Here is the project: Project:Rockhill Shipping & Transport Company

Allen, a manager of the South-Atlantic office of the Rockhill Shipping & Transport Company is negotiating a new shipping contract with Chimotoxic, a company that manufactures chemicals for industrial use. Chimotoxic wants Rockhill to pick up and transport waste products from its six plants to three waste disposal sites. Allen is very concerned about this proposal arrangement. The chemical wastes that will be hauled can be hazardous to humans or the environment if they leak. In addition, some of the communities in the regions where the plants are located may prohibit hazardous materials from being shipped through their municipal limits. Thus, not only the shipments have to be handled carefully and transported at reduced speeds, but they may also have to traverse in circuitous routes in some cases.

Allen has estimated the cost of shipping a barrel of waste from each of the six plants to each of the three waste disposal sites as shown in the following table.

Waste Proposal Site
Plant:	Orangeburg	Florence	Macon
Denver	$12	$15	$17
Morganton	14	9	10
Morrisville	13	20	11
Pineville	17	16	19
Rockhill	7	14	12
Statesville	22	16	18

The plants generate the following amounts of waste products each week:2

Plant:	Waste per Week (bbl)
Denver	45
Morganton	26
Morrisville	42
Pineville	53
Rockhill	29
Statesville	38

The three waste disposal sites at Orangeburg, Florence, and Macon can respectively accommodate a maximum of 65, 80, and 105 barrels per week.

In addition to shipping directly from each of the six plants to one of the three waste disposal sites, Allen is also considering using each of the plants and the waste disposal sites as intermediate shipping points. In this case, trucks would be able to drop a load at a plant or a disposal site to be picked up and carried on to the final destination by another truck. Furthermore, Rockhill would not incur any handling costs because Chimitoxic has agreed to take care of all the handling costs at the plants and at the disposal sites. In other words, Rockhill's only cists will be the transportation costs. Therefore, Allen wants to be able to consider the possibility that it may be cheaper to drop and pick up loads at intermediate points rather than ship them directly.

Allen has estimated the shipping costs per barrel between each of the six plants to be as follows:

Plant:	Denver	Morganton	Morrisville	Pineville	Rockhill	Statesville
Denver	$---	$3	$4	$9	$5	$4
Morganton	6	---	7	6	9	4
Morrisville	5	7	---	3	4	9
Pineville	5	4	3	---	3	11
Rockhill	5	9	5	3	---	14
Statesville	4	7	11	12	8	---

The estimated shipping cost per barrel between each of the waste disposal sites is as follows:

Waste Proposal Site

Waste Disposal Site:	Orangeburg	Florence	Macon
Orangeburg	$---	$12	$10
Florence	12	---	15
Macon	10	15	---

Allen wants to determine the shipping routes that will minimize Rockhill's total cost in order to develop a contract proposal to submit to Chimotoxic for waste proposal. He particularly wants to know if it would be cheaper to ship directly prom the plants to the waste sites or if he should drop and pick up some loads at the various plants and waste sites. 3

In the word document, explain the details of the solutions obtained for the optimal routes and their respective optimal costs for both cases. In particular for the case when loads are dropped and picked up at various plants and waste sites, explain how many barrels, in total, will be transported each week from a source to a destination. I want to get the optimal routes and optimal costs for both cases as well as answering this part of the question: [In particular for the case when loads are dropped and picked up at various plants and waste sites, explain how many barrels, in total, will be transported each week from a source to a destination. ] here also is part of my answers: [Introduction: This project involves determining the most cost-effective shipping routes for Rockhill Shipping & Transport Company to transport hazardous waste products from six plants owned by Chimotoxic to three waste disposal sites. The project also considers the possibility of using intermediate shipping points to reduce transportation costs.

The three waste disposal sites have different maximum capacities for waste products per week, and some of the communities where the plants are located may prohibit hazardous materials from being shipped through their municipal limits.

To solve this problem, a mathematical optimization model could be formulated to find the least-cost shipping routes subject to the constraints of the problem. This model will be solved using a linear programming solver.

Now, we start with defining the objective function, optimization variable and constraints.

Based on the project, the objective function for this problem could be to minimize the total transportation cost for all shipments.

If the transportation cost for shipping one barrel of waste from Plant i to Disposal Site j is cij. The decision variable xi,j represents the number of barrels shipped from Plant i to Disposal Site j. Then, the objective function can be written as:

• minimize Z = sum of (cij * xi,j) for all i and j

This objective function aims to find the values of xi,j that minimize the total cost of transportation.

xi,j represents the decision variable that determines the amount of waste to be shipped from plant i to waste disposal site j. The indices i and j are used to specify the specific plant and waste disposal site being referred to. So, the objective function is the sum of the product of the shipping cost cij and the amount of waste shipped xi,j for all possible combinations of plants and waste disposal sites.

Based on the objective function, the optimization variable would be:

• xi,j: the number of barrels of waste transported from Plant i to Waste Proposal Site j (directly or indirectly)

the following are the constraints:

1. The total amount of waste transported from Denver to all waste disposal sites cannot exceed the weekly waste generated by Denver:

xi,1 + xi,2 + xi,3 <= 45 for i=1

1. The total amount of waste transported from Morganton to all waste disposal sites cannot exceed the weekly waste generated by Morganton:

xi,1 + xi,2 + xi,3 <= 26 for i=2

1. The total amount of waste transported from Morrisville to all waste disposal sites cannot exceed the weekly waste generated by Morrisville:

xi,1 + xi,2 + xi,3 <= 42 for i=3

1. The total amount of waste transported from Pineville to all waste disposal sites cannot exceed the weekly waste generated by Pineville:

xi,1+ xi,2 + xi,3 <= 53 for i=4

1. The total amount of waste transported from Rockhill to all waste disposal sites cannot exceed the weekly waste generated by Rockhill:

xi,1 + xi,2 + xi,3 <= 29 for i=5

1. The total amount of waste transported from Statesville to all waste disposal sites cannot exceed the weekly waste generated by Statesville:

xi,1 + xi,2 + xi,3 <= 38 for i=6

1. The total amount of waste transported from each plant to each waste disposal site, directly or indirectly, cannot exceed the capacity of the respective waste disposal site:

xi,1 + xi,2 + xi,3 + xi,4 + xi,5 + xi,6 <= 65 for j=1

xi,1 + xi,2 + xi,3 + xi,4 + xi,5 + xi,6 <= 80 for j=2

xi,1 + xi,2 + xi,3 + xi,4 + xi,5 + xi,6 <= 105 for j=3

1. The number of barrels of waste transported from each plant to each waste disposal site, directly or indirectly, cannot be negative:

xi,j >= 0 for all i and j.

] can you please for the code part, do it in python and give me all the answer for this project step by step to be understand well? Thanks a lot