diff --git a/apps/backend/src/app/controllers/transport-problem/services/second-phase.service.test.ts b/apps/backend/src/app/controllers/transport-problem/services/second-phase.service.test.ts
new file mode 100644
index 0000000..c342af7
--- /dev/null
+++ b/apps/backend/src/app/controllers/transport-problem/services/second-phase.service.test.ts
@@ -0,0 +1,32 @@
+import {
+  secondPhaseFirstResultMock,
+  secondPhaseSecondResultMock,
+  vogelKordaFirstResultMock,
+  vogelKordaSecondResultMock,
+} from '@opres/shared/data/mocks';
+import {last} from 'lodash';
+
+import {SecondPhaseService} from './second-phase.service';
+
+describe('SecondPhaseService', () => {
+  let service: SecondPhaseService;
+
+  beforeEach(() => {
+    service = new SecondPhaseService();
+  });
+
+  it('should be created', () =>
+    expect(service).toBeInstanceOf(SecondPhaseService));
+
+  it("should calculate a transport problem's second phase part 1", () => {
+    const firstPhaseResult = last(vogelKordaFirstResultMock)?.transportation;
+    const result = service.calculate(firstPhaseResult, 'explanations');
+    expect(result).toEqual(secondPhaseFirstResultMock);
+  });
+
+  it("should calculate a transport problem's second phase part 2", () => {
+    const firstPhaseResult = last(vogelKordaSecondResultMock)?.transportation;
+    const result = service.calculate(firstPhaseResult, 'explanations');
+    expect(result).toEqual(secondPhaseSecondResultMock);
+  });
+});
diff --git a/apps/backend/src/app/controllers/transport-problem/services/second-phase.service.ts b/apps/backend/src/app/controllers/transport-problem/services/second-phase.service.ts
index 6ab4666..92dce8d 100644
--- a/apps/backend/src/app/controllers/transport-problem/services/second-phase.service.ts
+++ b/apps/backend/src/app/controllers/transport-problem/services/second-phase.service.ts
@@ -1,10 +1,22 @@
 import {Injectable} from '@nestjs/common';
 
 import {
+  AuxiliaryVariables,
   CalculationMode,
   SecondPhaseStep,
+  SelectedCell,
   TransportTable,
 } from '@opres/shared/types';
+import {
+  cloneDeep,
+  findIndex,
+  first,
+  forEach,
+  last,
+  map,
+  minBy,
+  some,
+} from 'lodash';
 
 @Injectable()
 export class SecondPhaseService {
@@ -12,7 +24,167 @@ export class SecondPhaseService {
     transportTable: TransportTable,
     mode: CalculationMode,
   ): Array<SecondPhaseStep> {
-    // todo wip
-    return [];
+    const process: Array<SecondPhaseStep> = [];
+    let nextBase: SelectedCell = {x: 0, y: 0};
+    do {
+      const variables = this.getAuxiliaryVariable(transportTable);
+      nextBase = this.getNextNewBase(transportTable, variables);
+      if (nextBase?.value !== undefined) {
+        const circle = this.getCircle(transportTable, nextBase);
+        process.push({
+          transportation: cloneDeep(transportTable),
+          auxiliaryVariables: variables,
+          nextBase: cloneDeep(nextBase),
+          circle,
+        });
+        this.reduceTransportTable(transportTable, cloneDeep(circle));
+        continue;
+      }
+      process.push({
+        transportation: cloneDeep(transportTable),
+        auxiliaryVariables: variables,
+      });
+    } while (nextBase?.value !== undefined);
+    return process;
+  }
+
+  private getAuxiliaryVariable(
+    transportTable: TransportTable,
+  ): AuxiliaryVariables {
+    const variables: AuxiliaryVariables = {
+      x: map(transportTable[0], () => null),
+      y: map(transportTable, () => null),
+    };
+    const initialRow = findIndex(transportTable, (row) =>
+      some(row, (cell) => !!cell.transported),
+    );
+    variables.y[initialRow] = 0;
+
+    const containsNull = (variable) => variable === null;
+    while (variables.x.some(containsNull) || variables.y.some(containsNull)) {
+      forEach(transportTable, (row, rowIndex) =>
