Understanding the Problem

To develop a linear programming model for minimizing delivery time, we need to consider the following factors:

• Delivery Locations: The specific locations where deliveries need to be made.
• Travel Time: The time taken to travel between different locations.
• Delivery Time: The time required to complete the delivery at each location.
• Vehicle Capacity: The maximum number of deliveries a vehicle can make in one trip.
• Vehicle Speed: The average speed of the delivery vehicle.

Defining Decision Variables

Let’s define the decision variables as follows:

• xij: A binary variable that equals 1 if the delivery to location j is made after location i, and 0 otherwise.

Objective Function

The objective is to minimize the total delivery time, which can be expressed as:

Minimize Z = Σ Σ (tij * xij)

Where:

• Z is the total delivery time
• tij is the travel time between locations i and j

Constraints

1. Each location must be visited exactly once:

   Σ xij = 1 for all j
   Σ xji = 1 for all i
   

2. Vehicle Capacity Constraints: Ensure that the number of deliveries in a route does not exceed the vehicle capacity.
3. Subtour Elimination Constraints: To prevent suboptimal solutions where the vehicle returns to the starting point before completing all deliveries, we can use subtour elimination constraints. These constraints ensure that a complete tour is formed.

Solving the Model

This linear programming model can be solved using various optimization techniques, such as the simplex method or specialized algorithms for vehicle routing problems. Software tools like Excel Solver, MATLAB, or commercial optimization software can be used to efficiently solve the model.

Example: A Simple Delivery Scenario

Suppose we have a delivery person who needs to deliver packages to three locations: A, B, and C. The travel times between the locations are as follows:

The objective is to minimize the total travel time.

Decision Variables:

• xAB: 1 if B is visited after A, 0 otherwise
• xAC: 1 if C is visited after A, 0 otherwise
• xBA: 1 if A is visited after B, 0 otherwise
• xBC: 1 if C is visited after B, 0 otherwise
• xCA: 1 if A is visited after C, 0 otherwise
• xCB: 1 if B is visited after C, 0 otherwise

Objective Function: Minimize Z = 10xAB + 20xAC + 10xBA + 15xBC + 20xCA + 15xCB

Constraints:

• xAB + xAC = 1
• xBA + xBC = 1
• xCA + xCB = 1
• xAB + xBA = 1
• xAC + xCA = 1
• xBC + xCB = 1

By solving this linear program, we can determine the optimal delivery route that minimizes the total travel time.