Introduction

It is always preferred to use
uniform high grade ore for production of metallic alloy. But in real life
situation, it is not possible to get a homogeneous ore. Chrome ore is one of
the precious natural resources having limited availability. The chemical
composition of the ore varies horizontally and vertically across the ore body
in mines. The ore quality is not uniform in all the faces of the mines. The
of ore is required at the mines head before using it in the furnaces for
smelting. The production requirement is to have a uniform grade metal
production to meet the customer specification with minimum or zero deviation.

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Further, Chrome ore being one of the precious
natural resources need to be preserved and used properly; the low and or medium
of misusing.

Present
Practice

The important chemical parameters of the
chrome ore are Cr2O3, Cr/Fe, FeO,
Al2O3, Phos, SiO2, CaO, MgO and
Cost ore. Each chemical parameter is having some lower and upper limit for
being suitable to be used for Ferro Chrome production. The requirement being uniform, higher grade of ore having uniform
chemical composition, this requirement forced to mine only high-grade ore, thereby
creating a huge stock of other medium and lower grade of ore at mines pit. This
not only affects the mines operation but also leads to the wastage of natural
resources.

Considering the practical problems
of reduction in cost of alloy production, preservation and use of high-grade
ore and enhanced value addition by using the medium and low-grade ores, a
mathematical model of non-linear nature has been developed. The model was
formulated for blending of chrome ore from different lots with three different
objective functions. The objective functions are

1.      Maximizing the quantity of ore
supply

2.      Minimizing the cost of Ore.

3.      Maximizing the satisfaction index

Maximize
the quantity of supply:

The problem formulated
with an objective of
maximizing the quantity of supply from the available lots of ore subject to
fulfilling the chemical parameters. The particulars of the model are as under.

Objective Function   Maximize
Z=  (Maximize the quantity of supply)

Qi = Quantity of ore from Lot (i)

The Constraints being

(Quantity considered for blending
should be less than the availability)

Qsi= Availability (stock) of ore in lot (i)

and

Weighted average Cr2O3
% in ore should be less than the upper limit and more than the lower limit.

and

Weighted average FeO % in ore
should be less than the upper limit and more than the lower limit.

and

Weighted average SiO2 % in ore should be less than the upper
limit and more than the lower limit.

and

Weighted average Al2O3 % in ore should be less than
the upper limit and more than the lower limit.

and

Weighted average CaO % in ore should be less than the upper limit and
more than the lower limit.

and

Weighted average MgO % in ore should be less than the upper limit and
more than the lower limit.

and

Weighted average Phos % in ore should be less than the upper limit and
more than the lower limit.

and

Weighted average Cr/Fe in ore should be less than the upper limit and more
than the lower limit.

Typical stocks of different lots
of chromium ore available with the upper and lower limit specification for
acceptance are as shown in table 1

Table 1 Different grade of chromium ore and its chemical composition

The problem formulation in case of maximizing the quantity available
considering the above available data can be as under.

Z= Maximize (Q1+Q2+Q3+……………. +Q10)

Where Q1, Q2,
Q3………Q10 are the quantity of material selected for
blending from lot-1, lot-2, Lot-3 and lot-10 respectively

Constraints for quantity availability (Quantity considered for blending should be
less than or equal with the availability)

Q1=0.46

(0.427Q1+0.48Q2+0.46Q3+0.43Q4+0.46Q5+0.44Q6+0.48Q7+0.46Q8+0.43Q9
+0.50Q10)  / (Q1+Q2+Q3…….+Q10)
=0.16

(0.188Q1+0.1523Q2+0.1802Q3+0.1717Q4+0.1681Q5+0.231Q6+0.1681Q7+0.231Q8+0.1142Q9+0.1257Q10)
/(Q1+Q2+Q3+…….+Q10) =0.035

(0.032Q1+0.032Q2+0.05Q3+0.032Q4+0.0368Q5+0.0368Q6+0.0368Q7+0.0368Q8+0.1515Q9+0.06Q10)
/ (Q1+Q2+Q3……+Q10) =0.09

(0.1336Q1+0.1346Q2+0.139Q3+0.1223Q4+0.1704Q5+0.1487Q6+0.1704Q7+0.1487Q8+0.0892Q9+0.0892Q10)
/ (Q1+Q2+Q3…….+Q10) =0.01

(0.0225Q1+0.0375Q2+0.015Q3+0.225Q4+0.0098Q5+0.0098Q6+0.0.0098Q7+0.0098Q8+0.0102Q9+0.0102Q10)
/ (Q1+Q2+Q3…….+Q10) =0.07

(0.0802Q1+0.0725Q2+0.08Q3+0.08Q4+0.0958Q5+0.0958Q6+0.0.0958Q7+0.0958Q8+0.1881Q9+0.1881Q10)
/ (Q1+Q2+Q3…….+Q10) =0.00001

(0.00013Q1+0.00013Q2+0.0001Q3+0.00013Q4+0.0001Q5+0.0001Q6+0.0.0001Q7+0.0001Q8+0.00007Q9+0.00007Q10)
/ (Q1+Q2+Q3…….+Q10) =2.2

(2Q1+2.78Q2+2.25Q3+2.21Q4+2.41Q5+1.68Q6+2.52Q7+1.76Q8+3.32Q9+3.51Q10)/
(Q1+Q2+Q3…….+Q10) = Lower
Limit

= Lower
Limit + Rj/5

= Lower
Limit + 2Rj/5

= Lower
Limit + 3Rj/5

= Lower
Limit + 4Rj/5          (Di) =5                        (Di)=1

Weightages of
the Parameters

Based on the feed back from the furnace operators the weightages for the
different parameters (Wj) for
comfortable operation has been decided as under.

