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Operations Management and Strategic Management · Transportation

Transportation Problem: Introduction and Formulation

Updated 10 October 2026 · Fact-checked

The transportation problem finds the cheapest way to ship a single product from several sources with fixed supply to several destinations with fixed demand. To formulate it, draw a cost matrix with supply on the right and demand at the bottom, then check whether total supply equals total demand.

Understand Transportation Problem: Introduction and Formulation

A transportation problem is a special linear programming problem. A firm has a product at a few sources (factories or warehouses) and needs to send it to a few destinations (markets or depots). Each source has a limited supply (availability). Each destination has a required demand. Each route has a unit transportation cost. You decide how many units to send on each route so that total cost is minimum.

The model rests on a few assumptions:
- The product is homogeneous, so any source can serve any destination.
- Supply and demand quantities are known and fixed.
- The cost per unit on a route is constant and the total cost on a route is cost per unit × units shipped (cost is linear).
- Shipping from any source to any destination is possible, unless a route is stated as prohibited.
- The objective is to minimise total transportation cost.

The data is shown in a cost matrix. Rows are sources, columns are destinations, and each cell holds the unit cost. The last column shows supply and the last row shows demand. The bottom-right corner holds the totals, so you can compare them at once.

A problem is balanced when total supply equals total demand. It is unbalanced when they differ. If supply exceeds demand, add a dummy destination with demand equal to the excess. If demand exceeds supply, add a dummy source with supply equal to the shortfall. Dummy routes carry zero cost, unless the question gives a penalty or storage cost. Units allocated to a dummy are not really shipped. They show unused stock or unmet demand.

Formulation is the first step. After it, you find an initial feasible solution and test it for optimality. Those methods are in the related topics. If you formulate wrongly, everything after it is wrong.

Key rules to remember

Objective function
Minimise Z = Σ Σ cij × xij (i = 1 to m sources, j = 1 to n destinations)
cij is the unit cost from source i to destination j. xij is the number of units shipped on that route.
Supply constraints
Σj xij = ai for each source i
Units sent out of a source equal its supply. In a balanced problem, all supply is used.
Demand constraints
Σi xij = bj for each destination j
Units received at a destination equal its demand.
Non-negativity
xij ≥ 0 for all i, j
You cannot ship a negative quantity.
Balance condition
Σ ai = Σ bj
If true, the problem is balanced. If not, add a dummy row or column.
Number of allocated cells in a basic feasible solution
m + n − 1
A non-degenerate basic feasible solution has exactly this many occupied cells. It is used in the later solution methods.

How to solve Transportation Problem: Introduction and Formulation questions

Use this method to formulate any transportation problem from a word statement.

  1. 1Identify the sources and destinations. Note the supply of each source and the demand of each destination.
  2. 2Draw the matrix with sources as rows and destinations as columns. Enter the unit cost in each cell.
  3. 3Write supply in the right-hand column and demand in the bottom row.
  4. 4Add total supply and total demand. Compare them.
  5. 5If they are equal, the problem is balanced. If supply is greater, add a dummy destination. If demand is greater, add a dummy source. Fill the dummy with the difference and zero costs, unless a penalty is given.
  6. 6Handle special conditions. Show a prohibited route with a very large cost M, or mark it clearly as not allowed.
  7. 7Write the objective function and constraints if the question asks for the mathematical model.
  8. 8State the final balanced matrix, which is ready for finding an initial solution.

Quickest way: Total-first check

When to use it: Use this in MCQs and in the first two minutes of a written answer.

  1. Add the supply column and the demand row before anything else.
  2. If the totals match, say balanced and move on.
  3. If not, the difference is the size of the dummy. Larger supply gives a dummy column. Larger demand gives a dummy row.
  4. Put zeros in the dummy cells.
  5. For an MCQ on the number of constraints, count m + n. For occupied cells in a basic solution, count m + n − 1.

Common mistakes in Transportation Problem: Introduction and Formulation

  • Not checking whether total supply equals total demand.

    Students jump straight into allocating units.

    Fix: Always total both sides first and write the result on the matrix.

  • Adding the dummy on the wrong side.

    Students mix up which side is short.

    Fix: Excess supply needs a dummy destination (column). Excess demand needs a dummy source (row).

  • Putting non-zero costs in the dummy row or column.

    Students copy costs from other cells.

