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Business Mathematics and Statistics · Class 11 Commerce

Ch 10Operations Research (Linear Programming Problem, Network Analysis) — Class 11 Business Mathematics and Statistics, concept-first.

Every earlier chapter of this course dealt with a single formula applied to a fixed set of numbers. Real management decisions are rarely that simple — a factory manager deciding how much of two products to make is constrained by limited machine hours, limited labour, and limited raw material all at once, and must find…

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Key concepts

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Solving an LPP by the Graphical (Corner-Point) Method

The Corner-Point Theorem states that on a bounded feasible region, the objective function's maximum and minimum both occur at one of the region's corner points, so only these (found by solving pairs of boundary equations…

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Chapter contents

The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.

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Introduction to Operations Research and Linear Programming

Every earlier chapter of this course dealt with a single formula applied to a fixed set of numbers. Real management decisions are rarely that simple — a factory manager deciding how much of two produc…

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Graphing Linear Inequalities and the Feasible Region

Solving an LPP graphically first requires plotting each constraint as a region on a graph, exactly as a linear inequality in two variables is graphed.

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Solving an LPP by the Graphical (Corner-Point) Method

Once the feasible region is graphed, the graphical (corner-point) method finds the optimal value of the objective function using one remarkable fact, stated here without proof, since this syllabus's o…

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Network Analysis — Diagram, Terms and Construction Rules

The second technique of this chapter, Network Analysis, plans and schedules a project made up of many separate but interdependent activities — the tasks involved in constructing a building, launching…

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Critical Path Method (CPM)

Once a project's network diagram is built and each activity's duration is known, the Critical Path Method (CPM) finds the shortest possible time in which the whole project can be completed, and identi…

Exercises

+Show 5 questions5 questions
  1. Q5A toy company manufactures two types of toys, A and B. Each toy A needs 2 hours of cutting and 1 hour of assembling; each toy B needs 1 hour…Free
  2. Q6A farmer wants his crop to receive at least 10 units of nitrogen and at least 15 units of phosphorus. Fertiliser $P$ costs ₹4 per bag and su…Free
  3. Q7A project has activities A, B, C, D, E with the following predecessors and durations: A (—, 3 days), B (—, 4 days), C (A, 2 days), D (B, 6 d…Preview
  4. Q8Graph the feasible region of the system of linear inequalities $x+y\le5$, $x\ge0$, $y\ge0$, and identify its corner points. If $Z=3x+5y$ is…Preview
  5. Q9In a project network, activities P and Q both start at event 1 and both end at event 4; P takes 3 days and Q takes 5 days. A third activity,…Preview

Sample & Board Papers

Sample papers and previous-year board questions for this subject.

+Show 17 questions17 questions
  1. Q1The longest path connected by the activities in the network is called : (a) Solution (b) Longest duration (c) Feasible region (d) Critical p…Preview
  2. Q2One of the conditions for the activity $(i, j)$ to lie on the critical path is : (a) $E_j - E_i = L_j - L_i \neq t_{ij}$ (b) $E_j - E_i = L_…Preview
  3. Q3Draw the event oriented network for the following data : | Events | 1 | 2 | 3 | 4 | 5 | 6 | 7 | | --- | --- | --- | --- | --- | --- | --- |…Preview
  4. Q4One of the conditions for the activity $(i,\ j)$ to lie on the critical path is : (a) $E_j - E_i = L_i - L_j = t_{ij}$ (b) $E_j - E_i = L_j…Preview
  5. Q5Draw the logic network for the following : Activities C and D both follow A, activity E follows C, activity F follows D, activity E and F pr…Preview
  6. Q6The objective of network analysis is to : (a) Minimize the total project duration (b) Minimize the production delays, interruption and confl…Preview
  7. Q7The maximum value of the objective function $Z = 3x + 5y$ subject to the constraints $2x + 5y \le 10$, $x \ge 0$, $y \ge 0$ is : (a) 25 (b)…Preview
  8. Q8Develop a network based on the following information. | Activity | A | B | C | D | E | F | G | H | | --- | --- | --- | --- | --- | --- | ---…Preview
  9. Q9A furniture dealer deals only in two items viz., tables and chairs. He has to invest ₹ 10,000 and a space to store atmost 60 pieces. The cos…Preview
  10. Q10(a) A project has the following time schedule. | Activity | 1-2 | 1-6 | 2-3 | 2-4 | 3-5 | 4-5 | 6-7 | 5-8 | 7-8 | | --- | --- | --- | --- |…Preview
  11. Q11One of the conditions for the activity $(i,j)$ to lie on the *critical* path is : (a) $E_j-E_i=L_i-L_j=t_{ij}$ (b) $E_j-E_i=L_j-L_i=t_{ij}$…Preview
  12. Q12Draw the event oriented network for the following data. | Events | 1 | 2 | 3 | 4 | 5 | 6 | 7 | | --- | --- | --- | --- | --- | --- | --- | -…Preview
  13. Q13Solve the following linear programming problems by graphical method. Maximize $z=40x_1+50x_2$ subject to constraints $3x_1+x_2\le 9$; $x_1+2…Preview
  14. Q14(a) A project schedule has the following characteristics. | Activity | 1-2 | 1-3 | 2-4 | 3-4 | 3-5 | 4-9 | 5-6 | 5-7 | 6-8 | 7-8 | 8-10 | 9-…Preview
  15. Q15The maximum value of the objective function $Z=3x+5y$ subject to the constraints $x\geq 0$, $y\geq 0$ and $2x+5y\leq 10$ is : (a) $25$ (b) $…Preview
  16. Q16Construct the network for the projects consisting of various activities and their precedence relationships are as given below : A, B, C can…Preview
  17. Q17(a) Solve the following LPP Maximize $Z=2x_1+5x_2$ subject to the conditions $x_1+4x_2\leq 24$, $3x_1+x_2\leq 21$, $x_1+x_2\leq 9$ and $x_1,…Preview

More questions

+Show 4 questions4 questions
  1. Example 1A furniture company manufactures tables and chairs. Each table needs 2 hours of carpentry and 1 hour of finishing; each chair needs 1 hour o…Free
  2. Example 2Solve the linear programming problem formulated in Worked Example 1 by the graphical (corner-point) method, and state the maximum weekly pro…Free
  3. Example 3A dietician wants to prepare a diet providing at least 8 units of vitamin A and at least 10 units of vitamin B daily. Food $F_1$ costs ₹6 pe…Preview
  4. Example 4A small project has the following activities, with their durations (in days) and immediate predecessors: | Activity | Predecessor(s) | Durat…Preview