+        forEach(row, (cell, columnIndex) => {
+          const currentXVariable = variables.x[columnIndex];
+          const currentYVariable = variables.y[rowIndex];
+
+          if (cell.transported === undefined) return;
+          if (currentXVariable === null && currentYVariable === null) return;
+          if (currentXVariable !== null && currentYVariable !== null) return;
+
+          if (currentYVariable === null)
+            variables.y[rowIndex] = cell.cost - variables.x[columnIndex];
+          else if (currentXVariable === null)
+            variables.x[columnIndex] = cell.cost - variables.y[rowIndex];
+        }),
+      );
+    }
+    return variables;
+  }
+
+  private getNextNewBase(
+    transportTable: TransportTable,
+    variables: AuxiliaryVariables,
+  ): SelectedCell {
+    let newBase: SelectedCell = {value: undefined, x: 0, y: 0};
+    forEach(transportTable, (row, rowIndex) =>
+      forEach(row, (cell, columnIndex) => {
+        if (cell.transported !== undefined)
+          return (cell.reducedCost = undefined);
+        const reducedCost =
+          cell.cost - (variables.x[columnIndex] + variables.y[rowIndex]);
+        transportTable[rowIndex][columnIndex].reducedCost = reducedCost;
+        if (
+          (newBase?.value === undefined && reducedCost < 0) ||
+          newBase?.value > reducedCost
+        )
+          newBase = {value: reducedCost, x: +columnIndex, y: rowIndex};
+      }),
+    );
+    return newBase;
+  }
+
+  private getCircle(
+    transportTable: TransportTable,
+    nextBase: SelectedCell,
+  ): Array<SelectedCell> {
+    let baseCells: Array<SelectedCell> = this.getBaseCellPositions(
+      transportTable,
+      nextBase,
+    );
+    let removedCells = Infinity;
+    while (removedCells > 0) {
+      const filteredBaseCells = baseCells.filter((value, _, array) => {
+        const xIndexes = array.filter((v) => v.x === value.x).length;
+        const yIndexes = array.filter((v) => v.y === value.y).length;
+        return xIndexes > 1 && yIndexes > 1;
+      });
+      removedCells = baseCells.length - filteredBaseCells.length;
+      baseCells = filteredBaseCells;
+    }
+    return this.sortCircleCells(baseCells);
+  }
+
+  private getMaxReducableValueFromCircle(circle: Array<SelectedCell>): number {
+    return minBy(
+      circle?.filter((cell, index) => cell?.value !== -1 && index % 2 !== 0),
+      'value',
+    )?.value;
+  }
+
+  private sortCircleCells(circle: Array<SelectedCell>): Array<SelectedCell> {
+    const comparingIndex = (isX: boolean) => (isX ? 'x' : 'y');
+    let isX = true;
+    const sortedCircle = [circle[0]];
+    for (let i = 1; i < circle.length; i++) {
+      const lastSortedCell = last(sortedCircle);
+      const baseCell = circle?.find(
+        (value) =>
+          value?.[comparingIndex(isX)] ===
+            lastSortedCell?.[comparingIndex(isX)] &&
+          value?.[comparingIndex(!isX)] !==
+            lastSortedCell?.[comparingIndex(!isX)],
+      );
+      sortedCircle.push(baseCell);
+      isX = !isX;
+    }
+    return sortedCircle;
+  }
+
+  private reduceTransportTable(
+    transportTable: TransportTable,
+    circle: Array<SelectedCell>,
+  ): void {
+    const maxReducableValue = this.getMaxReducableValueFromCircle(circle);
+    first(circle).value = maxReducableValue;
+    for (let i = 1; i < circle.length; i++) {
+      if (!circle[i]) continue;
+      if (i % 2) circle[i].value -= maxReducableValue;
+      else circle[i].value += maxReducableValue;
+    }
+    forEach(circle, (cell) => {
+      transportTable[cell?.y][cell?.x].transported =
+        cell?.value === 0 ? undefined : cell?.value;
+    });
+  }
+
+  private getBaseCellPositions(
+    transportTable: TransportTable,
+    nextBase?: SelectedCell,
+  ): Array<SelectedCell> {
