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

production requirement being uniform grade of ore, blending of different grades

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.

Further, Chrome ore being one of the precious

natural resources need to be preserved and used properly; the low and or medium

grade ore should be used by suitably blending with the high grade ore instead

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<= 1000 MT, Q2<=2000 MT, Q3<=3000 MT, Q4<=1500 MT, Q5<=7000 MT Q6<=2000 MT , Q7<=3000 MT, Q8<=1500 MT, Q9<=7000 MT, Q10<=10000 MT Constraints for Cr2O3 % in the Ore (0.427Q1+0.48Q2+0.46Q3+0.43Q4+0.46Q5+0.44Q6+0.48Q7+0.46Q8+0.43Q9+ 0.50Q10) / (Q1+Q2+Q3…….+Q10) >=0.46

(0.427Q1+0.48Q2+0.46Q3+0.43Q4+0.46Q5+0.44Q6+0.48Q7+0.46Q8+0.43Q9

+0.50Q10) / (Q1+Q2+Q3…….+Q10)

<=0.47
Weighted average Cr2O3
% in ore should be less than the upper limit and more than the lower limit.
Constraints for FeO % in the Ore
(0.188Q1+0.1523Q2+0.1802Q3+0.1717Q4+0.1681Q5+0.231Q6+0.1681Q7+0.231Q8+0.1142Q9+0.1257Q10) / (Q1+Q2+Q3+……+Q10)
>=0.16

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

/(Q1+Q2+Q3+…….+Q10) <=0.18
Weighted average FeO % in ore
should be less than the upper limit and more than the lower limit.
Constraints for SiO2 % in the Ore
(0.032Q1+0.032Q2+0.05Q3+0.032Q4+0.0368Q5+0.0368Q6+0.0368Q7+0.0368Q8+0.1515Q9+0.06Q10)
/ (Q1+Q2+Q3……+Q10) >=0.035

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

/ (Q1+Q2+Q3……+Q10) <=0.04
Weighted average SiO2 % in ore should be less than the upper
limit and more than the lower limit.
Constraints for Al2O3% in the Ore
(0.1336Q1+0.1346Q2+0.139Q3+0.1223Q4+0.1704Q5+0.1487Q6+0.1704Q7+0.1487Q8+0.0892Q9+0.0892Q10)
/ (Q1+Q2+Q3…….+Q10) >=0.09

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

/ (Q1+Q2+Q3…….+Q10) <=0.135
Weighted average Al2O3 % in ore should be less than
the upper limit and more than the lower limit.
Constraints for CaO% in the Ore
(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) >=0.01

(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) <=0.025
Weighted average CaO % in ore should be less than the upper limit and
more than the lower limit.
Constraints for MgO % in the Ore
(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) >=0.07

