Goal Programming Application

1. Goal Programming of
Bauxite Mines Company

2. Problem Statement
The Bauxite Mines Company owns three different mines which are Mine 1, Mine 2, and Mine 3,
which produces ore that after being crushed, is graded into three classes:
Low, Medium, and High grades.
KGF Smelting plant has offered a contract to smelt the obtained ore whose requirement per week is
in such a way:
• Low Grade: 12 tonnes
• Medium Grade: 8 tonnes
• High Grade: 24 tonnes

3. The three mines have different operating characteristics as
detailed below:
Now the Bauxite Mines company is in a state of decision-making
whether to accept the contract which is offered by the KGF smelting
company or make changes to the existing contract to make profits…?

4. Goal Programming
Decision Variables:
A= No: of Days per week Mine 1 is operated
B= No: of Days per week Mine 2 is operated
C= No: of Days per week Mine 3 is operated
Objective Function: Min 180A+108B+72C
Functional Constraints:
5A+B+C >= 12
2A+B+C >= 8
5A+3B+2C >= 24
A<=5
B<=5
C<=5
A, B, C >=0
*Note: Assuming Mine 1, Mine 2, and Mine 3 operate only for 5 days a week.

5. Objective function Value

6. Here we observed that
A = 0.7,
B = 3.5,
C = 5
thereby we are getting the objective function value as 864 (in lakhs)

7. ● But the problem arising here while considering the optimal
solution is the excess production of medium grade i.e., 1.9 tons in a
week.
● Initially, the problem is in minimizing the cost and now a new
problem has arisen to reduce the excess grade production in our
optimal solution because the company cannot handle excess as
such.
● where excess ore might be obtained as just referring to
medium-grade ore a moment’s thought will reveal if we can
reduce or cut down the medium-grade production through our
current solution. we might find an excess of another grade of
ore. Hence, we need to consider all grades and their excess
rather than just medium grades.

8. Goal Programming Formulation
*As programmed earlier through the optimal solution we inferred that the total production cost incurred
was 864 Lakhs whereas the company is also trying to cut down the production further. Hence, the
company chose to limit the production cost to 810 Lakhs.
Now we have two objectives:
 minimize cost – our previously calculated minimum cost was 864L (per week). The company is now
interested in decreasing the production cost that they incur.
Therefore, the company is now limiting the production cost to a maximum of 810 Lakhs.
 Excess Production of Ore – Referring to three grades of the ore for
which the respective goals are the amounts that we need to produce are 12, 8, and 24 tonsrespectively.

9. Bauxite mines company sells grade wise as low-grade 20 lakh/ton, Medium grade 60 lakh/ton, and
High grade 20 lakh/ton. As per selling the allotted weights are designed as:
Hence for our Bauxite Mines problem where we wish to reconcile our conflictingobjectives, we have the
equations:
180A+ 108B+72C = 810 + E+ - E-
5A+B+C = 12 + L+ - L-
2A+B+C= 8 + M+ - M-
5A+3B+2C = 24 + H+ - H-
A  5
B  5
C 
A,B,C,E+,E-,H+,H-,M+,M-,L+,L-  0
These equations are, given our variables, equations that must be satisfied.

10. To proceed this we are going with weighted goal approaches:
 to attach weights to each of the variables associated with upward/downward
deviation and then to solve a single problem whose objective is to minimize this weighted deviation sum. T
approach is sometimes known as weighted goal programming
Weighted Goal Programming:(Objective)
Minimize E+ + 20L+ + 20L- + 60M+ +60M- + 20H+ +20H-
+ Subject to
180A+ 108B+72C - E+ + E- = 810
5A+B+C - L++ L- = 12
2A+B+C- M++M- = 8
5A+3B+2C- H++ H- = 24
A  5
B  5
C 
A, B, C, E+, E-, H+, H-, M+, M-, L+, L-  0

11. Solution

12. • The production cost obtained from the solution of goal programming is 804 Lakhs and the goal which
the company asked to reduce (810 lakhs) is achieved and the days to be operated by mine 1, mine 2, Mine 3
is 1.333,5,0.333 days in a week
• As per the contract, the company wanted to produce a Low of 12 tons, a Medium of 8 tons, High of 24
tons but the company can achieve the capacity of Low-12 tons, Medium-8 tons, and High-22.333 tons.
• By linear programming 1.9 tons of medium grade get wasted so indirectly, it incurs the loss of
1.9*60=138 lakhs and the cost incurred is 864 lakhs. Total profit observed as 1200-864=336 lakhs for
production of Low-12 tons, Medium-8 tons, High-24 tons
• By Weighted pre-emptive goal programming, the total cost incurred is 804 lakhs for the production of
Low 12 tons, Medium-8 tons, and High-22.33 tons, and total profit observed as 1166.6-804=362 lakhs,
Which is a profitable condition for the company compared to producing Low-12 tons, Medium-8, and
Hia22.33 tons. So, Bauxite mines company make a contract with KGF smelting company for the quantities
of Low 12 tons, Medium-8 tons, High-22.333 tons instead of 12,8,24 tons
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