+    const baseCellPositions: Array<SelectedCell> = [];
+    if (nextBase) baseCellPositions.push({...nextBase, value: -1});
+
+    forEach(transportTable, (row, rowIndex) =>
+      forEach(row, (cell, columnIndex) => {
+        if (cell.transported !== undefined)
+          baseCellPositions.push({
+            x: +columnIndex,
+            y: rowIndex,
+            value: cell.transported,
+          });
+      }),
+    );
+
+    return baseCellPositions;
   }
 }
diff --git a/apps/backend/src/app/controllers/transport-problem/utils/base-cells-counter.util.ts b/apps/backend/src/app/controllers/transport-problem/utils/base-cells-counter.util.ts
new file mode 100644
index 0000000..a1ee001
--- /dev/null
+++ b/apps/backend/src/app/controllers/transport-problem/utils/base-cells-counter.util.ts
@@ -0,0 +1,22 @@
+import {TransportTable} from '@opres/shared/types';
+import {countBy} from 'lodash';
+
+import {getColumnUtil} from './get-column.util';
+
+export function getBaseCellsInARow(
+  transportTable: TransportTable,
+  rowIndex: number,
+): number {
+  return countBy(
+    transportTable[rowIndex],
+    (cell) => cell.transported !== undefined,
+  ).true;
+}
+
+export function getBaseCellsInColumn(
+  transportTable: TransportTable,
+  columnIndex: number,
+): number {
+  const column = getColumnUtil(transportTable, columnIndex);
+  return countBy(column, (cell) => cell.transported !== undefined).true;
+}
diff --git a/apps/backend/src/app/controllers/transport-problem/utils/get-column.util.ts b/apps/backend/src/app/controllers/transport-problem/utils/get-column.util.ts
new file mode 100644
index 0000000..37372c0
--- /dev/null
+++ b/apps/backend/src/app/controllers/transport-problem/utils/get-column.util.ts
@@ -0,0 +1,8 @@
+import {Cell, TransportTable} from '@opres/shared/types';
+
+export function getColumnUtil(
+  table: TransportTable,
+  columnIndex: number,
+): Array<Cell> {
+  return table.map((row) => row[columnIndex]);
+}
diff --git a/apps/frontend/src/app/pages/transport-problem/transport-problem.service.ts b/apps/frontend/src/app/pages/transport-problem/transport-problem.service.ts
index 58392be..45d8ead 100644
--- a/apps/frontend/src/app/pages/transport-problem/transport-problem.service.ts
+++ b/apps/frontend/src/app/pages/transport-problem/transport-problem.service.ts
@@ -83,7 +83,7 @@ export class TransportProblemService {
           return this.getEpsilonResult(
             last(steps)?.transportation || [],
             mode === 'explanations',
-          ).pipe(map((epsilon) => ({steps, epsilon} as FirstPhaseResult)));
+          ).pipe(map((epsilon) => ({steps, epsilon} as SecondPhaseResult)));
         }),
       );
   }
diff --git a/libs/shared/data/mocks/src/lib/transport-problem/index.ts b/libs/shared/data/mocks/src/lib/transport-problem/index.ts
index 8aa34b6..e631368 100644
--- a/libs/shared/data/mocks/src/lib/transport-problem/index.ts
+++ b/libs/shared/data/mocks/src/lib/transport-problem/index.ts
@@ -2,3 +2,4 @@ export * from './transport-problem.mock';
 export * from './north-west-result.mock';
 export * from './table-minimum-result.mock';
 export * from './vogel-korda-result.mock';
+export * from './second-phase-result.mock';
diff --git a/libs/shared/data/mocks/src/lib/transport-problem/second-phase-result.mock.ts b/libs/shared/data/mocks/src/lib/transport-problem/second-phase-result.mock.ts
new file mode 100644
index 0000000..ea6edc1
--- /dev/null
+++ b/libs/shared/data/mocks/src/lib/transport-problem/second-phase-result.mock.ts
@@ -0,0 +1,251 @@
+import {SecondPhaseStep} from '@opres/shared/types';
+
+export const secondPhaseFirstResultMock: Array<SecondPhaseStep> = [
+  {
+    transportation: [
+      {
+        '0': {cost: 8, reducedCost: 11},
+        '1': {cost: 7, reducedCost: 9},
