--- title: "Duality LPP min:Z=21x+50y" subject: "Mathematics III" chapter: 5 key: "semester_3/mathematics_iii:duality-lpp-min-z-21x-50y" source: https://www.bctengineer.com/q/semester_3/mathematics_iii/duality-lpp-min-z-21x-50y site: BCT Engineering --- # Duality LPP min:Z=21x+50y Mathematics III · Chapter 5: Linear Programming - Asked in: 2081 Bhadra, 7 marks; 2078 Bhadra, 8 marks; 2071 Shrawan, 10 marks; 2068 Chaitra, 7.5 marks (asked 4 times) - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_3/mathematics_iii/duality-lpp-min-z-21x-50y - Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-5 ## Question Solve the following linear programming problem by constructing the dual: Minimize Z = 21x + 50y subject to 2x + 5y ≥ 12, 3x + 7y ≥ 17 and x, y ≥ 0. ## Short answer The dual problem is max 12y1+17y2 subject to 2y1+3y2≤21, 5y1+7y2≤50. Solving by simplex yields y1=3, y2=5, max W=121. Primal variables read from slack Cj-Zj row: x=1, y=2, min Z=121. ## Answer **Step 1: Write the dual.** For the primal (minimisation, all constraints ≥), the dual is: ```text Primal: Minimize Z = 21x + 50y subject to 2x + 5y ≥ 12 3x + 7y ≥ 17 x, y ≥ 0 Dual: Maximize W = 12y₁ + 17y₂ subject to 2y₁ + 3y₂ ≤ 21 5y₁ + 7y₂ ≤ 50 y₁, y₂ ≥ 0 ``` Here y₁, y₂ are the dual variables, one for each primal constraint. **Step 2: Standard form of the dual.** Add slack variables s₁, s₂ (one per dual constraint). Since all right sides are positive, the slack variables give the starting basis. ```text Maximize W = 12y₁ + 17y₂ subject to 2y₁ + 3y₂ + s₁ = 21 5y₁ + 7y₂ + s₂ = 50 ``` **Step 3: Simplex tables for the dual** **Table 1** ```text Cj 12 17 0 0 Basis CB y₁ y₂ s₁ s₂ b Ratio --------------------------------- s₁ 0 2 3 1 0 21 7 s₂ 0 5 7 0 1 50 50/7 --------------------------------- Zj 0 0 0 0 0 Cj-Zj 12 17 0 0 Entering: y₂ Leaving: s₁ Pivot = 3 ``` **Table 2** ```text Cj 12 17 0 0 Basis CB y₁ y₂ s₁ s₂ b Ratio --------------------------------------- y₂ 17 2/3 1 1/3 0 7 21/2 s₂ 0 1/3 0 -7/3 1 1 3 --------------------------------------- Zj 34/3 17 17/3 0 119 Cj-Zj 2/3 0 -17/3 0 Entering: y₁ Leaving: s₂ Pivot = 1/3 ``` **Table 3** ```text Cj 12 17 0 0 Basis CB y₁ y₂ s₁ s₂ b ---------------------------- y₂ 17 0 1 5 -2 5 y₁ 12 1 0 -7 3 3 ---------------------------- Zj 12 17 1 2 121 Cj-Zj 0 0 -1 -2 ``` **Step 4: Read the answer.** All Cj − Zj ≤ 0, so the dual is solved: y₁ = 3, y₂ = 5, Max W = 121. By the duality theorem, Min Z = Max W. The values of the primal variables are the numbers (without sign) in the Cj − Zj row under the slack columns: x = 1 (under s₁), y = 2 (under s₂). Check: Z = 21(1) + 50(2) = 121. **Answer:** x = 1, y = 2, Minimum Z = 121. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. The dual of a min problem with >= constraints is? A. Minimise W with >= constraints B. Maximise W with <= constraints C. Minimise W with <= constraints D. Maximise W with >= constraints Answer: B. The dual flips objective and constraint direction. (Key point: Dual form) 2. What are the dual objective and constraint right sides? A. W = 2y1 + 5y2 B. W = 12y1 + 17y2; right sides 12 and 17 C. W = 21y1 + 50y2; right sides 12 and 17 D. W = 12y1 + 17y2; right sides 21 and 50 Answer: D. Primal RHS become dual costs and primal costs become dual RHS. (Key point: Dual objective) 3. What are the dual optimal values? A. y1 = 3, y2 = 5, W = 121 B. y1 = 5, y2 = 3, W = 121 C. y1 = 3, y2 = 5, W = 100 D. y1 = 7, y2 = 0, W = 84 Answer: A. 12(3) + 17(5) = 121. (Key point: Dual optimum) 4. How are the primal x, y found? A. From Cj - Zj entries under the slack columns, without sign B. By solving the primal graphically C. From the b column D. From the Zj row of y columns Answer: A. The values are 1 and 2, giving Z = 21 + 100 = 121. (Key point: Reading primal solution) ## Close variants set in other exams - [Duality LPP min:Z=8x+9y](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Aduality-lpp-min-z-8x-9y) — Mathematics III, chapter 5, asked 6× ## Related in Mathematics III - [Duality LPP min:Z=20x+50y](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Aduality-lpp-min-z-20x-50y) — Mathematics III, chapter 5 - [Duality LPP min:Z=3x₁+2x₂](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Aduality-lpp-min-z-3x-2x) — Mathematics III, chapter 5, asked 3× - [Duality LPP min:Z=3x₁+2x₂](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Aduality-lpp-min-z-3x-2x-two) — Mathematics III, chapter 5, asked 3× - [Duality LPP min:Z=8x₁+9x₂](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Aduality-lpp-min-z-8x-9x) — Mathematics III, chapter 5, asked 4× - [Duality LPP min:C=16x₁+45x₂](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Aduality-lpp-min-c-16x-45x) — Mathematics III, chapter 5 - [Linear programming problem](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Alinear-programming-problem) — Mathematics III, chapter 5