Question

The Surest Concrete Company produces cement in a continuous process.  Two ingredients in the cement are sand, which Surest purchases for $8 per ton, and gravel, which costs $10 per ton.  The other ingredients are lumped together and referred to as “Other”. The cost of the other ingredients is $7 per ton. No more than 40 percent of the cement can be sand, and at least 30 percent of the cement must be gravel.  The maximum amount of gravel available is 100 tons.  The maximum amount available of the “Other” ingredients is 125 tons. The cement sells for $62 per ton. Fractional values of the inputs and outputs are allowed.  

The company wants to maximize its profit.   

a)  Formulate the liner programming problem to maximize the profit.  

b)  Using linear programming, solve for the optimal solution, identifying the value of the objective function and the variables at optimality. To receive credit for this answer, you must provide a copy of your linear programming output, identifying the value of the objective function and the values of the variables at optimality. You must submit your linear programming formulation and show the linear programming software solution to this problem to receive credit.  

c)  Using the sensitivity analysis output from part b,provide any two sensitivity analysis interpretations.  One of the interpretations must relate to the objective function and the second one must relate to one of the constraints.

d) Write the constraints ensuring the following two conditions are satisfied. You only need to formulate these constraints.  You do not need to solve the linear programming problem for the solution.

·       Write the constraint(s) that ensures that the cost of  the “Gravel” ingredient is not less than 25% of the total cost of the cement being provided.

·       Write the constraint that ensures that the amount of the “Other” ingredients is not more than 20% of the amount of all of the ingredients being provided.

Answer 1

a) Let x1= ton of sand to be purchased

x2= ton of gravel to be purchased

x3= tons of others to be purchased

1 ton of cement sells for $62

Revenue= 62(x1+x2+x3)

cost = 8x1+10x2+7x3

So profit to maximize

62(x1+x2+x3)-(8x1+10x2+7x3)

=54x1+52x2+55x3

Constraints

No more than 40% of the cement can be sand

40% of the cement= 0.4*(x1+x2+x3)

x1<= 0.4(x1+x2+x3)

At least 30% of cement must be gravel

x2>=0.3(x1+x2+x3)

x2<=100 tons

x3<=125 tons

x1,x2,x3>=0

b)

solver output

?"- Sign in Data Analysis s olver clear 5. Book1 - Excel FILE HOME INSERT PAGE LAYOUT FORMULAS DATA REVIEW VIEW From Access 21 22 d From Web Properties Reapply i dis From Other Existing Refresh Sort Filter Text to Flash Remove Data Consolidate What If Relationships Group Ungroup Subtotal From Text Source - Connections All Edit Links Advanced Columns Fill Duplicates Validation Analysis Get External Data Connections Sort & Filter Data Tools Outline K12 = X fo S erties Analysis к N O P Q R S X2 54 Objective constraints X3 52 -0.4 0.7 Values 5517925 -0.4 -25 <= 3 0. 0 > 0 100 <= 1 125 = 100 0 125 Solution 108.3333 100 125 Sheet1 READY BE - + 100%

?"- Sign in Data Analysis 2 Solver S Book1 - Excel REVIEW VIEW Clear SE ES E ESE - Text to Flash Remove Data Consolidate What If Relationships Group Ungroup Subtotal Advanced Columns Fill Duplicates Validation Analysis Sort & Filter Data Tools Outline FILE HOME INSERT PAGE LAYOUT FORMULAS DATA From Access Connections From Web From Other Existing Refresh From Text Sources Connections Get External Data Connections K12 = X for eperties Sort Filter Ply Connections ources All Analysis Objective constraints -0.4 -0.4 -0.3 0.7 Values =SUMPRODUCT(G4:14,$G$11:$I$11) =SUMPRODUCT(65:15,$G$11:$1$11) < =SUMPRODUCT(G6:16,$G$11:$1$11) >> =SUMPRODUCT(G7:17,$G$11:$1$11) = =SUMPRODUCT(G8:18,$G$11:$I$11) <= 1 Solution 108.333333333333 100 125 Sheet1 READY BE - + 100%

Solver Parameters Set Objective: $J$4 To: O Max O Min O Value of: By Changing Variable Cells: $G$11:$1$11 Subject to the Constraints: $J$5 <= $L$5 $J$6 >= $L$6 $J$7 <= $L$7 $J$8 <= $L$8 Add Change Delete Reset All Load/Save v Make Unconstrained Variables Non-Negative Select a Solving Method: Simplex LP Options Solving Method Select the GRG Nonlinear engine for Solver Problems that are smooth nonlinear. Select the LP Simplex engine for linear Solver Problems, and select the Evolutionary engine for Solver problems that are non-smooth. Help Solve Close
objective is $17925

optimal values= x1=108.33

x2=100

x3=125

c)

the shadow price of constraint 1 is zero i.e. if we increase the right-hand side by one unit it will affect the objective by zero unit

X ? 4 - 5 Sign in Data Analysis Clear -3 = 2 Solver Book1 - Excel FILE HOME INSERT PAGE LAYOUT FORMULAS DATA REVIEW VIEW From Access mnections 21 A - [From Web Reapply From Other Existing Properties Refresh Sort Filter Text to Flash Remove Data Consolidate What If Relationships Group Ungroup Subtotal Sources - Connections All - Edit Links Advanced Columns Fill Duplicates Validation Analysis Get External Data Connections Sort & Filter Data Tools Outline A1 X F x Microsoft Excel 15.0 Sensitivity Report Analysis E F G H I J K L M N O P Q R ЗА В 1 Microsoft Excel 15.0 Sensitivity Report 2 Worksheet: [Book]]Sheet1 3 Report Created: 8/11/2019 6:22:49 AM 6 Variable Cells Final Value Reduced Objective Cost Coefficient Allowable increase Allowable Decrease Cell Name $G$11:$1$11 15 Constraints 17 18 Cell $1$5 $J$6 $1$7 $J$8 19 Final Shadow Constraint Allowable Allowable Value Price R.H. Side Increase Decrease -250 1E+30 25 0 180 0 32.5 12.5 100 178 100 25 46.42857143 1251 125 108.3333333 2 5 Name constraints Values Values Values Values 24 Sensitivity Report 1 Sheet1 READY E -- + 100%

the second constraint is having allowable increase 32.5 and decrease 12.5

so the constraint RHS can vary between -12.5 to 32.5 without affecting the current solution

d)

cost of gravel of x2 units = $10 x2

total cost of cement= 8x1+10x2+7x3

constraint

10x2>=0.25*(8x1+10x2+7x3)

other ingredients can not be more than 20%

x3<=0.2(x1+x2+x3)

please comment if any doubt

rate me

thanks