Cavity Candies mass produces an assortment of hard candies,consisting of three f

Question

Cavity Candies mass produces an assortment of hard candies,consisting of three flavors:cinnamon,root beer, and grape.Bag 1 must have between 75 and 100 pieces of candy,with at least 40% being cinnamon,at least 30% being grape and between 15% and 30% being root beer.Bag 2 must have between 50 and 75 candies,with no more than 30% cinnamon,at least 40% grape ,and at least 20% root beer.Each bag can contain at most 60 cinnamon,75 grape,and 75 root beer candies at costs of $0.01,$0.02, and $0.015 per candy,respectively.If bag 1 generates $0.03 per piece of candy in revenue,and bbag 2 generates $0.025 per piece in revenue,formulate and solve a linear program to help maximize Cavity's per-bag profits?

Note:This problem is in a deterministic operation research(david j rader).Exercise of 2.11 in chapter 2.

Explanation / Answer

Decision Variables

To begin with we are required to define our decision variables. Let c1, r1 and g1 represents number of candies of cinnamon, root beer and grape flavors respectively in Bag 1 and c2, r2. g2 are number of cinnamon, root beer and grape flavors respectively in Bag 2.

Objective Function

As given in the question we want to maximize Cavity's per-bag profit. Given bag 1 generates $0.03 per piece in revenue whereas bag 2 generates $0.025 per piece in revenue. Per piece costs of cinnamon, grape and root beer candies are $0.01, $0.02 and $0.015, therefore for bag 1 profit per piece are $0.02(.03-.01), $0.01(.03-.02) and $0.015 for cinnamon, grape and root beer candies. Similarly per unit profits for bag 2 are $0.015, $0.005 and $0.01.

Therefore the objective function is to Maximize .02c1 +.015c2 +.01g1 +.005g2 + .015r1 + .01r2

Lastly Constraints:

Bag 1 must have between 75 and 100 pieces ,therefore 75 <= c1+g1+r1 <= 100

     atleast 40% being cinnamon, therefore c1 >= .40*(c1+g1+r1)

     atleast 30% being grape, therefore g1 >= .30*(c1+g1+r1)

between 15% and 30% being root beer, therefore .15*(c1+g1+r1) <= r1 <= .30*(c1+g1+r1)

Similarly constraints for Bag 2 are as follows:

     50<= c2+g2+r2 <= 75; c2 <= .30*(c2+g2+r2); g2 >= .40*(c2+g2+r2) and r2 >= .20*(c2+g2+r2)

Non-negativity constraints c1, c2, g1, g2, r1, r2 being quantities can not be negative, all variables >= 0

This completes the formulation of the problem

Solution of the problem using simplex method is as follows:

    c1   c2   g1   g2   r1   r2      RHS   Dual
Maximize   .02   .015   .01   .005   .015   .01          
constraint1   1   0   1   0   1   0   >=   75   0
constraint2   1   0   1   0   1   0   <=   100   .0163
constraint3   .6   0   -.4   0   -.4   0   >=   0   0
constraint4   -.3   0   .7   0   -.3   0   >=   0   -.01
constraint5   -.15   0   -.15   0   .85   0   >=   0   -.005
constraint6   .3   0   .3   0   -.7   0   >=   0   0
constraint7   0   1   0   1   0   1   >=   50   0
constraint8   0   1   0   1   0   1   <=   75   .0095
constraint9   0   -.7   0   .3   0   .3   >=   0   -.005
constraint10   0   -.4   0   .6   0   -.4   >=   0   -.005
NEW Constraint 11   0   -.2   0   -.2   0   .8   >=   0   0
Solution->   55   22.5   30   30   15   22.5       2.3375  
The range for the optimal solution is as follows

note: Bag 2 can have maximum of 75 candies and 30% of that is 22.5 that is why solution is showing 22.5 each for cinnamon and root beer candies, grapes being at least 40% has to be 30 pieces. Therefore cinnamon should be 22 and root beer should be 23 for rounding.

Variable   Value   Reduced Cost   Original Val   Lower Bound   Upper Bound
c1   55   0   .02   .015   Infinity
c2   22.5   0   .015   .01   Infinity
g1   30   0   .01   -.0442   .02
g2   30   0   .005   -.0188   .01
r1   15   0   .015   -.0933   .02
r2   22.5   0   .01   .005   .015
Constraint   Dual Value   Slack/Surplus   Original Val   Lower Bound   Upper Bound
constraint1   0   25   75   -Infinity   100
constraint2   .0163   0   100   75   Infinity
constraint3   0   15   0   -Infinity   15
constraint4   -.01   0   0   -30   15
constraint5   -.005   0   0   -15   15
constraint6   0   15   0   -Infinity   15
constraint7   0   25   50   -Infinity   75
constraint8   .0095   0   75   50   Infinity
constraint9   -.005   0   0   -7.5   22.5
constraint10   -.005   0   0   -30   7.5
NEW Constraint 11   0   7.5   0   -Infinity   7.5
Final solution Bag 1 should have maximum number 100 candies out of which at least 30% grape so grape flavor 30 in number , 15% next best that is 15 of root beer and 55 highest number for the highest profit margin giver cinnamon.

Bag 2 should also have maximum number of 75 out of which 30 ( minimum 40%) for grape and 22 ( no more than 30%) of cinnamon and remaining 23 of root beer,

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