PROBLEM: Green Cross-docking

Suppose that you are an Operations Manager at a cross-docking facility, where trucks from different origins come for unloading, and loaded trucks leave for different destinations. In particular, you are responsible for managing the Inbound Operations, that is, unloading of the incoming trucks to the facility.

Currently, there is one dock assigned for unloading the incoming trucks. At this dock, there is a forklift that completes the unloading operations. Since different incoming trucks have different sizes and different loads, you observed that it takes different times to unload the incoming trucks. From the past data on truck unloading operations at the dock, you estimate that the forklift at the dock can unload 5 incoming trucks per hour on average and the unloading time is exponentially distributed.

The incoming trucks form a single line in front of the dock and the forklift starts unloading the truck at the beginning of the line. Using the past arrival data of the incoming trucks, you estimate that a new incoming truck arrives in the inbound operations area every 15 minutes on average and the interarrival time between incoming trucks is exponentially distributed.

a) Define the above queueing system by describing the customers, arrival rate, expected interarrival time, server(s), service rate, and expected service time. Use Kendall's notation to define the queueing system.

Currently, your company wishes to reduce the carbon emissions generated at the cross-docking facility. Carbon emissions are generated because the incoming trucks waiting for unloading are not turning off their engines as they need to move towards the dock frequently. You know that, on average, an incoming truck generates 20 grams of CO₂ per hour. On the other hand, since the forklift has smaller engine, its engine is turned off for the times the forklift is not unloading the incoming truck; hence, it does not generate CO₂ when it is inactive. On the other hand, while the forklift is unloading incoming trucks, it generates 5 grams of CO₂ per hour.

b) Calculate the expected amount of CO₂ generated per hour due to inbound operations? You can use templates for the values of L, L_q, W, W_q, P_r(W>t), P_r(W_q>t), and P_n; however, for other equations you use, please show your calculations and how you reach to your final answer.

The company now targets that the average CO₂ emissions from inbound operations should not exceed 80 grams per hour and you observed that you are exceeding this limit. Therefore, you are considering some changes in the inbound operations to reduce the CO₂ emissions generated from inbound operations.

Specifically, since you do not have control over the incoming trucks, you want to make changes in the forklift operations. In particular, you have two other alternative forklifts as detailed below:

- Alternative forklift 1: It can unload an incoming truck in 11 minutes on average and unloading time is exponentially distributed. Since it is faster, it generates more CO₂ emissions. Particularly, it generates 20 grams of CO₂ per hour when it is active for unloading. When it is inactive, the engine can be turned off.

- Alternative forklift 2: It can unload an incoming truck in 10 minutes on average and unloading time is exponentially distributed. Since it is faster, it generates more CO₂ emissions. Particularly, it generates 30 grams of CO₂ per hour when it is active for unloading. When it is inactive, the engine cannot be turned off (as frequent engine turn offs can create mechanic issues) and it generates 20 grams of CO₂ per hour when it is inactive.

c) Does forklift alternative 1 satisfy the company's new target for not exceeding 80 grams of CO₂ per hour on average? Show you calculations on how you determine the expected amount of CO₂ generated per hour due to inbound operations when you start using forklift alternative 1?

d) Does forklift alternative 2 satisfy the company's new target for not exceeding 80 grams of CO₂ per hour on average? Show you calculations on how you determine the expected amount of CO₂ generated per hour due to inbound operations when you start using forklift alternative 1?

e) Based on the expected carbon emissions per hour, which alternative you would select if you want to minimize carbon emissions from inbound operations?