[SOLVED] C math ME 44206 Assignment Q1

ME 44206 Assignment Q1
Steel Recycling
Due Date: November 1, 2019

Steel is one of the easiest metals to be recycled. Every particle of scrap iron or scrap steel would actually help in making steel. The problem is when different metals are mixed, as it is very difficult to separate the metals in the scrap once it is mixed. Some metals are required, others are undesired. In 18/10 steel (stainless steel for cookware), you need to have 18% chromium and 10% nickel. On the other hand, copper is an impurity which makes it look less shining.

You have currently 10 heaps (A to J) of scrap iron, identified in the table below. Each heap contains a consistent mix of chromium, nickel, copper and iron. An arbitrary portion of each heap can be used in a mix. Pure chromium can also be bought on the market and added at the cost of 125 euro/ton.

Heap
A
B
C
D
E
F
G
H
I
J
Chromium (%)
10
25
25
14
0
18
10
2
25
10
Nickel (%)
0
15
10
16
15
1
12
8
19
2
Copper (%)
4
0
2
5
3
0
1
3
0
15
Amount (ton)
30
90
50
70
20
30
10
20
5
100
Cost (euro/ton)
50
100
90
105
85
115
90
80
70
15

The objective is to produce, at minimum costs, 5 batches of steel of 18/10 quality: at least 18% chromium and 10% nickel. However, the content of iron must be at least 65% for each batch. The required weight for each batch (in tons) and the maximum of impurities (% copper) for each batch is different and given in the table below.

Batch
1
2
3
4
5
Weight (tons)
125
50
25
80
20
Max. copper (%)
5.0
2.5
1.0
0.5
0.0

Electrolysis can be used to remove copper from a batch. The weight reduction of a batch caused by electrolysis is equal to the weight of the copper that is removed. The costs for such a treatment consist of a fixed and a variable part. Currently the fixed costs for electrolysis are 1000 euro per batch. The variable costs are 0.50 euro per kg of copper that is removed this way. The weight of a batch in above table must be the weight after electrolysis, if applied.

It is unsecure whether the fixed costs for this electrolysis process will go up or down in the future. Furthermore the price of pure chromium can change. A model can help in analyzing these issues.
(A research question for this problem will be given later)

Formulate a mathematical model for this problem
Implement the model and verify your implementation
Solve the problem using the given data; calculate the optimal value of the objective function and the related decision variables.
Use your model to study the relation between the fixed costs for electrolysis and the optimal decisions for this problem.

P.S. Also read the Guidelines for the me44206 assignment; posted separately!