LPP: Problem formulation

Mathematical Formulations of Linear Programming Models for Product-Mix Problems

Maximization Case

Problem Statement: A firm produces two products, A and B. The production requirements and constraints are as follows:

• Product A:
  • Requires 4 kg of raw material per unit.
  • Requires 6 labor hours per unit.
  • Sells for Rs 35 per unit.
• Product B:
  • Requires 4 kg of raw material per unit.
  • Requires 5 labor hours per unit.
  • Sells for Rs 40 per unit.
• Constraints:
  • Total availability of raw material: 60 kg per week.
  • Total availability of labor hours: 90 hours per week.

Objective: Maximize profit by determining the optimal number of units of products A and B to produce.

Formulation:

• Decision Variables:
  • x1​: Number of units of Product A produced per week.
  • x2​: Number of units of Product B produced per week.
• Objective Function:
  • Maximize profit Z: $Z=35x_1+40x{}_2$
• Constraints:
  • Raw material constraint: $4x_1+4x{}_2\le 60$
  • Labor hours constraint: $6x_1+5x{}_2\le 90$
  • Non-negativity restriction: $x_1,x_2\ge 0$

Thus, the linear programming problem can be summarized as:

Maximize $Z=35x{}_1+40x_2$​

Subject to $4x_1+4x{}_2\le 60$

                 $6x_1+5x{}_2\le 90$

                 $x_1,x_2\ge 0$

Minimization Case

Problem Statement: An agricultural research institute recommends a farmer to use at least 5000 kg of phosphate fertilizer and at least 7000 kg of nitrogen fertilizer. The fertilizers can be obtained from two mixtures, A and B, each weighing 100 kg:

• Mixture A:
  • Contains 40 kg of phosphate.
  • Contains 60 kg of nitrogen.
  • Costs Rs 40 per bag.
• Mixture B:
  • Contains 60 kg of phosphate.
  • Contains 40 kg of nitrogen.
  • Costs Rs 25 per bag.

Objective: Minimize the cost of purchasing the required fertilizers.

Formulation:

• Decision Variables:
  • x1​: Number of bags of Mixture A.
  • x2​: Number of bags of Mixture B.
• Objective Function:
  • Minimize cost Z: $Z=40x_1+25x{}_2$​
• Constraints:
  • Phosphate constraint: $40x{}_1+60x_2\ge 5000$
  • Nitrogen constraint: $60x{}_1+40x_2\ge 7000$
  • Non-negativity restriction: $x_1,x{}_2\ge 0$

Thus, the linear programming problem can be summarized as:

Minimize $Z=40x_1+25x{}_2$

Subject to $40x{}_1+60x_2\ge 5000$

                 $60x{}_1+40x_2\ge 7000$

                 $x_1,x{}_2\ge 0$

Mathematical Formulations of Linear Programming (LP) Models for Product-Mix Problems

Linear Programming (LP) is a powerful mathematical technique used to optimize an objective function subject to constraints. It finds extensive application in production planning, transportation, and finance. Below, we explore LP formulations for both maximization and minimization cases through illustrative examples.

Maximization Case: Product Mix Problem

Problem Statement:

A company produces two products, A and B. Each unit of product A requires 4 kg of raw material and 6 labor hours, while each unit of product B requires 4 kg of raw material and 5 labor hours. The weekly availability of raw material and labor hours is 60 kg and 90 hours, respectively. The unit prices of products A and B are Rs 35 and Rs 40, respectively. The goal is to determine the number of units of each product to produce weekly to maximize profit.

Step-by-Step Solution:

1. Define Decision Variables:

   • $x_1$: Number of units of product A produced per week.
   • $x_2$: Number of units of product B produced per week.

