Linear Programming – Mathematics Optional PYQs (2008–2026)

Here are all 55 Linear Programming questions asked in UPSC Mathematics Optional Paper II from 2008 to 2026, arranged under the 3 topics of the official syllabus. Within each topic the latest questions come first.

Each question ends with a code such as [2024/10m/1a] – the year, the marks and the question number in the paper.

4.1] LPP, graphical and simplex methods

30 questions

1. Solve the dual problem of the following linear programming problem by the graphical method : Minimize $Y=10y_1+6y_2+2y_3+y_4$ subject to the constraints : $\begin{array}{cc}\hfill y_1+y_2+y_3+y_4\hfill & \hfill \ge 2\hfill \\ \hfill 2y_1+y_2-y_3-2y_4\hfill & \hfill \ge 1\hfill \\ \hfill y_1,y_2,y_3,y_4\hfill & \hfill \ge 0\hfill \end{array}$ [2026/10m/1e]

2. Solve the following linear programming problem by the Simplex method : Maximize $Z=4x_1+10x_2$ subject to the constraints $\begin{array}{cc}\hfill 2x_1+x_2\hfill & \hfill \le 10\hfill \\ \hfill 2x_1+5x_2\hfill & \hfill \le 20\hfill \\ \hfill 2x_1+3x_2\hfill & \hfill \le 18\hfill \\ \hfill x_1,x_2\hfill & \hfill \ge 0\hfill \end{array}$ Obtain an alternate optimal solution, if it exists, with explanation. [2026/20m/4c]

3. How many basic solutions are there for the following system of equations? $\begin{array}{cc}\hfill 2x_1-x_2+3x_3+x_4\hfill & \hfill =6\hfill \\ \hfill 4x_1-2x_2-x_3+2x_4\hfill & \hfill =10\hfill \end{array}$ Find all of them. Furthermore, find the number of basic solutions, which are feasible/non-feasible/non-degenerate. [2025/10m/1e]

4. Use two phase method to solve the following linear programming problem : Maximize $z=x_1+2x_2$ subject to $\begin{array}{cc}\hfill x_1-x_2\hfill & \hfill \ge 3\hfill \\ \hfill 2x_1+x_2\hfill & \hfill \le 10\hfill \\ \hfill x_1,x_2\hfill & \hfill \ge 0\hfill \end{array}$ [2024/10m/1e]

5. A person requires 24, 24 and 20 units of chemicals $A$, $B$ and $C$ respectively for his garden. Product $P$ contains 2, 4 and 1 units of chemicals $A$, $B$ and $C$ respectively per jar and product $Q$ contains 2, 1 and 5 units of chemicals $A$, $B$ and $C$ respectively per jar. If a jar of $P$ costs ₹ 30 and a jar of $Q$ costs ₹ 50, then how many jars of each should be purchased in order to minimize the cost and meet the requirements? [2023/10m/1e]

6. Solve the following linear programming problem by Big M method : Minimize $Z=2x_1+3x_2$ subject to $\begin{array}{cc}\hfill x_1+x_2\hfill & \hfill \ge 9\hfill \\ \hfill x_1+2x_2\hfill & \hfill \ge 15\hfill \\ \hfill 2x_1-3x_2\hfill & \hfill \le 9\hfill \\ \hfill x_1,x_2\hfill & \hfill \ge 0\hfill \end{array}$ Is the optimal solution unique? Justify your answer. [2023/20m/3c]

7. Use two-phase method to solve the following linear programming problem : Minimize $Z=x_1+x_2$ subject to $\begin{array}{cc}\hfill 2x_1+x_2\hfill & \hfill \ge 4\hfill \\ \hfill x_1+7x_2\hfill & \hfill \ge 7\hfill \\ \hfill x_1,x_2\hfill & \hfill \ge 0\hfill \end{array}$ [2022/10m/1e]

8. Solve the following linear programming problem by the simplex method. Write its dual. Also, write the optimal solution of the dual from the optimal table of the given problem : Maximize $Z=x_1+x_2+x_3$ subject to $\begin{array}{cc}\hfill 2x_1+x_2+x_3\hfill & \hfill \le 2\hfill \\ \hfill 4x_1+2x_2+x_3\hfill & \hfill \le 2\hfill \\ \hfill x_1,x_2,x_3\hfill & \hfill \ge 0\hfill \end{array}$ [2022/15m/3c]

9. Solve the following linear programming problem using Big M method : Maximize $Z=4x_1+5x_2+2x_3$ subject to $\begin{array}{cc}\hfill 2x_1+x_2+x_3\hfill & \hfill \ge 10\hfill \\ \hfill x_1+3x_2+x_3\hfill & \hfill \le 12\hfill \\ \hfill x_1+x_2+x_3\hfill & \hfill =6\hfill \\ \hfill x_1,x_2,x_3\hfill & \hfill \ge 0\hfill \end{array}$ [2021/15m/4c]