+        '2': {cost: 3, reducedCost: 5},
+        '3': {cost: 2, transported: 15},
+      },
+      {
+        '0': {cost: 1, transported: 8},
+        '1': {cost: 4, reducedCost: 2},
+        '2': {cost: 2, transported: 35},
+        '3': {cost: 5, reducedCost: -1},
+      },
+      {
+        '0': {cost: 2, transported: 10},
+        '1': {cost: 3, transported: 13},
+        '2': {cost: 4, reducedCost: 1},
+        '3': {cost: 7, transported: 5},
+      },
+      {
+        '0': {cost: 1, reducedCost: 1},
+        '1': {cost: 1, transported: 19},
+        '2': {cost: 4, reducedCost: 3},
+        '3': {cost: 4, reducedCost: -1},
+      },
+    ],
+    auxiliaryVariables: {
+      x: [-3, -2, -2, 2],
+      y: [0, 4, 5, 3],
+    },
+    nextBase: {value: -1, x: 3, y: 1},
+    circle: [
+      {value: -1, x: 3, y: 1},
+      {x: 3, y: 2, value: 5},
+      {x: 0, y: 2, value: 10},
+      {x: 0, y: 1, value: 8},
+    ],
+  },
+  {
+    transportation: [
+      {
+        '0': {cost: 8, reducedCost: 10},
+        '1': {cost: 7, reducedCost: 8},
+        '2': {cost: 3, reducedCost: 4},
+        '3': {cost: 2, transported: 15},
+      },
+      {
+        '0': {cost: 1, transported: 3},
+        '1': {cost: 4, reducedCost: 2},
+        '2': {cost: 2, transported: 35},
+        '3': {cost: 5, transported: 5},
+      },
+      {
+        '0': {cost: 2, transported: 15},
+        '1': {cost: 3, transported: 13},
+        '2': {cost: 4, reducedCost: 1},
+        '3': {cost: 7, reducedCost: 1},
+      },
+      {
+        '0': {cost: 1, reducedCost: 1},
+        '1': {cost: 1, transported: 19},
+        '2': {cost: 4, reducedCost: 3},
+        '3': {cost: 4, reducedCost: 0},
+      },
+    ],
+    auxiliaryVariables: {
+      x: [-2, -1, -1, 2],
+      y: [0, 3, 4, 2],
+    },
+  },
+];
+
+export const secondPhaseSecondResultMock: Array<SecondPhaseStep> = [
+  {
+    transportation: [
+      {
+        '0': {cost: 1, transported: 12},
+        '1': {cost: 5, reducedCost: 3},
+        '2': {cost: 4, reducedCost: 7},
+        '3': {cost: 3, transported: 3},
+        '4': {cost: 2, reducedCost: 3},
+      },
+      {
+        '0': {cost: 1, reducedCost: -3},
+        '1': {cost: 1, reducedCost: -4},
+        '2': {cost: 0, transported: 3},
+        '3': {cost: 3, reducedCost: -3},
+        '4': {cost: 7, reducedCost: 5},
+      },
+      {
+        '0': {cost: 6, reducedCost: 0},
+        '1': {cost: 3, reducedCost: -4},
+        '2': {cost: 2, transported: 3},
+        '3': {cost: 8, transported: 1},
+        '4': {cost: 4, transported: 4},
+      },
+      {
+        '0': {cost: 1, reducedCost: 1},
+        '1': {cost: 1, transported: 7},
+        '2': {cost: 5, reducedCost: 9},
+        '3': {cost: 2, transported: 6},
+        '4': {cost: 3, reducedCost: 5},
+      },
+    ],
+    auxiliaryVariables: {
+      x: [1, 2, -3, 3, -1],
+      y: [0, 3, 5, -1],
+    },
+    nextBase: {value: -4, x: 1, y: 1},
+    circle: [
+      {value: -1, x: 1, y: 1},
+      {x: 1, y: 3, value: 7},
+      {x: 3, y: 3, value: 6},
+      {x: 3, y: 2, value: 1},
+      {x: 2, y: 2, value: 3},
+      {x: 2, y: 1, value: 3},
+    ],
+  },
+  {
+    transportation: [
+      {
+        '0': {cost: 1, transported: 12},
+        '1': {cost: 5, reducedCost: 3},
+        '2': {cost: 4, reducedCost: 3},
+        '3': {cost: 3, transported: 3},
+        '4': {cost: 2, reducedCost: -1},
+      },
+      {
+        '0': {cost: 1, reducedCost: 1},
+        '1': {cost: 1, transported: 1},
+        '2': {cost: 0, transported: 2},
+        '3': {cost: 3, reducedCost: 1},
+        '4': {cost: 7, reducedCost: 5},
+      },
+      {
+        '0': {cost: 6, reducedCost: 4},
+        '1': {cost: 3, reducedCost: 0},
+        '2': {cost: 2, transported: 4},
+        '3': {cost: 8, reducedCost: 4},
+        '4': {cost: 4, transported: 4},
