Mathematical formulation of Linear Programming Problem

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Discussion Overview

The discussion revolves around formulating a linear programming (LP) problem related to cargo distribution on a ship. Participants are exploring how to maximize profit while adhering to capacity constraints for different cargo loads (forward, center, and after) and ensuring the distribution maintains the ship's trim. The conversation includes aspects of decision variables, objective functions, and constraints.

Discussion Character

  • Homework-related
  • Mathematical reasoning
  • Technical explanation

Main Points Raised

  • One participant presents the problem, detailing the cargo loads and their respective capacity limits, as well as the offered cargoes with weights, volumes, and profits.
  • Another participant requests to see the original poster's attempts to identify where they are struggling, suggesting a collaborative approach to problem-solving.
  • Multiple participants outline the steps involved in formulating an LP problem, emphasizing the identification of decision variables, the objective function, and constraints.
  • A participant suggests defining decision variables for each cargo type and load position, indicating that there are more decisions to consider than initially presented.
  • Constraints are discussed, including a proposed constraint for the total weight of cargo in the forward position, along with non-negativity conditions for the decision variables.

Areas of Agreement / Disagreement

There is no consensus yet, as participants are still in the process of formulating the problem and discussing various aspects of the LP formulation. Some participants agree on the steps to take, while others introduce additional considerations regarding decision variables and constraints.

Contextual Notes

Participants have not yet resolved the complete set of constraints or the full formulation of the LP problem, and there may be dependencies on how the cargo distribution is defined relative to the ship's trim.

Suvadip
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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.

Please help
 
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Can you show us what you have tried so that our helpers know where you are stuck and how best to offer help?
 
Hi suvadip!

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?
 
I like Serena said:
Hi suvadip!

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.
 
suvadip said:
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:
$$x_{AF} \ge 0$$
$$x_{BF} \ge 0$$
$$x_{CF} \ge 0$$
 

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