10. UPSC maintenance section has purchased sufficient number of curtain cloth pieces to meet the curtain requirement of its building. The length of each piece is 17 feet. The requirement according to curtain length is as follows : $\begin{array}{cc}\hfill \mathrm{\text{Curtain length (in feet)}}\hfill & \hfill \mathrm{\text{Number required}}\hfill \\ \hfill 5\hfill & \hfill 700\hfill \\ \hfill 9\hfill & \hfill 400\hfill \\ \hfill 7\hfill & \hfill 300\hfill \end{array}$ The width of all curtains is same as that of available pieces. Form a linear programming problem in standard form that decides the number of pieces cut in different ways so that the total trim loss is minimum. Also give a basic feasible solution to it. [2020/10m/1e]

11. Solve the linear programming problem using simplex method : Minimize $z=-6x_1-2x_2-5x_3$ subject to $\begin{array}{cc}\hfill 2x_1-3x_2+x_3\hfill & \hfill \le 14\hfill \\ \hfill -4x_1+4x_2+10x_3\hfill & \hfill \le 46\hfill \\ \hfill 2x_1+2x_2-4x_3\hfill & \hfill \le 37\hfill \\ \hfill \hfill & \hfill x_1\ge 2,x_2\ge 1,x_3\ge 3\hfill \end{array}$ [2020/15m/3b]

12. Use graphical method to solve the linear programming problem. Maximize $Z=3x_1+2x_2$ subject to $\begin{array}{cc}\hfill x_1-x_2\hfill & \hfill \ge 1,\hfill \\ \hfill x_1+x_3\hfill & \hfill \ge 3\hfill \end{array}$ and $x_1,x_2,x_3\ge 0$ [2019/10m/1e]

13. Solve the linear programming problem using Simplex method. Minimize $Z=x_1+2x_2-3x_3-2x_4$ subject to $\begin{array}{cc}\hfill x_1+2x_2-3x_3+x_4\hfill & \hfill =4\hfill \\ \hfill x_1+2x_2+x_3+2x_4\hfill & \hfill =4\hfill \end{array}$ and $x_1,x_2,x_3,x_4\ge 0$ [2019/15m/3b]

14. An agricultural firm has 180 tons of nitrogen fertilizer, 250 tons of phosphate and 220 tons of potash. It will be able to sell a mixture of these substances in their respective ratio $3:3:4$ at a profit of Rs. 1500 per ton and a mixture in the ratio $2:4:2$ at a profit of Rs. 1200 per ton. Pose a linear programming problem to show how many tons of these two mixtures should be prepared to obtain the maximum profit. [2018/10m/1e]

15. Solve the following linear programming problem by Big M-method and show that the problem has finite optimal solutions. Also find the value of the objective function : Minimize $z=3x_1+5x_2$ subject to $\begin{array}{cc}\hfill x_1+2x_2\hfill & \hfill \ge 8\hfill \\ \hfill 3x_1+2x_2\hfill & \hfill \ge 12\hfill \\ \hfill 5x_1+6x_2\hfill & \hfill \le 60,\hfill \\ \hfill x_1,x_2\hfill & \hfill \ge 0.\hfill \end{array}$ [2018/20m/2b]

16. How many basic solutions are there in the following linearly independent set of equations ? Find all of them. $\begin{array}{cc}\hfill 2x_1-x_2+3x_3+x_4\hfill & \hfill =6\hfill \\ \hfill 4x_1-2x_2-x_3+2x_4\hfill & \hfill =10.\hfill \end{array}$ [2018/15m/3c]

17. Using graphical method, find the maximum value of $2x+y$ subject to $\begin{array}{cc}\hfill 4x+3y\hfill & \hfill \le 12\hfill \\ \hfill 4x+y\hfill & \hfill \le 8\hfill \\ \hfill 4x-y\hfill & \hfill \le 8\hfill \\ \hfill x,y\hfill & \hfill \ge 0.\hfill \end{array}$ [2017/10m/1e]

18. Solve the following linear programming problem by simplex method : Maximize $z=3x_1+5x_2+4x_3$ subject to $\begin{array}{cc}\hfill 2x_1+3x_2\hfill & \hfill \le 8\hfill \\ \hfill 2x_2+5x_3\hfill & \hfill \le 10\hfill \\ \hfill 3x_1+2x_2+4x_3\hfill & \hfill \le 15\hfill \\ \hfill x_1,x_2,x_3\hfill & \hfill \ge 0.\hfill \end{array}$ [2017/20m/3c]

19. Find the maximum value of $5x+2y$ with constraints $x+2y\ge 1,2x+y\le 1,x\ge 0\mathrm{\text{and}}y\ge 0$ by graphical method. [2016/10m/1e]

20. Maximize $z=2x_1+3x_2+6x_3$ subject to $\begin{array}{cc}\hfill 2x_1+x_2+x_3\hfill & \hfill \le 5\hfill \\ \hfill 3x_2+2x_3\hfill & \hfill \le 6\hfill \\ \hfill x_1\ge 0,x_2\ge 0,x_3\hfill & \hfill \ge 0.\hfill \end{array}$ Is the optimal solution unique ? Justify your answer. [2016/20m/2c]