(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) <=0.14
Weighted average MgO % in ore should be less than the upper limit and
more than the lower limit.
Constraints for Phos % in the Ore
(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) >=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) <=0.00013
Weighted average Phos % in ore should be less than the upper limit and
more than the lower limit.
Constraints for Cr/Fe in the Ore
(2Q1+2.78Q2+2.25Q3+2.21Q4+2.41Q5+1.68Q6+2.52Q7+1.76Q8+3.32Q9+3.51Q10)/
(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) <=2.8
Weighted average Cr/Fe in ore should be less than the upper limit and
more than the lower limit.
The problem was solved by using excel solver
optimizer and the output of the optimization has been summarized in table 2
Table 2 Output of the optimization with objective function of maximizing
the availability
The table shows that the optimum quantities (Qi) of ore selected from different lots in order to
maximize the availability are
Lot-1 Q1 1000 MT Lot-2 Q2 2000 MT
Lot-3 Q3 0 MT Lot-4 Q4 1500 MT
Lot-5 Q5 2874 MT Lot-6 Q6 200 MT
Lot-7 Q7 26 MT Lot-8 Q8 1500 MT
Lot-9 Q9 0 MT Lot-10 Q10 2824 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 46.26 17.50 4.00 13.50 1.62 10.85 0.010 2.46
ore mix
The average cost of the above
blended ore is Rs 277 per metric tonne.
Minimize the
cost of Ore
The problem formulated
with an objective of
minimizing the cost of ore supply from the available lots of ore subject to
fulfilling the minimum quantity and chemical parameters requirement. The
particulars of the model are as under.
Objective Function = Minimize the cost of Ore
Objective Function Minimize Z=
Qi = Quantity of ore from Lot (
i)
Ci is the
cost of Mining of the 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)
Constraints for all quality
parameters being same as in the same of maximizing the quantity of ore supply
The problem formulation in case of
minimizing the cost of ore supply from the available lots of ore subject to
fulfilling the minimum quantity and chemical parameters requirement. The
problem formulation using the data available in Table-10 can be as shown below.
Z=Min (250Q1+275Q2+275Q3+250Q4+250Q5+250Q6+275Q7+275Q8+250Q9+350Q10)
Where Q1, Q2,
Q3……Q10 are the quantity of material from lot-1, lot-2,
lot-3 and lot-10 respectively.
The constraints of the model being
same as that of the maximizing the quantity problem. The output of the
optimization has been summarized in table 3
Table 3 Output of the optimization with objective function of minimizing
the average cost of ore.
The table shows that the optimum quantities (Qi) of ore selected from different lots in order to
minimize the cost are
Lot-1 Q1 1000 MT Lot-2 Q2 2000 MT
Lot-3 Q3 0 MT Lot-4 Q4 1500
MT
Lot-5 Q5 2541 MT Lot-6 Q6 1893 MT
Lot-7 Q7 0 MT Lot-8 Q8 0 MT
Lot-9 Q9 97
MT Lot-10 Q10 1969
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 46.00 17.03 4.00 13.50 1.78 10.53 0.01 2.49
ore mix
The average cost of the above blended ore is
Rs 272 per metric tonne. Where the present cost of ore is about Rs.290
Maximizing the
User satisfaction Index:
The user requirement is a
homogeneity ore with higher metallic content and less of impurities available
at low cost. There are few elements of ore, higher the value of which results
in higher level of user satisfaction. Parameters Cr2O3,
FeO, Cr/Fe, CaO, SiO2 and MgO fall in this category. The other
category where lower the value, higher is the satisfaction are Al2O3,
Phos and Cost of Ore.
Objective
Function –
Maximizing the User satisfaction Index.
Objective Function Maximize Z=
Dj – Satisfaction index
of the parameter (j)
Wj – Weighatage
of the Parameter (j)
The Constraints being
(Quantity considered for blending
should be less than the availability)
Qsi= Availability (stock) of ore in lot (i)
Qi = Quantity of ore from Lot ( i)
Constraints for all quality
parameters being same as in the same of maximizing the quantity of ore supply
To decide on the user satisfaction
index the parameters of the ore be divided in to two broad categories as
Category-1: Higher the value of the parameter higher is
the level of user satisfaction. Parameters falling in category- 1 are Cr2O3,
FeO, Cr/Fe, CaO, SiO2 and MgO
Category-2: Higher the value of the parameter Lower is
the level of user satisfaction. Parameters falling in category- 2 are Al2O3,
Phos and Cost of Ore
Each parameter is having lower and
upper limit. Let Rj be the range of
the parameter, i.e., (difference of the maximum and minimum limit). The
satisfaction index for each parameter has been divided in a five-point scale as
follows.
Satisfaction index Satisfaction index
Category-1 Category-2
Actual Parameter >= Lower

Limit

<= Lower Limit + Rj/5 (Di)=1 (Di) =5
Actual Parameter >= Lower

Limit + Rj/5

<= Lower Limit + 2Rj/5 (Di)=2 (Di) =4
Actual Parameter >= Lower

Limit + 2Rj/5

<= Lower Limit + 3Rj/5
Di) =3 Di) =3
Actual Parameter >= Lower

Limit + 3Rj/5

<= Lower Limit + 4Rj/5
(Di) =4 (Di)=2
Actual Parameter >= 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%

The impact of additional chromium

input will lead to higher volume of production which can be derived as

(Additional chromium input x Chromium recovery) / (% chromium in the finished

goods)

Where Additional chromium input =

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.

By valuing the additional

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.