    Fix: Use zero unless the question gives a shortage penalty or storage cost. If it does, use that figure.

  • Switching the rows and columns, so supply is read as demand.

    The question lists data in an unclear order.

    Fix: Read the statement: what a plant makes or a warehouse holds is supply. What a market needs is demand.

  • Ignoring a prohibited route.

    The condition is hidden in a line of text.

    Fix: Underline such conditions and put a very large cost M in that cell so it is never chosen.

  • Writing the total cost as the sum of unit costs.

    Students forget to multiply by quantity.

    Fix: Total cost = Σ (unit cost × units shipped) for the occupied cells only.

Worked examples

Example 1

A company has three plants, P1, P2 and P3, with capacities of 40, 60 and 50 units. It has three depots, D1, D2 and D3, needing 30, 70 and 50 units. Unit costs (₹) are: P1: 4, 6, 8. P2: 5, 3, 7. P3: 6, 4, 5 (for D1, D2, D3). Formulate the problem.

Show the solution
  1. Total supply = 40 + 60 + 50 = 150 units.
  2. Total demand = 30 + 70 + 50 = 150 units.
  3. Supply equals demand, so the problem is balanced. No dummy is needed.
  4. Let xij be the units sent from plant i to depot j.
  5. Objective: Minimise Z = 4x11 + 6x12 + 8x13 + 5x21 + 3x22 + 7x23 + 6x31 + 4x32 + 5x33.
  6. Supply constraints: x11 + x12 + x13 = 40; x21 + x22 + x23 = 60; x31 + x32 + x33 = 50.
  7. Demand constraints: x11 + x21 + x31 = 30; x12 + x22 + x32 = 70; x13 + x23 + x33 = 50.
  8. Non-negativity: all xij ≥ 0.

Answer: Balanced problem (150 units each side). Minimise Z = Σ cij xij with 3 supply and 3 demand equality constraints and xij ≥ 0.

Example 2

Three warehouses W1, W2, W3 hold 50, 40 and 60 units. Four shops S1, S2, S3, S4 need 20, 30, 50 and 30 units. Unit costs (₹) are: W1: 8, 6, 10, 9. W2: 9, 12, 13, 7. W3: 14, 9, 16, 5 (for S1 to S4). Make the problem ready for solution.

Show the solution
  1. Total supply = 50 + 40 + 60 = 150 units.
  2. Total demand = 20 + 30 + 50 + 30 = 130 units.
  3. Supply exceeds demand by 150 − 130 = 20 units, so the problem is unbalanced.
  4. Add a dummy destination S5 with demand 20 units.
  5. Set the cost from every warehouse to S5 as ₹0, as no penalty is given.
  6. New demand total = 130 + 20 = 150 units, equal to supply.
  7. The matrix is now 3 × 5. Each row has costs followed by a 0 in the S5 column. Supplies are 50, 40, 60 and demands are 20, 30, 50, 30, 20.
  8. In the final answer, units allocated to S5 represent unused stock at a warehouse.

Answer: Unbalanced, with excess supply of 20 units. Add dummy destination S5 (demand 20, zero cost) to get a balanced 3 × 5 problem with total 150 units.

Exam tips

  • Write the totals check as the first line of your answer. It earns marks even if later steps go wrong.
  • In MCQs, expect questions on the dummy rule, the assumptions and the m + n − 1 rule. Learn these cold.
  • Draw the matrix neatly with supply and demand labelled. Examiners give marks for correct presentation.
  • If the question mentions a penalty, storage cost or prohibited route, use it in the dummy or cell. Do not default to zero.

Practice questions from Transportation

Transportation Problem: Introduction and Formulation in other exams

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

Transportation Problem: Introduction and Formulation: frequently asked questions

What is a transportation problem in operations research?

It is a linear programming problem about shipping a homogeneous product from sources with fixed supply to destinations with fixed demand at least total cost. The data is shown in a cost matrix.

What is the difference between balanced and unbalanced transportation problems?

In a balanced problem, total supply equals total demand. In an unbalanced problem they differ. You balance it by adding a dummy destination for excess supply or a dummy source for excess demand.

What cost do I give to a dummy row or column?

Use zero, as nothing is actually shipped. If the question gives a shortage penalty or holding cost, use that value instead.

How many constraints does a transportation problem have?

With m sources and n destinations, there are m supply constraints and n demand constraints, so m + n in total. The number of variables is m × n.