Question

A coal mining company examines the option of buying two types of dumpers with the following details.

In order to minimize the operating cost, the optimum fleet of dumpers of Type-1 and Type-2, respectively are

• A. 20,15
• B. 0,30
• C. 0,31
• D. 40,0

Hint:

When costs per tonne differ, prioritize the dumper with lower operating cost as long as capital and tonnage constraints are satisfied.

Answer 1

The Correct Option is A

Let the number of Type-1 dumpers be $x$ and Type-2 dumpers be $y$.
1. Capital cost constraint:
\[ 3x + 4y \le 120 \] 2. Daily tonnage requirement:
Daily tonnage hauled by one dumper:
Type-1: $40 \times 20 = 800$ tonnes/day
Type-2: $50 \times 20 = 1000$ tonnes/day
Thus,
\[ 800x + 1000y \ge 31000 \] 3. Objective: Minimize operating cost
Cost per tonne:
Type-1 = 300 ₹/tonne, Type-2 = 200 ₹/tonne.
Therefore, to minimize operating cost, use more Type-2 dumpers since they are cheaper per tonne.
Check feasible combinations from options:
Option (A): $x=20,y=15$
Capital cost:
\[ 3(20) + 4(15) = 60 + 60 = 120 \quad \checkmark \] Daily capacity:
\[ 800(20) + 1000(15) = 16000 + 15000 = 31000 \quad \checkmark \] Option (B): $x=0,y=30$
Capital cost = 120 (OK)
Daily tonnage = $30000$ (NOT enough) ✗
Option (C): $x=0,y=31$
Capital cost = $124$ (exceeds limit) ✗
Option (D): $x=40,y=0$
Capital cost = $120$ (OK)
Daily tonnage = $32000$ (OK)
But operating cost is much higher because Type-1 is more expensive. ✗
Thus, the minimum-cost feasible combination is:
\[ \boxed{x=20,y=15} \] Final Answer: 20,15