2. Formulate the Objective Function:

   • The objective is to maximize profit.
   • Profit from product A: $35x_1$
   • Profit from product B: $40x_2$
   • Total profit ($Z$): $$Z=35x_1+40x_2$$

3. Formulate the Constraints:

   • Raw Material Constraint: Each unit of A and B requires 4 kg of raw material. $$4x_1+4x_2\le 60$$
   • Labor Hours Constraint: Each unit of A requires 6 labor hours, and each unit of B requires 5 labor hours. $$6x_1+5x_2\le 90$$
   • Non-Negativity Restriction: $$x_1,x_2\ge 0$$

4. Linear Programming Model:

   • Objective Function: $$\text{Maximize }Z=35x_1+40x_2$$

Subject to: $$4x_1+4x_2\le 60\text{ (Raw Material Constraint)}$$ $$6x_1+5x_2\le 90\text{ (Labor Hours Constraint)}$$ $$x_1,x_2\ge 0\text{ (Non-Negativity Restriction)}$$

Explanation:

• The objective function $Z$ represents the total profit from producing $x_1$ units of product A and $x_2$ units of product B.
• The constraints ensure that the total consumption of raw materials and labor hours does not exceed the available limits.
• Non-negativity restrictions ensure that production quantities cannot be negative.

Minimization Case: Agricultural Fertilizer Problem

Problem Statement:

An agricultural research institute recommends that a farmer use at least 5000 kg of phosphate and 7000 kg of nitrogen fertilizers to improve crop productivity. The farmer can choose between two mixtures, A and B, each weighing 100 kg. Mixture A costs Rs 40 per bag and contains 40 kg of phosphate and 60 kg of nitrogen, while Mixture B costs Rs 25 per bag and contains 60 kg of phosphate and 40 kg of nitrogen. The goal is to determine the number of bags of each mixture to buy to meet the fertilizer requirements at minimum cost.

Step-by-Step Solution:

1. Define Decision Variables:

   • $x_1$: Number of bags of mixture A.
   • $x_2$: Number of bags of mixture B.

2. Formulate the Objective Function:

   • The objective is to minimize cost.
   • Cost of mixture A: $40x_1$
   • Cost of mixture B: $25x_2$
   • Total cost ($Z$): $$Z=40x_1+25x_2$$

3. Formulate the Constraints:

   • Phosphate Constraint: Mixture A contains 40 kg of phosphate, and mixture B contains 60 kg of phosphate. $$40x_1+60x_2\ge 5000$$
   • Nitrogen Constraint: Mixture A contains 60 kg of nitrogen, and mixture B contains 40 kg of nitrogen. $$60x_1+40x_2\ge 7000$$
   • Non-Negativity Restriction: $$x_1,x_2\ge 0$$

4. Linear Programming Model:

   • Objective Function: $$\text{Minimize }Z=40x_1+25x_2$$

Subject to: $$40x_1+60x_2\ge 5000\text{ (Phosphate Constraint)}$$ $$60x_1+40x_2\ge 7000\text{ (Nitrogen Constraint)}$$ $$x_1,x_2\ge 0\text{ (Non-Negativity Restriction)}$$

Explanation:

• The objective function $Z$ represents the total cost of purchasing bags of mixtures A and B.
• The constraints ensure that the minimum requirements for phosphate and nitrogen are met.
• Non-negativity restrictions ensure that negative quantities of bags are not feasible.

Key Points and Additional Information

• Objective Function: Represents the goal of the problem, either maximizing profit or minimizing cost. Formulated based on the contribution of each decision variable (e.g., unit price or cost).

• Constraints: Represent limitations or requirements that must be satisfied. Expressed as inequalities to indicate maximum (≤) or minimum (≥) allowable levels.

• Non-Negativity Restriction: Ensures that decision variables (e.g., quantities produced or purchased) are non-negative, which is crucial for practical feasibility.

• Formulation Process:

  1. Define decision variables.
  2. Establish the objective function.
  3. Identify and formulate constraints.
  4. Ensure non-negativity conditions.

• Usage in Business: LP models are used for optimizing resource allocation, production planning, cost minimization, and profit maximization, helping businesses make informed decisions through a mathematical framework.

By following these steps and considerations, businesses can effectively apply linear programming to address various optimization problems and improve decision-making processes.

Summary

Linear Programming (LP) models provide a systematic and efficient approach to solving product-mix problems, enabling optimal allocation of resources. By defining decision variables, objective functions, and constraints, LP helps businesses maximize profits or minimize costs while adhering to resource limitations. The non-negativity condition ensures that solutions are realistic and practical.