Linear Programming
CASE PROBLEM ( PRODUCT MIX) Prepared for DR Taqi AlFaraj
Group J 225742 Bara'a AlJumbaz 4 226352 Mwuaffag Baswaid Ahmad AlAbdul Muney" 205943
30 May 2006
Case Problem PRODUCT MIX
Background:
TJ's Inc., makes 3 nut mixes for sale to grocery chains located in the Southeast. The three mixes are: Regular Mix, Deluxe Mix and Holding Mix. They are made by mixing different percentages of five types of nuts as follow:
Regular Mix Deluxe Mix Holding Mix
Almond 15% 20% 25%
Brazil 25% 20% 15%
Filbert 25% 20% 15%
Pecan 10% 20% 25%
Walnut 25% 20% 20%
TJ's has just purchased the following shipments of nuts at the prices shown: Type of Nut Shipment Cost per Shipments($) Amount(Pounds) Almond 6000 7500 I I Brazil 7500 7125 Filbert 6750 7500 Pecan 6000 7200 7875 Walnut 7500 I
1
The ~ r o f icontribution Der Pound and Customer orders are: t Profit contribution per Pound Customer Orders (Pounds) Regular mix $1.65 10000 Deluxe mix $2 3000 Holding Mix $ 2.25 5000
Solution:
Xij = quantity of nut (i) to mix j i= 1,2,3,4,5 j= 1,2,3 Profit 1$1.65
I $2.00
D
I $2.25
H
X13
R A
F P

Available
6,000 pounds
I
I
I
XI1 X21
I
I
I
X12 X22
I X23
I X33
1 7,500 pounds I
1 7,500 pounds 1 6,000 pounds
I
I
w
Max s.t.
I X51
I X31 I X41
I X32 I X42
Orders
I 10.000 pounds
I X52
I X43 I X53
I
I
I
( 3.000 pounds
1 5,000 pounds
1 7,500 pounds
1.65(Xll+X21+X3 1+X41+X51) + 2(X12+X22+X32+X42+X52) + 2.25(Xl3+X23+X33+X43+X53) X1 l+X12+X1316,000 X2 1+X22+X2317,500 X3 1+XX?+X3317,5OO X4 1+X42+X4316,000 X5 1+X52+X5317,5OO 0.15X11+0.25X21+0.25X31+0.10X41+0.25X51>10,000 0.20X12+0.20x22+0.20X32+0.20x42+0.20X52>3,000 0.25X13+0.15X23+0.15X33+0.25X43+0.20X53>5,000 Xij>O for all ij
1 Cost per pound of the nuts ( cost per shipmentslshipments Amount)
I Almond
Brazil Filbert Pecan Walnut
750016000 = 712517500 = 675017500 = 720016000 = 787517500 =
$1.25 $0.95 $0.9 $ 1.2 $ 1.05
1
The cost per pound of the nuts included in:
Regular Mix = 1.25(l5%) + 0.95(25%) + 0.9(25%) + 1.2 (10%) + 1.O5(25%) = 1.032 Deluxe Mix = 1.25(20%) + 0.95(20%) + 0.9(20%) + 1.2 (20%) + 1.05(20%) = 1.07 Holding Mix = 1.25(25%) + O.%(l5%) + O.9(15%) + 1.2 (25%) + 1.O5(2O%) = 1.10
2. Total Profit Contribution Regular Mix Deluxe Mix Holding Mix Total
$33,750
Almond
Brazil
Filbert
Pecan
Walnut
Available Regular Mix Deluxe Mix Holding Mix nut available after saling
6000
1500 600 1250 2650
7500
2500 600 7 50 3650
7500
2500 600 750 3650
6000
1000 600 1250 3150
7500
2500 600 1000 3400
*
So, purchasing more nuts will not increase profit because we already have enough of nuts. 4. TJ's does not have to purchase an additional 1000 pounds of almonds, because there are 2650 pounds are not used yet.
5. If that happen, then the profit contribution could be increased by concentrate on producing Holding Mix, because it has high profit than other.
Dr. Taqi N. AlFaraj Quantitative Methods for Management OM 310 Case Studies
Case Problem
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