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# The company has a factory and several warehouses located in different cities.

INSTRUCTIONS TO CANDIDATES

Ques.) The company has a factory and several warehouses located in different cities. The company wants to determine the optimal allocation of its trucks to transport products from the factory to the warehouses. The demand for each product is normally distributed with a mean of 10,000 units per day and a standard deviation of 2,500 units per day. The company has 60 trucks available to allocate to the warehouses.

The objective is to minimize the total cost of transportation, which is the sum of the fixed cost and the variable cost for each truck, as well as the cost of fuel and driver salaries based on distance and travel time.

Solution) In today's globalized and competitive market, efficient logistics management has become crucial for the success of any business. Effective management of logistics involves the optimization of resources, including transportation, storage, and distribution, to ensure that products are delivered to customers in a timely and cost-effective manner. One important aspect of logistics management is the allocation of transportation resources to various destinations. This is particularly important in the case of companies that have multiple warehouses located in different parts of the country or even in different parts of the world.

Consider a company that has a factory and several warehouses located in different cities. The company wants to determine the optimal allocation of its trucks to transport products from the factory to the warehouses. The demand for each product is normally distributed with a mean of 10,000 units per day and a standard deviation of 2,500 units per day. The company has 60 trucks available to allocate to the warehouses.

The objective of the problem is to minimize the total cost of transportation, which is the sum of the fixed cost and the variable cost for each truck, as well as the cost of fuel and driver salaries based on distance and travel time.

To solve this problem, we will use linear programming, a mathematical optimization technique that is widely used in logistics management. Linear programming is a method used to find the best outcome in a mathematical model whose requirements are represented by linear relationships. In other words, it is used to find the optimal solution to a problem where we want to minimize or maximize a linear objective function, subject to a set of linear constraints.

In the next section, we will discuss the mathematical model for the problem, including the objective function and constraints. We will also discuss how to solve the problem using Excel Solver, a powerful optimization tool that is widely used in logistics management.

There are several constraints that must be satisfied:

1. Demand Constraint: The total demand for each product must be met by the assigned trucks, which can be expressed as:

∑i=17xij=dj, for j∈{A,B,C,D,E,F,G}

where \$d_j\$ is the mean demand for each product in city \$j\$.

2. Capacity Constraint: The total capacity of the assigned trucks must not exceed the total demand, which can be expressed as:

∑j=AGxij≤ci, for i∈{1,2,3,4,5,6,7}

where \$c_i\$ is the truck capacity for warehouse \$i\$.

3. Non-negativity Constraint: The number of assigned trucks must be non-negative, which can be expressed as:

xij≥0, for i∈{1,2,3,4,5,6,7} and j∈{A,B,C,D,E,F,G}

4. Integer Constraint: The number of assigned trucks must be integer, which can be expressed as:

xij∈Z, for i∈{1,2,3,4,5,6,7} and j∈{A,B,C,D,E,F,G}

where \$\mathbb{Z}\$ is the set of integer numbers.

The objective is to minimize the total cost of transportation, which is the sum of the fixed cost and the variable cost for each truck, as well as the cost of fuel and driver salaries based on distance and travel time. We can express this objective function as:

minimize Z = ∑(i=1)^7 (f_i * x_i) + ∑(i=1)^7 ∑(j=1)^7 (d_ij * c * q_ij * x_i)

where f_i is the fixed cost per truck per day for warehouse i, d_ij is the distance between warehouse i and warehouse j, c is the cost per unit distance traveled by the truck, q_ij is the quantity of products transported from warehouse i to warehouse j, and x_i is the number of trucks assigned to warehouse i.

The objective function has two parts: the fixed cost and the variable cost. The fixed cost is the cost of maintaining the trucks and does not depend on the amount of products transported. The variable cost is the cost of fuel and driver salaries, which depends on the distance traveled and the quantity of products transported.

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