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Transportation Problem: Northwest Corner Method & Linear Programming Formulation
Published on May 28, 2026
Linear-Algebra
Solve a transportation problem using the Northwest Corner Method to find the minimum shipping cost and formulate it as a linear programming problem.
Question
Answer
Based on the provided image, here is the step-by-step solution for Question One.
Question One Solution
Given Data:
- Supplies (Factories):
- Factory A:
boxes - Factory B:
boxes - Total Supply =
boxes
- Factory A:
- Demands (Retailers):
- Retailer 1:
boxes - Retailer 2:
boxes - Retailer 3:
boxes - Total Demand =
boxes
- Retailer 1:
- Unit Shipping Costs (Tanzania shillings per box):
Factory Retailer 1 Retailer 2 Retailer 3 A 14 13 11 B 13 13 12
(a) Northwest Corner Method
The Northwest Corner Method starts allocations from the top-left cell and moves right or down as supply/demand is exhausted.
- Cell (A, 1): Demand for Retailer 1 is
, Supply from A is . Allocate . - Retailer 1 demand is satisfied.
- Remaining supply at A:
.
- Cell (A, 2): Demand for Retailer 2 is
, Remaining supply from A is . Allocate . - Factory A supply is exhausted.
- Remaining demand for Retailer 2:
.
- Cell (B, 2): Remaining demand for Retailer 2 is
, Supply from B is . Allocate . - Retailer 2 demand is satisfied.
- Remaining supply at B:
.
- Cell (B, 3): Demand for Retailer 3 is
, Remaining supply from B is . Allocate . - All demands and supplies are satisfied.
Shipping Schedule:
- Factory A
Retailer 1: boxes - Factory A
Retailer 2: boxes - Factory B
Retailer 2: boxes - Factory B
Retailer 3: boxes
Total Cost Calculation:
(b) Linear Programming Formulation
Let
Objective Function:
Minimize the total shipping cost
Minimize the total shipping cost
Constraints:
-
Supply Constraints:
-
(Factory A) -
(Factory B)
-
-
Demand Constraints:
-
(Retailer 1) -
(Retailer 2) -
(Retailer 3)
-
-
Non-negativity Constraints:
-
for all and .
-