# Mathematical formulation of Linear Programming Problem

##### Member
A ship has three cargo loads -forward, centre and after. The capacity limits are given:

Commodity Weight (in tonne) Volume (in cu. feet)

Forward 2000 100000
Centre 3000 135000
After 1500 30000

The following cargoes are offered. The ship owner may accept all or any part of each commodity:

Commodity Weight (in tonne) Volume (in cu. feet) Profit per tonne (in Rs)

A 6000 60 150
B 4000 50 200
C 2000 25 125

In order to preserve the trim of the ship, the weight in each load must be proportional to the capacity in tonne. The cargo is to be distributed
so as to maximize the profit. Formulate the problem as LPP model.

#### MarkFL

Staff member
Can you show us what you have tried so that our helpers know where you are stuck and how best to offer help?

#### Klaas van Aarsen

##### MHB Seeker
Staff member

An LP problem consists of 3 steps:
1. Identify the decision variables.
2. Identify the target function in terms of the decision variables.
3. Identify the constraints.

How far do you get?

##### Member

An LP problem consists of 3 steps:
1. Identify the decision variables.
2. Identify the target function in terms of the decision variables.
3. Identify the constraints.

How far do you get?
Let x1 tonne of A, x2 tonne of B and x3 tonne of C

Objective function: Max Z=150 x1 +200 x2+125 x3

Constraints:

Non-negativity conditions: x1, x2, x3>=0

Please give me hints about a single constraint. Rest I can do the rest.

#### Klaas van Aarsen

##### MHB Seeker
Staff member
Let x1 tonne of A, x2 tonne of B and x3 tonne of C

Objective function: Max Z=150 x1 +200 x2+125 x3

Constraints:

Non-negativity conditions: x1, x2, x3>=0

Please give me hints about a single constraint. Rest I can do the rest.
I'm afraid that you have more decisions to make: whether cargo should go forward, center, or aft.

Let $x_{AF}$ be the tonne of A that goes Forward, $x_{BF}$ the tonne of B that goes Forward, and $x_{CF}$ the tonne of C that goes Forward.
In total you will have 9 decision variables.

Then the first constraint is that:
$$x_{AF} + x_{BF} + x_{CF} \le 2000$$

Extra constraints are the non-negativity constraints.
For these 3 decision variables, those are:
$$x_{AF} \ge 0$$
$$x_{BF} \ge 0$$
$$x_{CF} \ge 0$$