21. Consider the following linear programming problem : $\begin{array}{cc}\hfill \hfill & \hfill \mathrm{\text{Maximize}}Z=x_1+2x_2-3x_3+4x_4\hfill \\ \hfill \hfill & \hfill \mathrm{\text{subject to}}\hfill \\ \hfill \hfill & \hfill x_1+x_2+2x_3+3x_4=12\hfill \\ \hfill \hfill & \hfill x_2+2x_3+x_4=8\hfill \\ \hfill \hfill & \hfill x_1,x_2,x_3,x_4\ge 0\hfill \end{array}$ \(i\) Using the definition, find its all basic solutions. Which of these are degenerate basic feasible solutions and which are non-degenerate basic feasible solutions? (ii) Without solving the problem, show that it has an optimal solution. Which of the basic feasible solution(s) is/are optimal? [2015/20m/3c]

22. Solve the following linear programming problem by the simplex method. Write its dual. Also, write the optimal solution of the dual from the optimal table of the given problem : $\begin{array}{cc}\hfill \hfill & \hfill \mathrm{\text{Maximize}}Z=2x_1-4x_2+5x_3\hfill \\ \hfill \hfill & \hfill \mathrm{\text{subject to}}\hfill \\ \hfill \hfill & \hfill x_1+4x_2-2x_3\le 2\hfill \\ \hfill \hfill & \hfill -x_1+2x_2+3x_3\le 1\hfill \\ \hfill \hfill & \hfill x_1,x_2,x_3\ge 0\hfill \end{array}$ [2015/20m/4c]

23. Solve graphically : Maximize $Z=6x_1+5x_2$ subject to $\begin{array}{cc}\hfill 2x_1+x_2\hfill & \hfill \le 16\hfill \\ \hfill x_1+x_2\hfill & \hfill \le 11\hfill \\ \hfill x_1+2x_2\hfill & \hfill \ge 6\hfill \\ \hfill 5x_1+6x_2\hfill & \hfill \le 90\hfill \\ \hfill x_1,x_2\hfill & \hfill \ge 0\hfill \end{array}$ [2014/10m/1e]

24. Find all optimal solutions of the following linear programming problem by the simplex method : Maximize $Z=30x_1+24x_2$ subject to $\begin{array}{cc}\hfill 5x_1+4x_2\hfill & \hfill \le 200\hfill \\ \hfill x_1\hfill & \hfill \le 32\hfill \\ \hfill x_2\hfill & \hfill \le 40\hfill \\ \hfill x_1,x_2\hfill & \hfill \ge 0\hfill \end{array}$ [2014/20m/4c]

25. Maximize $z=2x_1+3x_2-5x_3$ subject to $\begin{array}{cc}\hfill \hfill & \hfill x_1+x_2+x_3=7\hfill \\ \hfill \mathrm{\text{and}}\hfill & \hfill 2x_1-5x_2+x_3\ge 10,x_i\ge 0.\hfill \end{array}$ [2013/10m/1e]

26. Minimize $z=5x_1-4x_2+6x_3-8x_4$ subject to the constraints $\begin{array}{cc}\hfill x_1+2x_2-2x_3+4x_4\hfill & \hfill \le 40\hfill \\ \hfill 2x_1-x_2+x_3+2x_4\hfill & \hfill \le 8\hfill \\ \hfill 4x_1-2x_2+x_3-x_4\hfill & \hfill \le 10\hfill \\ \hfill x_i\hfill & \hfill \ge 0\hfill \end{array}$ [2013/20m/4c]

27. For each hour per day that Ashok studies mathematics, it yields him 10 marks and for each hour that he studies physics, it yields him 5 marks. He can study at most 14 hours a day and he must get at least 40 marks in each. Determine graphically how many hours a day he should study mathematics and physics each, in order to maximize his marks ? [2012/12m/1d]

28. Solve by Simplex method, the following LP Problem : Maximize, $Z=5x_1+3x_2$ Constraints, $\begin{array}{cc}\hfill 3x_1+5x_2\hfill & \hfill \le 15\hfill \\ \hfill 5x_1+2x_2\hfill & \hfill \le 10\hfill \\ \hfill x_1,x_2\hfill & \hfill \ge 0\hfill \end{array}$ [2011/12m/1d]

29. A paint factory produces both interior and exterior paint from two raw materials $M_1$ and $M_2$. The basic data is as follows : $\begin{array}{cccc}\hfill \hfill & \hfill \mathrm{\text{Tons of raw material per ton of}}\hfill & \hfill \hfill & \hfill \mathrm{\text{Maximum}}\hfill \\ \hfill \hfill & \hfill \mathrm{\text{Exterior paint}}\hfill & \hfill \mathrm{\text{Interior paint}}\hfill & \hfill \mathrm{\text{daily availability}}\hfill \\ \hfill \mathrm{\text{Raw Material}}{M}_1\hfill & \hfill 6\hfill & \hfill 4\hfill & \hfill 24\hfill \\ \hfill \mathrm{\text{Raw Material}}{M}_2\hfill & \hfill 1\hfill & \hfill 2\hfill & \hfill 6\hfill \\ \hfill \mathrm{\text{Profit per ton (Rs. 1,000)}}\hfill & \hfill 5\hfill & \hfill 4\hfill & \hfill \hfill \end{array}$ A market survey indicates that the daily demand for interior paint cannot exceed that of exterior paint by more than 1 ton. The maximum daily demand of interior paint is 2 tons. The factory wants to determine the optimum product mix of interior and exterior paint that maximizes daily profits. Formulate the LP problem for this situation. [2009/12m/1f]

