This summary outlines the process of delivering a package from one destination to another, focusing on key logistics and mathematical principles. The process begins with route optimization to determine the most efficient path, followed by the selection of an appropriate transportation mode based on cost and speed. The total delivery time is calculated, factoring in travel time, handling, and processing. Cost optimization is achieved using models like the Economic Order Quantity (EOQ) to minimize expenses. Queueing theory is applied to manage potential delays at various stages, and the overall success probability is assessed, considering risks of delays or failures.
Below is a breakdown of the process along with the mathematical aspects involved:
Concepts
Route Optimization
 Objective: Find the most efficient route from A to B, minimizing cost, time, or distance.
 Mathematical Model:
 Dijkstra’s Algorithm: Used to find the shortest path between two points in a graph, where nodes represent locations and edges represent routes with associated costs (e.g., distance, time).

Formula:
\[\text{Cost}(A, B) = \min \left( \sum_{i=1}^{n1} d_i \right)\]where \(d_i\) is the distance between consecutive points on the route from A to B.
Transportation Mode Selection
 Objective: Choose the mode of transport (e.g., air, road, rail, sea) that balances speed and cost.
 Mathematical Model:

Cost Function:
\[C = C_{\text{trans}} + C_{\text{handling}} + C_{\text{time}}\]where:
 \(C_{\text{trans}}\) is the transportation cost,
 \(C_{\text{handling}}\) is the cost associated with loading/unloading,
 \(C_{\text{time}}\) is the timerelated cost (e.g., penalties for delays).

Time Calculation
 Objective: Estimate the time taken to deliver the package.
 Mathematical Model:

Time Calculation Formula:
\[T = \frac{D}{v} + T_{\text{processing}} + T_{\text{loading/unloading}} + T_{\text{delays}}\]where:
 \(D\) is the total distance,
 \(v\) is the average speed of the transport,
 \(T_{\text{processing}}\) is the time for processing (sorting, customs),
 \(T_{\text{loading/unloading}}\) is the time spent on loading and unloading the package,
 \(T_{\text{delays}}\) accounts for any unexpected delays.

Inventory and Warehousing
 Objective: Manage storage if the package is held in transit.
 Mathematical Model:
 EOQ (Economic Order Quantity) Model: To minimize total inventory costs.

Formula:
\[EOQ = \sqrt{\frac{2DS}{H}}\]where:
 \(D\) is the demand rate,
 \(S\) is the order cost,
 \(H\) is the holding cost per unit.
Queueing Theory for Handling Delays
 Objective: Model delays at sorting centers, hubs, or delivery stations.
 Mathematical Model:
 M/M/1 Queue Model: Assumes a single queue with Poisson arrivals and exponential service times.

Formula:
\[L_q = \frac{\lambda^2}{\mu(\mu  \lambda)}\]where:
 \(L_q\) is the average number of packages in the queue,
 \(\lambda\) is the arrival rate,
 \(\mu\) is the service rate.
Cost Optimization
 Objective: Minimize the total cost of delivery while meeting service requirements.
 Mathematical Model:
 Linear Programming: To optimize costs subject to constraints (e.g., budget, time).

Objective Function:
\[\text{Minimize } Z = \sum_{i=1}^n c_ix_i\]
subject to:
\[\sum_{j=1}^m a_{ij}x_j \leq b_i, \quad x_j \geq 0\]where:
 \(Z\) is the total cost,
 \(c_i\) is the cost coefficient,
 \(x_i\) is the decision variable,
 \(a_{ij}\) are the coefficients in the constraint equations,
 \(b_i\) are the constraint bounds.
Final Delivery
 Objective: Ensure the package reaches destination B.
 Mathematical Consideration:
 Success Rate Model: Estimating the probability that a package is delivered successfully.

