3. Lara See\'s is a pie shop that specializes in custard and fruit pies. The sho
ID: 3111921 • Letter: 3
Question
3. Lara See's is a pie shop that specializes in custard and fruit pies. The shop is so popular that each day, it can sell all the pies made that day. The table below presents resource requirements and pro t information for every dozen pies.
Pie Type Flour (lbs) Eggs Sugar (lbs) Fruit Mix (lbs) Net Profit
Custard 12 50 5 0 $15
Fruit 10 40 10 15 $25
Available Today 150 500 90 120
(a) Formulate a linear program that will tell Lara See how many dozen custard and fruit pies should be made today.
(b) Convert the problem in part (a) into standard form, and solve using Matlab's linprog function.
(c) Which variables are in the basis in the optimal solution you found in part (b)? Based upon your answer, use the formulas to construct an optimal final tableau.
Explanation / Answer
Solution:
For Part A :
X1 = the number of dozen custard pies baked
X2 = the number of dozen fruit pies baked
MAX 15X1+25X2
S.T. 12X1 + 10X2 <= 150 (Flour)
50X1 + 40X2 <= 500 (Eggs)
5X1 + 10X2 <= 90 (Sugar)
15X2 <= 120 (Fruit mixture)
XI, X2 >= 0
X1 = the number of dozen custard pies baked
X2 = the number of dozen fruit pies baked
B) We can introduce slack variables s1, s2 ,s3 and s4 into the constraints (one for each constraint) and re-write the problem in standard form as:
MAX Z(X1,X2)= 15X1+25X2
S.T. 12X1 + 10X2 +S1= 150 (Flour)
50X1 + 40X2 +S2= 500 (Eggs)
5X1 + 10X2 +S3= 90 (Sugar)
15X2 +S4= 120 (Fruit mixture)
XI >= 0
X2 >= 0
X1 = the number of dozen custard pies baked
X2 = the number of dozen fruit pies baked
C)
Adjustable Cells
Final
Reduced
Objective
Allowable
Allowable
Value
Cost
Coefficient
Increase
Decrease
x1
4.67
0
15
0.5
7
x2
6.67
0
25
14
0.4
Optimal solution: Bake 56 (4 2/3 dozen) custard pies and 80 (6 2/3 dozen) fruit pies; Profit = $236.67
Adjustable Cells
Final
Reduced
Objective
Allowable
Allowable
Value
Cost
Coefficient
Increase
Decrease
x1
4.67
0
15
0.5
7
x2
6.67
0
25
14
0.4
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