30. Maximize : $Z=3x_1+5x_2+4x_3$ subject to : $\begin{array}{cc}\hfill 2x_1+3x_2\hfill & \hfill \le 8,\hfill \\ \hfill 3x_1+2x_2+4x_3\hfill & \hfill \le 15,\hfill \\ \hfill 2x_2+5x_3\hfill & \hfill \le 10,\hfill \\ \hfill x_i\hfill & \hfill \ge 0.\hfill \end{array}$ [2009/30m/4b]

4.2] Duality

10 questions

1. Solve the dual problem of the following linear programming problem by the graphical method : Minimize $Y=10y_1+6y_2+2y_3+y_4$ subject to the constraints : $\begin{array}{cc}\hfill y_1+y_2+y_3+y_4\hfill & \hfill \ge 2\hfill \\ \hfill 2y_1+y_2-y_3-2y_4\hfill & \hfill \ge 1\hfill \\ \hfill y_1,y_2,y_3,y_4\hfill & \hfill \ge 0\hfill \end{array}$ [2026/10m/1e]

2. Apply the principle of duality to solve the following linear programming problem : Maximize $Z=3x_1+4x_2$ subject to the constraints $\begin{array}{cc}\hfill x_1-x_2\hfill & \hfill \le 1\hfill \\ \hfill x_1+x_2\hfill & \hfill \ge 4\hfill \\ \hfill x_1-3x_2\hfill & \hfill \le 3\hfill \\ \hfill x_1,x_2\hfill & \hfill \ge 0\hfill \end{array}$ [2025/15m/3c]

3. Using duality principle, solve the following linear programming problem : Minimize $z=4x_1+3x_2+x_3$ subject to $\begin{array}{cc}\hfill x_1+2x_2+4x_3\hfill & \hfill \ge 12\hfill \\ \hfill 3x_1+2x_2+x_3\hfill & \hfill \ge 8\hfill \\ \hfill x_1,x_2,x_3\hfill & \hfill \ge 0\hfill \end{array}$ [2024/15m/3c]

4. Solve the following linear programming problem by the simplex method. Write its dual. Also, write the optimal solution of the dual from the optimal table of the given problem : Maximize $Z=x_1+x_2+x_3$ subject to $\begin{array}{cc}\hfill 2x_1+x_2+x_3\hfill & \hfill \le 2\hfill \\ \hfill 4x_1+2x_2+x_3\hfill & \hfill \le 2\hfill \\ \hfill x_1,x_2,x_3\hfill & \hfill \ge 0\hfill \end{array}$ [2022/15m/3c]

5. Convert the following LPP into dual LPP : Minimize $Z=x_1-3x_2-2x_3$ subject to $\begin{array}{cc}\hfill 3x_1-x_2+2x_3\hfill & \hfill \le 7\hfill \\ \hfill 2x_1-4x_2\hfill & \hfill \ge 12\hfill \\ \hfill -4x_1+3x_2+8x_3\hfill & \hfill =10\hfill \end{array}$ where $x_1$, $x_2\ge 0$ and $x_3$ is unrestricted in sign. [2021/15m/3c]

6. Consider the following LPP, Maximize $Z=2x_1+4x_2+4x_3-3x_4$ subject to $\begin{array}{cc}\hfill x_1+x_2+x_3\hfill & \hfill =4\hfill \\ \hfill x_1+4x_2+x_4\hfill & \hfill =8\hfill \end{array}$ and $x_1,x_2,x_3,x_4\ge 0$ Use the dual problem to verify that the basic solution $(x_1,x_2)$ is not optimal. [2019/10m/4d]

7. Solve the following linear programming problem by the simplex method. Write its dual. Also, write the optimal solution of the dual from the optimal table of the given problem : $\begin{array}{cc}\hfill \hfill & \hfill \mathrm{\text{Maximize}}Z=2x_1-4x_2+5x_3\hfill \\ \hfill \hfill & \hfill \mathrm{\text{subject to}}\hfill \\ \hfill \hfill & \hfill x_1+4x_2-2x_3\le 2\hfill \\ \hfill \hfill & \hfill -x_1+2x_2+3x_3\le 1\hfill \\ \hfill \hfill & \hfill x_1,x_2,x_3\ge 0\hfill \end{array}$ [2015/20m/4c]

