A manufacturer of edible oils has two plants P1, P2 at different locations. They
ID: 350863 • Letter: A
Question
A manufacturer of edible oils has two plants P1, P2 at different locations. They buy raw oils from 4 different suppliers S1 to S4 and process it. The processing yields two grades of oils, grades 1 and 2 and some residue. There are companies which buy the residue for use in manufacturing soap etc. Here is the data. All prices and costs are quoted in the same money units.
mu/u = money units/unit
Approximately 4% of the raw oil purchased is lost in transit from the suppliers to the plants. The two plants operate with different equipment and refining procedures, and hence the yields are different, and the outputs command different market prices.
The production costs at plants P1, P2 are 25, 24 mu/u of raw oil processed respectively. The plants at P1, P2 have capacities of processing 7000, 5000 units of raw oil respectively per month. The operating range of each plant is 50 to 100% of its capacity. Formulate the problem of finding a production plan
that maximizes the company’s net profit as an LP(Find constraints of this LP problem)
Explanation / Answer
We have to find the amount to be shipped from each supplier to each plant. We are assuming that
Xij be the amount shipped form supplied form supplier i (i=1,2,3,4) to plant j (j=1,2)
STEP 1
Decision variables
X11,X12,X21,X22,X31,X32,X41,X42
STEP 2
Profit = sales - raw oil purchase cost - shipment cost - processing cost
Sales = ( sale of grade 1 at p1 + sale of grade 2 at p1+ sale of residue at P1 ) + (sale of grade 1 at p2 + sale of grade 2 at p2+ sale of residue at P2)
For e.g.
total shipment to P1 = X11+X21+X31+X41
total shipment to P2 = X12+X22+X32+X42
Total oil processed at P1
= total shipment to P1* 0.96( after transit loss of 4%)
= (X11+X21+X31+X41) *0.96
Recovered volume of grade 1 at P1
= total oil processed at P1 * Recovery rate for grade 1
= (X11+X21+X31+X41) *0.96 *0.6
Sale of grade 1 at P1= recovered volume of grade 1 at P1 * price of grade 1 at P1
= (X11+X21+X31+X41) *0.96 *0.6 *200
Similarly we find sales for other grades and residue at P1 and P2
So
Sale
=total shipment to P1*0.96*0.6*200 +
total shipment to P1*0.96*0.3*150+
total shipment to P1*0.96*0.1*29 +
total shipment to P2*0.96*0.51*190 +
total shipment to P2*0.96*0.29*145+
total shipment to P2*0.96*0.20*29
= ( X11+X21+X31+X41)*0.96*0.6*200 +
(X11+X21+X31+X41)*0.96*0.3*150+
(X11+X21+X31+X41)*0.96*0.1*29 +
(X12+X22+X32+X42 )*0.96*0.51*190 +
(X12+X22+X32+X42) *0.96*0.29*145+
(X12+X22+X32+X42 )*0.96*0.20*29
Simplifying we get ,
Sale =[ 161.84(X11+X21+X31+X41 ) + 138.96(X12+X22+X32+X42 )]
Also,
purchase cost = (35.5X11+ 34.5X21+33.7X31+35 X41) + ( 35.5 X12+ 34.5X22+33.7X32+35 X42)
shipment cost = [(2X11+6X21+3.2X31+4X41 ) +(5.5X12+5X22+4.3X32+3.7X42 )]
processing cost= [ 25*0.96((X11+X21+X31+X41 ) + 24*0.96* (X12+X22+X32+X42 )]
=[ 24 *(X11+X21+X31+X41 ) + 23.04* (X12+X22+X32+X42 )]
Objective function
Maximize,
Z=Profit
Z = [ 161.84(X11+X21+X31+X41 ) + 138.96(X12+X22+X32+X42 )] -
[ (35.5X11+ 34.5X21+33.7X31+35 X41) + ( 35.5 X12+ 34.5X22+33.7X32+35 X42)] -
[(2X11+6X21+3.2X31+4X41 ) +(5.5X12+5X22+4.3X32+3.7X42 )] -
[ 24 *(X11+X21+X31+X41 ) + 23.04* (X12+X22+X32+X42 )]
Constraints
Plant processing capacity (50%-100%)
X11+X21+X31+X413500
X11+X21+X31+X417000
X12+X22+X32+X42 2500
X12+X22+X32+X425000
Supplier capacity
X11 + X121300
X21+X222300
X31+X322550
X41+ X421850
Non-negativity constraint
X11,X12,X21,X22,X31,X32,X41,X42 0
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