Assignment:
Your team is asked to analyze the finished product logistics for ACME which produces 3 different electronic car components (A, B, and C). Component A, B and C has a volume of 5m^3 10m^3, and 20m^3, respectively. ACME receives orders for its products and finishes the production of each order in X days. Orders arrive according to a Poisson process with an arrival rate of 2 orders per day. Each order contains a single component and the content of the order has a probability mass function P(x) where P(A) = 0.5. P(B) = 0.4, P(C) = 0.1. The production capacity of the firm is 3 components per day. If the company receives more than 3 orders in a day, the excess order is referred to another manufacturer. ACME is not responsible for the production of the excess demand. The company currently uses a third-party logistics (3-PL) provider to distribute its finished products to its customers, hi this system, whenever the 3-PL ships the products it charges a standard per m^3 transportation fee to the manufacturer. Previous analysis illustrated that the company considers "a shipment in every 4 business days"' to be optimal based on the minimization of the expected average transportation cost. The transportation cost is m^3), and inventory holding cost (Sc per day per m^3). The firm considers another logistics plan which allows them renting a truck for each shipment. Truck rental costs are $K_1 and $K_2 per shipment for 45m^3 and 100m^3 trucks, respectively. Your team aims to understand whether the new truck-for-rent-based logistics option can be more efficient than the existing 3-PL option. For this purpose:
Q: Assume that X = 0 day (production time is negligible) and the ACME's responsibility for its products ends when they are shipped. Build a Markov Chain model that can keep track of the production of the components and their transportation.