8. Write down the dual of the following LP problem and hence solve it by graphical method : Minimize, $Z=6x_1+4x_2$ Constraints, $\begin{array}{cc}\hfill 2x_1+x_2\hfill & \hfill \ge 1\hfill \\ \hfill 3x_1+4x_2\hfill & \hfill \ge 1.5\hfill \\ \hfill x_1,x_2\hfill & \hfill \ge 0\hfill \end{array}$ [2011/20m/4c]

9. Construct the dual of the primal problem : Maximize $z=2x_1+x_2+x_3$, subject to the constraints $x_1+x_2+x_3\ge 6$, $3x_1-2x_2+3x_3=3$, $-4x_1+3x_2-6x_3=1$, and $x_1,x_2,x_3\ge 0$. [2010/12m/1f]

10. Find the dual of the following linear programming problem: Max. $Z=2x_1-x_2+x_3$ such that $\begin{array}{cc}\hfill \hfill & \hfill x_1+x_2-3x_3\le 8\hfill \\ \hfill \hfill & \hfill 4x_1-x_2+x_3=2\hfill \\ \hfill \hfill & \hfill 2x_1+3x_2-x_3\ge 5\hfill \\ \hfill \hfill & \hfill x_1,x_2,x_3\ge 0\hfill \end{array}$ [2008/12m/1f]

4.3] Transportation and assignment problems

15 questions

1. The following table represents the estimated costs of assigning jobs to machines. Solve this assignment problem to minimize the total cost. $\begin{array}{ccccc}\hfill \mathrm{\text{Job}}\setminus \mathrm{\text{Machine}}\hfill & \hfill M_1\hfill & \hfill M_2\hfill & \hfill M_3\hfill & \hfill M_4\hfill \\ \hfill J_1\hfill & \hfill 8\hfill & \hfill 24\hfill & \hfill 28\hfill & \hfill 32\hfill \\ \hfill J_2\hfill & \hfill 8\hfill & \hfill 13\hfill & \hfill 17\hfill & \hfill 19\hfill \\ \hfill J_3\hfill & \hfill 8\hfill & \hfill 15\hfill & \hfill 19\hfill & \hfill 22\hfill \end{array}$ Is the assignment unique for the total minimum cost ? Justify your answer and obtain an alternate assignment, if it exists. [2026/15m/3c]

2. The following table shows all the necessary information on the available supply to each warehouse, the requirement of each market and the unit transportation cost from each warehouse to each market : $\begin{array}{cccccc}\hfill \mathrm{\text{Warehouse}}\setminus \mathrm{\text{Market}}\hfill & \hfill I\hfill & \hfill II\hfill & \hfill III\hfill & \hfill IV\hfill & \hfill \mathrm{\text{Supply}}\hfill \\ \hfill A\hfill & \hfill 5\hfill & \hfill 2\hfill & \hfill 4\hfill & \hfill 3\hfill & \hfill 22\hfill \\ \hfill B\hfill & \hfill 4\hfill & \hfill 8\hfill & \hfill 1\hfill & \hfill 6\hfill & \hfill 15\hfill \\ \hfill C\hfill & \hfill 4\hfill & \hfill 6\hfill & \hfill 7\hfill & \hfill 5\hfill & \hfill 8\hfill \\ \hfill \mathrm{\text{Requirement}}\hfill & \hfill 7\hfill & \hfill 12\hfill & \hfill 17\hfill & \hfill 9\hfill & \hfill \hfill \end{array}$ The shipping clerk has worked out the following schedule from experience : 12 units from $A$ to $II$, 1 unit from $A$ to $III$, 9 units from $A$ to $IV$, 15 units from $B$ to $III$, 7 units from $C$ to $I$ and 1 unit from $C$ to $III$. Find the optimal schedule and minimum total shipping cost. [2025/20m/4c]

3. The personnel manager of a company wants to assign officers $A$, $B$ and $C$ to the regional offices at Delhi, Mumbai, Kolkata and Chennai. The cost of relocation (in thousand Rupees) of the three officers at the four regional offices are given below : $\begin{array}{ccccc}\hfill \mathrm{\text{Officer}}\setminus \mathrm{\text{Office}}\hfill & \hfill \mathrm{\text{Delhi}}\hfill & \hfill \mathrm{\text{Mumbai}}\hfill & \hfill \mathrm{\text{Kolkata}}\hfill & \hfill \mathrm{\text{Chennai}}\hfill \\ \hfill A\hfill & \hfill 16\hfill & \hfill 22\hfill & \hfill 24\hfill & \hfill 20\hfill \\ \hfill B\hfill & \hfill 10\hfill & \hfill 32\hfill & \hfill 26\hfill & \hfill 16\hfill \\ \hfill C\hfill & \hfill 10\hfill & \hfill 20\hfill & \hfill 46\hfill & \hfill 30\hfill \end{array}$ Find the assignment which minimizes the total cost of relocation and also determine the minimum cost. [2024/20m/4c]

