Question

The commanding officer (CO) plans to move a part of his battalion to
another location. There are two types of vehicle available to him, a Ford
vehicle that can carry 25 m3 plus 5 personnel, and a Holden vehicle that
can carry 15 m3 plus 10 personnel. For security reason, all vehicles must
move together and they will be used for a single trip. The required
materials to be taken to the new location are organized in unit pallet
load of 2 m3. The CO requires transporting of a total of 60 pallet loads
and 40 personnel. There are a maximum of 30 Ford and 40 Holden
vehicles available. Each Ford vehicle is estimated to use 50 L of fuel per
trip, whereas the Holden vehicle will only use 30 L. If the CO wants to
move all the required materials and personnel at minimum fuel use,
what mix of Ford and Holden vehicles should the CO choose? Formulate
the problem as an LP model.

Answer 1

Suppose we need f number of ford vehicles and h number of Holden vehicles

Now ford uses 50L fuel and holden uses 30L fuel

so our objective function is min z = 50f + 30h

where f,h 0 as number of vehicles cannot be negative

Now according to question

f 30 and h 40

but CO requires 60 pallet loads and 40 personnel

and we know that 1 pallet = 2 m3

so 60 pallet = 120 m3

so constraints are

25f + 15h 120 (required number of m3)

5f + 10h 40 (required number of personnel)

The LPP thus formulated:

Minimization of fuel, z = 50f + 30h, subject to the constraints:

25f + 15h 120,
5f + 10h 40,

f 30,

h 40,

where f,h 0