is procedural and the steps are the same every time, so it is one of the more scoring numerical areas for a prepared student. Examiners can set a full descriptive question or MCQs on it, and partial marks are usually available for correct steps even if a later calculation slips. A student who practises the routine can finish a problem in good time and keep that time for longer, wordier questions. The effort is moderate compared with the return, but only if you practise on paper and not just read solutions.

Transportation: topics in the order to study them

  1. 1Transportation Problem Formulation and BasicsYou need the cost matrix, the supply and demand balance, and the idea of basic cells (m + n − 1) before any method makes sense.
  2. 2Initial Basic Feasible Solution MethodsEvery problem starts here, so master NWCR, least cost and VAM before you try to optimize.
  3. 3Optimality Test using MODI MethodMODI works on an IBFS, so it comes after you can build one reliably; it is the most calculation-heavy part.
  4. 4Special Cases: Degeneracy, Unbalanced and MaximizationThese are variations on the standard routine, so learn them last, once the base method is automatic.

How to prepare Transportation

Treat this chapter as a skill to drill, not a theory to read. Use pencil and paper and repeat the same cycle until it is smooth.

  1. Learn the setup: write supply, demand and the cost matrix cleanly, and check total supply against total demand first.
  2. Practise the three IBFS methods on the same problem. Compare the costs you get, and note that VAM usually starts closest to the optimum.
  3. Always count your allocated cells and confirm it equals m + n − 1 (rows + columns − 1) before moving on.
  4. Learn MODI as a fixed routine: set one u or v to 0, find the rest from allocated cells (u + v = cost), then compute the opportunity cost for each empty cell as cost − (u + v).
  5. Practise the loop (closed path) for the entering cell, the plus-minus signs, and the choice of the smallest minus-cell quantity.
  6. Drill the special cases separately: add a dummy row or column at zero cost, add a tiny quantity (ε) for degeneracy, and convert maximization to minimization.
  7. Finish with timed full problems, ending each with the total cost and a one-line conclusion.

Common mistakes in Transportation

  • Not balancing supply and demand before starting.

    Fix: Total supply and demand first. If they differ, add a dummy row or column with zero cost before any method.

  • Wrong number of allocated cells.

    Fix: Count cells against m + n − 1 after the IBFS. If short, add ε in a suitable independent cell.

  • Errors in VAM penalties.

    Fix: Recompute all penalties after each allocation, using only the rows and columns still open.

  • Mistakes in the MODI loop and signs.

    Fix: Use only allocated cells as corners besides the entering cell, alternate signs, and shift the smallest quantity from a minus cell.

  • Treating dummy and ε cells as real costs in the final answer.

    Fix: Dummy cells have zero cost and ε is negligible. Report total cost from real allocations only, and state unused or unmet quantity in words.

  • Mishandling maximization problems.

    Fix: Convert to an opportunity-loss matrix first, solve as minimization, and then calculate the final profit using the original profit table.

Last-day revision: Transportation

  • A balanced problem has total supply equal to total demand.
  • A basic feasible solution has exactly m + n − 1 allocated cells in non-degenerate cases.
  • NWCR starts at the top-left cell and ignores costs, so it is quick but usually costly.
  • Least cost method allocates first to the cheapest cell, then the next cheapest.
  • VAM: find row and column penalties (difference of two lowest costs), pick the highest penalty, allocate to its cheapest cell.
  • MODI: for allocated cells, u + v = cost.
  • For empty cells, opportunity cost = cost − (u + v).
  • For minimization, the solution is optimal when no empty cell has a negative value.
  • If a negative value exists, pick the most negative cell, draw a closed loop, and shift the smallest quantity on the minus cells.
  • Unbalanced problem: add a dummy destination or source with zero cost to balance it.
  • Degeneracy: allocated cells fewer than m + n − 1; place ε in an independent empty cell and treat it as allocated.
  • Maximization: subtract every profit from the highest profit, or maximize with reversed optimality test, and then solve as minimization; compute final value from the original profits.

Transportation practice questions

Transportation in other exams

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

Transportation: frequently asked questions

Which initial solution method should I use in the exam?

Use the method the question names. If it leaves the choice to you, VAM generally gives a better start and needs fewer MODI iterations. If the question asks for a specific method, using a different one will lose marks.

How many allocated cells should a solution have?

A non-degenerate basic feasible solution has m + n − 1 allocated cells, where m is the number of rows and n the number of columns. If you have fewer, the solution is degenerate and needs an ε allocation before MODI.

How do I know the solution is optimal?

For a minimization problem, compute the opportunity cost of every empty cell using MODI. If none is negative, the solution is optimal. A zero in an empty cell suggests an alternative optimal solution may exist.

Do I need to learn all three IBFS methods?

Yes. Any of them can be named in a question, and the chapter is only a few methods. Practise each on the same table so you can see why their starting costs differ.