+      },
+      {
+        '0': {cost: 1, reducedCost: 1},
+        '1': {cost: 1, transported: 6},
+        '2': {cost: 5, reducedCost: 5},
+        '3': {cost: 2, transported: 7},
+        '4': {cost: 3, reducedCost: 1},
+      },
+    ],
+    auxiliaryVariables: {
+      x: [1, 2, 1, 3, 3],
+      y: [0, -1, 1, -1],
+    },
+    nextBase: {value: -1, x: 4, y: 0},
+    circle: [
+      {value: -1, x: 4, y: 0},
+      {x: 4, y: 2, value: 4},
+      {x: 2, y: 2, value: 4},
+      {x: 2, y: 1, value: 2},
+      {x: 1, y: 1, value: 1},
+      {x: 1, y: 3, value: 6},
+      {x: 3, y: 3, value: 7},
+      {x: 3, y: 0, value: 3},
+    ],
+  },
+  {
+    transportation: [
+      {
+        '0': {cost: 1, transported: 12},
+        '1': {cost: 5, reducedCost: 3},
+        '2': {cost: 4, reducedCost: 4},
+        '3': {cost: 3, transported: 1},
+        '4': {cost: 2, transported: 2},
+      },
+      {
+        '0': {cost: 1, reducedCost: 1},
+        '1': {cost: 1, transported: 3},
+        '2': {cost: 0, reducedCost: 1},
+        '3': {cost: 3, reducedCost: 1},
+        '4': {cost: 7, reducedCost: 6},
+      },
+      {
+        '0': {cost: 6, reducedCost: 3},
+        '1': {cost: 3, reducedCost: -1},
+        '2': {cost: 2, transported: 6},
+        '3': {cost: 8, reducedCost: 3},
+        '4': {cost: 4, transported: 2},
+      },
+      {
+        '0': {cost: 1, reducedCost: 1},
+        '1': {cost: 1, transported: 4},
+        '2': {cost: 5, reducedCost: 6},
+        '3': {cost: 2, transported: 9},
+        '4': {cost: 3, reducedCost: 2},
+      },
+    ],
+    auxiliaryVariables: {
+      x: [1, 2, 0, 3, 2],
+      y: [0, -1, 2, -1],
+    },
+    nextBase: {value: -1, x: 1, y: 2},
+    circle: [
+      {value: -1, x: 1, y: 2},
+      {x: 1, y: 3, value: 4},
+      {x: 3, y: 3, value: 9},
+      {x: 3, y: 0, value: 1},
+      {x: 4, y: 0, value: 2},
+      {x: 4, y: 2, value: 2},
+    ],
+  },
+  {
+    transportation: [
+      {
+        '0': {cost: 1, transported: 12},
+        '1': {cost: 5, reducedCost: 4},
+        '2': {cost: 4, reducedCost: 4},
+        '3': {cost: 3, reducedCost: 1},
+        '4': {cost: 2, transported: 3},
+      },
+      {
+        '0': {cost: 1, reducedCost: 0},
+        '1': {cost: 1, transported: 3},
+        '2': {cost: 0, reducedCost: 0},
+        '3': {cost: 3, reducedCost: 1},
+        '4': {cost: 7, reducedCost: 5},
+      },
+      {
+        '0': {cost: 6, reducedCost: 3},
+        '1': {cost: 3, transported: 1},
+        '2': {cost: 2, transported: 6},
+        '3': {cost: 8, reducedCost: 4},
+        '4': {cost: 4, transported: 1},
+      },
+      {
+        '0': {cost: 1, reducedCost: 0},
+        '1': {cost: 1, transported: 3},
+        '2': {cost: 5, reducedCost: 5},
+        '3': {cost: 2, transported: 10},
+        '4': {cost: 3, reducedCost: 1},
+      },
+    ],
+    auxiliaryVariables: {
+      x: [1, 1, 0, 2, 2],
+      y: [0, 0, 2, 0],
+    },
+  },
+];
diff --git a/libs/shared/types/src/lib/transport-problem.type.ts b/libs/shared/types/src/lib/transport-problem.type.ts
index 2d22cee..2e9109d 100644
--- a/libs/shared/types/src/lib/transport-problem.type.ts
+++ b/libs/shared/types/src/lib/transport-problem.type.ts
@@ -14,6 +14,7 @@ export interface TPData {
 export interface Cell {
   cost: number;
   transported?: number;
+  reducedCost?: number;
 }
 
 export type TransportRow = Record<string, Cell>;
@@ -32,8 +33,16 @@ export interface FirstPhaseStep {
   explanation?: string;
 }
 
-// todo: wip
+export interface SelectedCell {
+  x: number;
+  y: number;
+  value?: number;
+}
+
 export interface SecondPhaseStep {
   transportation: TransportTable;
+  circle?: Array<SelectedCell>;
+  auxiliaryVariables?: AuxiliaryVariables;
+  nextBase?: SelectedCell;
   explanation?: string;
 }