4. A department head has 5 subordinates and 5 jobs to be performed. The time (in hours) that each subordinate will take to perform each job is given in the matrix below : $\begin{array}{cccccc}\hfill \mathrm{\text{Subordinates}}\setminus \mathrm{\text{Jobs}}\hfill & \hfill A\hfill & \hfill B\hfill & \hfill C\hfill & \hfill D\hfill & \hfill E\hfill \\ \hfill I\hfill & \hfill 4\hfill & \hfill 9\hfill & \hfill 4\hfill & \hfill 12\hfill & \hfill 4\hfill \\ \hfill II\hfill & \hfill 15\hfill & \hfill 11\hfill & \hfill 20\hfill & \hfill 5\hfill & \hfill 8\hfill \\ \hfill III\hfill & \hfill 17\hfill & \hfill 7\hfill & \hfill 15\hfill & \hfill 12\hfill & \hfill 18\hfill \\ \hfill IV\hfill & \hfill 9\hfill & \hfill 13\hfill & \hfill 11\hfill & \hfill 9\hfill & \hfill 14\hfill \\ \hfill V\hfill & \hfill 6\hfill & \hfill 11\hfill & \hfill 12\hfill & \hfill 9\hfill & \hfill 14\hfill \end{array}$ How should the jobs be assigned, one to each subordinate, so as to minimize the total time? Also, obtain the total minimum time to perform all the jobs if the subordinate $IV$ cannot be assigned job $C$. [2023/20m/4c]

5. Find the initial basic feasible solution of the following transportation problem by Vogel’s approximation method and use it to find the optimal solution and the transportation cost of the problem : $\begin{array}{cccccc}\hfill \mathrm{\text{Source}}\setminus \mathrm{\text{Destination}}\hfill & \hfill A\hfill & \hfill B\hfill & \hfill C\hfill & \hfill D\hfill & \hfill \mathrm{\text{Availability}}\hfill \\ \hfill S_1\hfill & \hfill 21\hfill & \hfill 16\hfill & \hfill 25\hfill & \hfill 13\hfill & \hfill 11\hfill \\ \hfill S_2\hfill & \hfill 17\hfill & \hfill 18\hfill & \hfill 14\hfill & \hfill 23\hfill & \hfill 13\hfill \\ \hfill S_3\hfill & \hfill 32\hfill & \hfill 27\hfill & \hfill 18\hfill & \hfill 41\hfill & \hfill 19\hfill \\ \hfill \mathrm{\text{Requirement}}\hfill & \hfill 6\hfill & \hfill 10\hfill & \hfill 12\hfill & \hfill 15\hfill & \hfill 43\hfill \end{array}$ [2022/20m/4c]

6. A department of a company has five employees with five jobs to be performed. The time (in hours) that each man takes to perform each job is given in the effectiveness matrix. Assign all the jobs to these five employees to minimize the total processing time : $\begin{array}{cccccc}\hfill \mathrm{\text{Jobs}}\setminus \mathrm{\text{Employees}}\hfill & \hfill I\hfill & \hfill II\hfill & \hfill III\hfill & \hfill IV\hfill & \hfill V\hfill \\ \hfill A\hfill & \hfill 10\hfill & \hfill 5\hfill & \hfill 13\hfill & \hfill 15\hfill & \hfill 16\hfill \\ \hfill B\hfill & \hfill 3\hfill & \hfill 9\hfill & \hfill 18\hfill & \hfill 13\hfill & \hfill 6\hfill \\ \hfill C\hfill & \hfill 10\hfill & \hfill 7\hfill & \hfill 2\hfill & \hfill 2\hfill & \hfill 2\hfill \\ \hfill D\hfill & \hfill 7\hfill & \hfill 11\hfill & \hfill 9\hfill & \hfill 7\hfill & \hfill 12\hfill \\ \hfill E\hfill & \hfill 7\hfill & \hfill 9\hfill & \hfill 10\hfill & \hfill 4\hfill & \hfill 12\hfill \end{array}$ [2021/10m/1e]

7. Find the initial basic feasible solution of the following transportation problem by Vogel’s approximation method and use it to find the optimal solution and the transportation cost of the problem. $\begin{array}{cccccc}\hfill \mathrm{\text{Sources}}\setminus \mathrm{\text{Destinations}}\hfill & \hfill D_1\hfill & \hfill D_2\hfill & \hfill D_3\hfill & \hfill D_4\hfill & \hfill \mathrm{\text{Availability}}\hfill \\ \hfill S_1\hfill & \hfill 10\hfill & \hfill 0\hfill & \hfill 20\hfill & \hfill 11\hfill & \hfill 15\hfill \\ \hfill S_2\hfill & \hfill 12\hfill & \hfill 8\hfill & \hfill 9\hfill & \hfill 20\hfill & \hfill 25\hfill \\ \hfill S_3\hfill & \hfill 0\hfill & \hfill 14\hfill & \hfill 16\hfill & \hfill 18\hfill & \hfill 10\hfill \\ \hfill \mathrm{\text{Demand}}\hfill & \hfill 5\hfill & \hfill 20\hfill & \hfill 15\hfill & \hfill 10\hfill & \hfill \hfill \end{array}$ [2020/20m/4c]

