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

Do it in Python and also 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.

Here also is part of my answers, use this for the rest of the question answer please:

To obtain the optimal routes and costs for both cases, we can formulate a linear programming model.

First, we define the decision variables as follows:

Let xij be the number of barrels shipped from Plant i to Waste Site j directly.

Let yijk be the number of barrels shipped from Plant i to Plant k and then to Waste Site j.

The objective function is to minimize the total transportation cost:

Minimize Z = i=1 to 6 j=1 to 3 (cij * xij) + i=1 to 6 j=1 to 3 k=1 to 6 (cij + cjk - cik) * yijk

subject to the following constraints:

For each Waste Site j, the total barrels shipped from all Plants i should not exceed the maximum capacity of the Waste Site: i=1 to 6 (xij + k=1 to 6 yijk) Mj for j = 1,2,3

For each Plant i, the total barrels shipped to all Waste Sites j should not exceed the weekly production of the Plant: j=1 to 3 (xij + k=1 to 6 yijk) Pi for i = 1,2,3,4,5,6

The decision variables should be non-negative: xij, yijk 0 for all i,j,k

Now, Using a linear programming solver, we can obtain the optimal routes and costs for both cases please.