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About Potential method calculator
Solve transportation problems using the potential method online — find an initial basic feasible solution and iterate with detailed step-by-step solution. with complete, detailed, step-by-step description of solutions, that solves linear-programming problems up to 20×20 in size with coefficients of this type: decimal numbers and fractions.
To start the calculation, you need to first enter the problem dimensions in the input fields at the very top of the screen, and also choose the desired operation from the side menu.
A little below you will find an input window where you need to enter the coefficients, constraints, and right-hand-side values using the keyboard. The input control panel is also located here, which simplifies work with LP problems and contains the following control elements:
- The first element allows you to expand the input window. This can be especially useful when the tableau does not fit completely on the screen. If the table is still not fully visible after expanding the window, you can change the scale using the + / - buttons;
- The second element copies the current problem input to the memory buffer. This can be useful when you frequently solve the same LP problem or need to transfer data between operations;
- And the last element pastes the previously copied input, which allows you to restore problem data in just a few clicks instead of re-entering it manually;
And further down you will find a toolbar that allows you to customize the calculator and make it easier to work with. It is visually divided into three parts, each of which is responsible for the following functionality:
- The first part lets you select the number format used when displaying the solution result. You can also turn off step-by-step comments if you already understand the method and only need to check your own calculations, or hide the step-by-step solution entirely if you only need the final answer;
- The second part contains buttons that let you modify the input table dimensions, clear individual coefficients or the entire input, and the main button with an equal sign that takes you to the solution screen. All these buttons are duplicated by keyboard shortcuts. Hover over a button to see the corresponding key in a tooltip. You can also use the arrow keys to move the cursor between input fields;
- And the last part allows you to choose the number of digits after the decimal point for rounding non-integer results. A live preview shows how the rounded values will appear;
What is the Potential method (transportation problem)?
The potential method (MODI method) solves the balanced transportation problem, which seeks to ship goods from n suppliers to m consumers at minimum total cost, subject to supply and demand constraints. The problem is balanced when total supply equals total demand. The method works with a basic feasible solution consisting of exactly n + m - 1 non-degenerate allocations and iteratively improves it by finding and reallocating along improvement cycles.
How to solve a transportation problem with the potential method?
Start by finding an initial basic feasible solution using the northwest-corner rule or a similar heuristic. Compute supplier potentials u_i and consumer potentials v_j by solving u_i + v_j = c_ij for each basic cell. Evaluate the reduced costs d_ij = c_ij - u_i - v_j for all non-basic cells; if all reduced costs are non-negative the current solution is optimal. Otherwise select the most negative reduced cost as the entering cell, trace a closed cycle through basic cells, and shift the maximum possible flow around the cycle to improve the objective. Repeat until all reduced costs are non-negative.
Example of solving a transportation problem with the potential method
Using the method of the north-western corner, we construct the first reference plan of the transportation problem:
Find the potentials for all consumers and suppliers from the relation Ui + Vj = Cij, where:
Ui is the potential of the i-th supplier;
Vj - potential of the j-th consumer;
Cij - the cost of goods transportation in the base cell, located in the i-th row, j-th column;
Then we calculate the estimates of non-basic cells using the relation Oij = Cij - (Ui + Vj), where
Oij - the cost of goods transportation in the corresponding non-basic cell;
Since there are negative values among the estimates, the support plan can be further improved;
To do this, enter the cell with the lowest negative estimate in the base;
To enter a cell in the base, a cycle is constructed relative to it;
The first vertex of the loop is in the cell, which is entered into the base;
All other vertices in basic cells;
The cycle shows the order of redistribution of resources for introducing a cell into the base;
The order of redistribution is determined by the + and - signs, which alternate in the cells where the cycle vertices are located;
In the cell that is entered in the basis is always +;
Find the potentials for all consumers and suppliers from the relation Ui + Vj = Cij, where:
Ui is the potential of the i-th supplier;
Vj - potential of the j-th consumer;
Cij - the cost of goods transportation in the base cell, located in the i-th row, j-th column;
Then we calculate the estimates of non-basic cells using the relation Oij = Cij - (Ui + Vj), where
Oij - the cost of goods transportation in the corresponding non-basic cell;
Since there are no negative values among the estimates, the support plan can be considered optimal;
(3 * 0) + (1 * 1) + (1 * 2) + (5 * 0) + (5 * 0) = 3
3
Frequently asked questions
What is the potential (transportation) method used for?
It solves the transportation problem — shipping goods from several suppliers to several consumers at minimum total cost, given supply amounts, demand amounts, and per-unit shipping costs.
What does "balanced" versus "unbalanced" mean?
A problem is balanced when total supply equals total demand. When they differ, the method adds a fictitious supplier or consumer to absorb the surplus, then solves the balanced version.
What are the potentials u and v?
They are dual variables, one per supplier (u) and one per consumer (v). They test optimality: if every empty cell's cost minus (u + v) is non-negative, the current shipment plan is optimal.
How do I read the optimal plan?
The result shows how many units to ship on each supplier-to-consumer route and the minimum total cost. Routes that are not used carry zero shipment.