8. $\begin{array}{cccccc}\hfill \mathrm{\text{Operator}}\setminus \mathrm{\text{Machine}}\hfill & \hfill M_1\hfill & \hfill M_2\hfill & \hfill M_3\hfill & \hfill M_4\hfill & \hfill M_5\hfill \\ \hfill O_1\hfill & \hfill 24\hfill & \hfill 29\hfill & \hfill 18\hfill & \hfill 32\hfill & \hfill 19\hfill \\ \hfill O_2\hfill & \hfill 17\hfill & \hfill 26\hfill & \hfill 34\hfill & \hfill 22\hfill & \hfill 21\hfill \\ \hfill O_3\hfill & \hfill 27\hfill & \hfill 16\hfill & \hfill 28\hfill & \hfill 17\hfill & \hfill 25\hfill \\ \hfill O_4\hfill & \hfill 22\hfill & \hfill 18\hfill & \hfill 28\hfill & \hfill 30\hfill & \hfill 24\hfill \\ \hfill O_5\hfill & \hfill 28\hfill & \hfill 16\hfill & \hfill 31\hfill & \hfill 24\hfill & \hfill 27\hfill \end{array}$ In a factory there are five operators $O_1,O_2,O_3,O_4,O_5$ and five machines $M_1,M_2,M_3,M_4,M_5$. The operating costs are given when the $O_i$ operator operates the $M_j$ machine $(i,j=1,2,\dots ,5)$. But there is a restriction that $O_3$ cannot be allowed to operate the third machine $M_3$ and $O_2$ cannot be allowed to operate the fifth machine $M_5$. The cost matrix is given above. Find the optimal assignment and the optimal assignment cost also. [2018/15m/4c]

9. Find the initial basic feasible solution of the following transportation problem using Vogel’s approximation method and find the cost. $\begin{array}{ccccccc}\hfill \hfill & \hfill D_1\hfill & \hfill D_2\hfill & \hfill D_3\hfill & \hfill D_4\hfill & \hfill D_5\hfill & \hfill \mathrm{\text{Supply}}\hfill \\ \hfill O_1\hfill & \hfill 4\hfill & \hfill 7\hfill & \hfill 0\hfill & \hfill 3\hfill & \hfill 6\hfill & \hfill 14\hfill \\ \hfill O_2\hfill & \hfill 1\hfill & \hfill 2\hfill & \hfill -3\hfill & \hfill 3\hfill & \hfill 8\hfill & \hfill 9\hfill \\ \hfill O_3\hfill & \hfill 3\hfill & \hfill -1\hfill & \hfill 4\hfill & \hfill 0\hfill & \hfill 5\hfill & \hfill 17\hfill \\ \hfill \mathrm{\text{Demand}}\hfill & \hfill 8\hfill & \hfill 3\hfill & \hfill 8\hfill & \hfill 13\hfill & \hfill 8\hfill & \hfill \hfill \end{array}$ [2017/15m/4b]

10. Solve the following assignment problem to maximize the sales : $\begin{array}{cccccc}\hfill \mathrm{\text{Salesmen}}\setminus \mathrm{\text{Territories}}\hfill & \hfill I\hfill & \hfill II\hfill & \hfill III\hfill & \hfill IV\hfill & \hfill V\hfill \\ \hfill A\hfill & \hfill 3\hfill & \hfill 4\hfill & \hfill 5\hfill & \hfill 6\hfill & \hfill 7\hfill \\ \hfill B\hfill & \hfill 4\hfill & \hfill 15\hfill & \hfill 13\hfill & \hfill 7\hfill & \hfill 6\hfill \\ \hfill C\hfill & \hfill 6\hfill & \hfill 13\hfill & \hfill 12\hfill & \hfill 5\hfill & \hfill 11\hfill \\ \hfill D\hfill & \hfill 7\hfill & \hfill 12\hfill & \hfill 15\hfill & \hfill 8\hfill & \hfill 5\hfill \\ \hfill E\hfill & \hfill 8\hfill & \hfill 13\hfill & \hfill 10\hfill & \hfill 6\hfill & \hfill 9\hfill \end{array}$ [2015/10m/1e]

11. Find the initial basic feasible solution to the following transportation problem by Vogel’s approximation method. Also, find its optimal solution and the minimum transportation cost : $\begin{array}{cccccc}\hfill \hfill & \hfill D_1\hfill & \hfill D_2\hfill & \hfill D_3\hfill & \hfill D_4\hfill & \hfill \mathrm{\text{Supply}}\hfill \\ \hfill O_1\hfill & \hfill 6\hfill & \hfill 4\hfill & \hfill 1\hfill & \hfill 5\hfill & \hfill 14\hfill \\ \hfill O_2\hfill & \hfill 8\hfill & \hfill 9\hfill & \hfill 2\hfill & \hfill 7\hfill & \hfill 16\hfill \\ \hfill O_3\hfill & \hfill 4\hfill & \hfill 3\hfill & \hfill 6\hfill & \hfill 2\hfill & \hfill 5\hfill \\ \hfill \mathrm{\text{Demand}}\hfill & \hfill 6\hfill & \hfill 10\hfill & \hfill 15\hfill & \hfill 4\hfill & \hfill \hfill \end{array}$ [2014/20m/2c]