Parameters                 Weightage

Cr2O3               0.30

Cr/Fe               0.20

FeO                 0.10

Cost                 0.10

Al2O3               0.075

Ph                    0.075

SiO2                 0.05

CaO                 0.05

MgO                0.05

The problem formulation in case of maximizing the satisfaction index can
be formulated as under

Z=Maximize (30% Satisfaction index
Cr2O3 +20% Satisfaction index Cr/Fe +10% Satisfaction index FeO+10% Satisfaction index Cost+7.5% Satisfaction index Al2O3 +7.5% Satisfaction index Ph+5% Satisfaction index SiO2+5%
Satisfaction index CaO)

The constraints are same as that
of the maximizing the quantity problem. The problem was solved and the output
of the model is presented in table 4

Table 4. Output of the model of
maximizing user satisfaction index.

(out of 5)

The table shows that the optimum quantity (Qi) of ore selected from different lots in order to
maximizing user satisfaction index.

Lot-1               Q1          1000 MT                     Lot-2               Q2          1670 MT

Lot-3               Q3          699   MT                     Lot-4               Q4          1500 MT

Lot-5               Q5          921   MT                     Lot-6               Q6          639   MT

Lot-7               Q7          1156 MT                     Lot-8               Q8          1433 MT

Lot-9               Q9          0       MT                     Lot-10             Q10         1982 MT

Average Chemical composition of ore against the upper and lower limits
under.

Cr2O3  FeO     SiO2     Al2O3   CaO    MgO    Ph        Cr/Fe

Upper limit                  47.00   18        4.00     13.50   2.50
14.00   0.013   2.80

Lower Limit                46.0     16.0     3.5       9.0       1.0       7.0       0.000   2.2

Actual with above       46.41   17.30   4.00     13.43   1.73     10.43   0.011   2.47

ore mix

The average cost of the above
blended ore is Rs 279 per metric tonne.

Summary and Observation

The output of the models having
different objective functions are summarized and presented in table 4.12

Table 5 Output comparison from different objective functions

There is a increase in cost of the
ore in the output of maximizing quantity and maximizing satisfaction index from
the out of the objective function of minimizing cost by Rs 5 and Rs 7 per MT
respectively, the impact of which is Rs 62477 and Rs 75237. The impact can be
derived as under

Impact of increasing in average
cost of ore from the objective function of minimizing cost to the objective
function of maximizing quantity = (272-277) Rs/MT x 13723MT = Rs (-)
62477/Month. Similarly the impact of increasing in average cost of ore from the
objective function of minimizing cost to the objective function of maximizing
satisfaction index =(272-279) Rs/MT x 11000MT
= Rs (-) 75237/ Month

There is an increase in Cr2O3
% of the ore in the output of maximizing quantity and maximizing satisfaction
index from the out of the objective function of minimizing cost by 0.26% and
0.41% respectively, the impact of which is Rs 8.08 lacs and Rs 12.80 lacs .

The impact can be derived as under

Cr2O3
% in the Ore                 % Increase
from the                                                                                                                Output of Minimizing Cost

Minimizing
Cost                                 46%                                         ——

Maximizing
Quantity                          46.26%                                    0.26%

Maximizing
Satisfaction index           46.41%                                    0.41%

input will lead to higher volume of production which can be derived as
(Additional chromium input x Chromium recovery) / (% chromium in the finished
goods)

Difference in Cr2O3 % x Conversion factor (0.685) x
Quantity of ore in MT

Chromium Recovery   = 83% & percentage (%) of Chromium in the
finished good= 60%

In case of the output of
maximizing quantity (0.0026 x 0.685 x 13723 x 0.83) / 0.6 = 26.94 MT and in
case of the output of maximizing satisfaction index (0.0041 x 0.685 x 11000 x
0.83) / 0.6  = 42.66 MT respectively for
maximizing quantity or maximizing satisfaction index respectively.

production at the current market price @ Rs 30000 per MT the impact will be Rs
8.08 lacs and Rs 12.80 lacs. The net benefit by deducting the increasing in
cost of ore from the output minimizing cost will be about Rs 7.45 lacs and Rs
12.04 lacs.

Conclusion

It is suggested to use the model
with the objective function of maximizing the user satisfaction index.
The model suggests a
blend of available ore matching to all the elemental requirement of production.
The low grade ore which was otherwise idle will get utilized in due course of
time there by improving in productive use of natural resources without
affecting the quality of the output. In addition, there is a significant
improvement in the profitability of the organization. 