Formula:
\[P(\text{success}) = 1  \sum P(\text{failure at step } i)\]This involves considering all potential failure points (loss, damage, misrouting).
Summary of the Process:
 Route is optimized to minimize travel time and cost.
 Transport mode is selected based on cost and speed considerations.
 Time is calculated for the delivery considering various factors.
 Inventory management is handled if the package is held temporarily.
 Queueing models are used to manage potential delays.
 Cost optimization ensures the delivery process is economically viable.
 Final delivery involves assessing and ensuring the package reaches its destination safely.
These mathematical models help in making informed decisions to ensure efficient and costeffective delivery of packages from A to B.
Example
Let’s walk through an example of delivering a package from destination A (say, New York City) to destination B (say, Los Angeles). We’ll apply the various mathematical models and concepts discussed to demonstrate the process.
Scenario Details:
 Distance (D): 4,500 km
 Average Speed (v): 80 km/h by truck
 Handling Time: 2 hours at both A and B
 Cost Per km: $1.5
 Loading/Unloading Cost: $50
 Processing Time: 5 hours at a central hub
 Probability of Delay: 10%
 Holding Cost per Day: $2 per package
 Demand Rate (D): 500 packages/day
 Order Cost (S): $100
 Service Rate (μ): 10 packages/hour
 Arrival Rate (λ): 8 packages/hour
Route Optimization
We use Dijkstra’s algorithm or other shortestpath algorithms to find the most efficient route from New York City to Los Angeles.
Given it’s a straight route on highways, the shortest path is simply the direct road distance of 4,500 km.
Transportation Mode Selection
We select truck transport because it’s costeffective for this distance. Now, let’s calculate the cost:

Transportation Cost (\(C_{trans}\)):
\[C_{\text{trans}} = \text{Distance} \times \text{Cost per km} = 4500 \times 1.5 = \$6750\] 
Handling Costs (\(C_{handling}\)):
\[C_{\text{handling}} = 50 (\text{at A}) + 50 (\text{at B}) = \$100\] 
Total Cost (C):
\[C = C_{\text{trans}} + C_{\text{handling}} = 6750 + 100 = \$6850\]
Time Calculation
To calculate the total time taken:

Travel Time (\(T_{travel}\)):
\[T_{\text{travel}} = \frac{D}{v} = \frac{4500 \text{ km}}{80 \text{ km/h}} = 56.25 \text{ hours}\] 
Processing Time (\(T_{processing}\)):
\[T_{\text{processing}} = 5 \text{ hours}\] 
Loading/Unloading Time (\(T_{loading/unloading}\)):
\[T_{\text{loading/unloading}} = 2 (\text{at A}) + 2 (\text{at B}) = 4 \text{ hours}\] 
Total Time (\(T\)):
\[T = T_{\text{travel}} + T_{\text{processing}} + T_{\text{loading/unloading}} = 56.25 + 5 + 4 = 65.25 \text{ hours} \approx 2.72 \text{ days}\]
Inventory and Warehousing
If the package is held at a warehouse for a day:

Holding Cost:
\[\text{Holding Cost per day} = \$2\]For 2.72 days:
\[\text{Total Holding Cost} = 2.72 \times 2 = \$5.44\]
Queueing Theory for Handling Delays
Considering delays at the central hub using the M/M/1 queue model:

Average Number of Packages in Queue (\(L_q\)):
\[L_q = \frac{\lambda^2}{\mu(\mu  \lambda)} = \frac{8^2}{10(10  8)} = \frac{64}{20} = 3.2 \text{ packages}\] 
Expected Waiting Time in Queue:
\[W_q = \frac{L_q}{\lambda} = \frac{3.2}{8} = 0.4 \text{ hours}\]
Cost Optimization
Using Economic Order Quantity (EOQ) to determine the optimal batch size for deliveries:

EOQ Calculation:
\[EOQ = \sqrt{\frac{2DS}{H}} = \sqrt{\frac{2 \times 500 \times 100}{2}} = \sqrt{50000} = 223.6 \text{ packages}\]
Thus, ordering around 224 packages at a time minimizes the total cost.
Final Delivery Success Probability
If there’s a 10% probability of delay at each step, and there are 3 major steps (transport, processing, delivery):

Probability of Delay:
\[P(\text{success}) = 1  \left( \text{Probability of Delay in Transport} + \text{Processing} + \text{Delivery} \right) = 1  0.3 = 0.7\]
Thus, there’s a 70% probability the package will be delivered without delay.
Summary of the Process
 Route Optimization determined a 4,500 km direct route.
 Transportation Cost using a truck was calculated at $6,850.
 Total Time was calculated to be approximately 2.72 days.
 Holding Cost was estimated at $5.44.
 Queueing Theory showed an average of 3.2 packages waiting in line at the hub.
 EOQ Model suggested ordering around 224 packages minimizes costs.
 Final Probability indicated a 70% chance of ontime delivery.
These calculations give a comprehensive understanding of the logistics, cost, and time associated with delivering a package from New York City (A) to Los Angeles (B).