12. Solve the minimum time assignment problem : $\begin{array}{ccccc}\hfill \mathrm{\text{Jobs}}\setminus \mathrm{\text{Machines}}\hfill & \hfill M_1\hfill & \hfill M_2\hfill & \hfill M_3\hfill & \hfill M_4\hfill \\ \hfill J_1\hfill & \hfill 3\hfill & \hfill 12\hfill & \hfill 5\hfill & \hfill 14\hfill \\ \hfill J_2\hfill & \hfill 7\hfill & \hfill 9\hfill & \hfill 8\hfill & \hfill 12\hfill \\ \hfill J_3\hfill & \hfill 5\hfill & \hfill 11\hfill & \hfill 10\hfill & \hfill 12\hfill \\ \hfill J_4\hfill & \hfill 6\hfill & \hfill 14\hfill & \hfill 4\hfill & \hfill 11\hfill \end{array}$ [2013/15m/4a]

13. By the method of Vogel, determine an initial basic feasible solution for the following transportation problem : Products $P_1$, $P_2$, $P_3$ and $P_4$ have to be sent to destinations $D_1$, $D_2$ and $D_3$. The cost of sending product $P_i$ to destinations $D_j$ is $C_{ij}$, where the matrix $\left[C_{ij}\right]=\left[\begin{array}{cccc}\hfill 10\hfill & \hfill 0\hfill & \hfill 15\hfill & \hfill 5\hfill \\ \hfill 7\hfill & \hfill 3\hfill & \hfill 6\hfill & \hfill 15\hfill \\ \hfill 0\hfill & \hfill 11\hfill & \hfill 9\hfill & \hfill 13\hfill \end{array}\right].$ The total requirements of destinations $D_1$, $D_2$ and $D_3$ are given by 45, 45, 95 respectively and the availability of the products $P_1$, $P_2$, $P_3$ and $P_4$ are respectively 25, 35, 55 and 70. [2012/20m/4c]

14. Determine an optimal transportation programme so that the transportation cost of 340 tons of a certain type of material from three factories $F_1$, $F_2$, $F_3$ to five warehouses $W_1$, $W_2$, $W_3$, $W_4$, $W_5$ is minimized. The five warehouses must receive 40 tons, 50 tons, 70 tons, 90 tons and 90 tons respectively. The availability of the material at $F_1$, $F_2$, $F_3$ is 100 tons, 120 tons, 120 tons respectively. The transportation costs per ton from factories to warehouses are given in the table below : $\begin{array}{cccccc}\hfill \hfill & \hfill W_1\hfill & \hfill W_2\hfill & \hfill W_3\hfill & \hfill W_4\hfill & \hfill W_5\hfill \\ \hfill F_1\hfill & \hfill 4\hfill & \hfill 1\hfill & \hfill 2\hfill & \hfill 6\hfill & \hfill 9\hfill \\ \hfill F_2\hfill & \hfill 6\hfill & \hfill 4\hfill & \hfill 3\hfill & \hfill 5\hfill & \hfill 7\hfill \\ \hfill F_3\hfill & \hfill 5\hfill & \hfill 2\hfill & \hfill 6\hfill & \hfill 4\hfill & \hfill 8\hfill \end{array}$ Use Vogel’s approximation method to obtain the initial basic feasible solution. [2010/30m/4c]

15. Solve the following transportation problem : $\begin{array}{cccccccc}\hfill \hfill & \hfill D_1\hfill & \hfill D_2\hfill & \hfill D_3\hfill & \hfill D_4\hfill & \hfill D_5\hfill & \hfill D_6\hfill & \hfill \mathrm{\text{Availability}}\hfill \\ \hfill F_1\hfill & \hfill 2\hfill & \hfill 1\hfill & \hfill 3\hfill & \hfill 3\hfill & \hfill 2\hfill & \hfill 5\hfill & \hfill 50\hfill \\ \hfill F_2\hfill & \hfill 3\hfill & \hfill 2\hfill & \hfill 2\hfill & \hfill 4\hfill & \hfill 3\hfill & \hfill 4\hfill & \hfill 40\hfill \\ \hfill F_3\hfill & \hfill 3\hfill & \hfill 5\hfill & \hfill 4\hfill & \hfill 2\hfill & \hfill 4\hfill & \hfill 1\hfill & \hfill 60\hfill \\ \hfill F_4\hfill & \hfill 4\hfill & \hfill 2\hfill & \hfill 2\hfill & \hfill 1\hfill & \hfill 2\hfill & \hfill 2\hfill & \hfill 30\hfill \\ \hfill \mathrm{\text{Demand}}\hfill & \hfill 30\hfill & \hfill 50\hfill & \hfill 20\hfill & \hfill 40\hfill & \hfill 30\hfill & \hfill 10\hfill & \hfill \hfill \end{array}$ (rows: Factories; columns: Destinations) by finding the initial solution by Matrix Minima method. [2008/30m/4c]

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Last updated: 30 October 2026