DSC3702 Assignment 2 Semester 2 |COMPLETE WORKED SOLUTIONS| 2025

DSC3702
Assignment 2

Semester 2

2025

, Question 1 — NOPCo paper production (LP + Sensitivity)
(1.1) LP formulation and optimal solution

Decision variables Let

• 𝑥1 = tonnes/day of computer paper,

• 𝑥2 = tonnes/day of writing paper,

• 𝑥3 = tonnes/day of brown paper.

Objective (maximize net profit)

max𝑧 = 4200𝑥 1 + 7800𝑥2 + 3120𝑥3

Constraints (daily limits) Labour: 4𝑥1 + 6𝑥2 + 3𝑥3 ≤ 120 Woodchips: 2𝑥1 + 2𝑥2 +
1.8𝑥3 ≤ 60 Chlorine: 1𝑥1 + 3𝑥2 + 0.2𝑥3 ≤ 47 Nonnegativity: 𝑥1 , 𝑥2 , 𝑥3 ≥ 0.

Solution (by corner-point evaluation / simplex): The optimal solution is:

• 𝑥1 = 0, 𝑥 2 = 15, 𝑥 3 = 10

• Optimal profit 𝑧 ∗ = 4200(0) + 7800(15) + 3120(10) = 𝐑 𝟏𝟒𝟖 𝟐𝟎𝟎 .

Binding constraints at optimum:

• Labour: 6(15) + 3(10) = 120 → binding

• Chlorine: 3(15) + 0.2(10) = 47 → binding

• Woodchips used = 2(15) + 1.8(10) = 48 ≤ 60 → slack = 12 (non-binding)

Dual (shadow prices) at optimum: Let (𝑦1 , 𝑦2 , 𝑦3 ) correspond to (Labour, Woodchips,
Chlorine). Solving 𝐴⊤𝐵 𝑦 = 𝑐 𝐵 for the basic variables 𝑥2 , 𝑥3 gives:

• 𝑦1 = 𝐑 𝟏𝟎𝟎𝟎 per unit (person-day) of labour

• 𝑦2 = 𝐑 𝟎 per tonne of woodchips

• 𝑦3 = 𝐑 𝟔𝟎𝟎 per kg of chlorine