ory Case Study (15 Marks) Bluegrass Farms, located in Lexington, Kentucky, has b
ID: 381631 • Letter: O
Question
ory Case Study (15 Marks) Bluegrass Farms, located in Lexington, Kentucky, has been experimenting with a special diet for its racehorses. The feed components available for the diet are a standard horse feed product, a vitamin-enriched oat product, and a new vitamin and mineral feed additive. The nutritional values in units per pound and the costs for the three feed components are summarized in the following Table, for example, each pound of the standard feed components contains 0.8 unit of ingredient A, 1 unit of ingredient B, and 0.1 unit of ingredient C. The minimum daily diet requirements for each horse are three units of ingredient A, six units of ingredient B, and four units of ingredient C.In addition, to control the weight of the horses, the total daily feed for a horse should not exceed 6 pounds. Bluegrass Farms would like to determine the minimum-cost mix that will satisfy the daily diet requirements Table: Nutritional Value and Cost Data for the Bluegrass Farms Problem Feed Component Ingredient A Ingredient B Ingredient C Enriched Oat Additive 0.2 1.5 0.6 S0.50 2.0 3.00 Cost S0.25 Microsoft Excel Sensitivity Report Adjustable Cells Final Reduced Objective Allowable Allowable Cell Name Value Cost Coefficient Increase SCS3 S SDS3E SES3 A E+30 0642857143 0.425 E+30 147826087 .514 0.000 0.9460.000 0.000 0.25 0.5 E+30 1541 Constraints Final Shadow Constraint Allowable Cell Name Value Price R.H. Side Increase SF$7 Decrease 1.216 0.000 1.959 6.00 -0919 3.000 9.554 4.000 3 0.368421053 1857142857 E+30 6 3.554054054 0.875 SF$9 FS1O a. Develop a LP model. What is the optimal solution, and wht is the total profit? What 0 6 2.478260969 0.4375 is the plan for the use of overtime? b. Referring to the sensitivity report, write down the intervals for profit coefficients and c. Referring to the sensitivity report, explain the Reduced cost and shadow PriceExplanation / Answer
Part A:
Formulation
S = number of pounds of the standard horse feed product.
E = number of pounds of the enriched oat product.
A = number of pounds of the vitamin and mineral feed additive
Objective is to minimize total cost of blend mix.
Min. Z = 0.25S + 0.50E + 3A
Subject to:
Constraint
Equation
Minimum requirement of A
0.8 S + 0.2 E >= 3
Minimum requirement of B
1.0 S + 1.5 E + 3.0 A >= 6
Minimum requirement of C
0.1 S + 0.6 E + 2.0 A >= 4
Maximum weight constraint
S + E + A <= 6
Nonnegative constraint
S, E, A >= 0
Optimal Solution:
S = 3.514, E = 0.946, and A = 1.541
Total optimal Cost = 0.25S + 0.50E + 3A = 0.25(3.514) + 0.50(0.946) + 3(1.541)
Total optimal Cost = $5.97
Part B:
Sensitivity Report:
Decision Variable
Obj. Cost coeff.
Allowable Increase
Allowable Decrease
Maximum Range
Minimum Range
S
0.25
infinite
0.6428
Infinite
0.25-0.6428
= - 0.3928
Cost cannot be negative, thus minimum cost range is zero.
E
0.5
0.425
Infinite
0.5+0.425 = 0.925
Zero
A
3
infinite
1.4782
Infinite
1.5218
Decision Variable
Cost range within which the optimal solution remains same
S
Zero to infinite
E
Zero to 0.925
A
1.5218 to infinite
Constraint
Constraint R. H. Side
Allowable Increase
Allowable Decrease
Maximum Range
Minimum Range
Minimum requirement of A
3
0.3684
1.8571
3+0.3684
= 3.3684
1.1429
Minimum requirement of B
6
3.5541
Infinite
6+3.5541
= 9.5541
Infinite or zero
Minimum requirement of C
4
0.875
1.9
4.875
4-1.9
= 2.1
Maximum weight constraint
6
2.4782
0.4375
8.4782
5.5625
Constraint
Resource range within which the optimal solution remains feasible
Minimum requirement of A
1.1429 to 3.3684
Minimum requirement of B
Zero to 0.95541
Minimum requirement of C
2.1 to 4.875
Maximum weight constraint
5.5625 to 8.4782
Part C:
For LPP problem, the reduced cost of the decision variable is zero if it is included in optimal solution otherwise it is positive or negative value. For minimization problem the reduced cost value represents the reduction in the objective cost coefficient so that variable will be included in optimal solution. For given solution all variables have zero reduced cost value, it means all the variables are considered in optimal solution.
For the shadow price of constraint represent the effect on the objective function value by the change in the unit value of constraint RH side. For the constraints which are completely utilized have shadow price more than of less than zero. Constraints those are not completely utilized have zero shadow price value. The shadow price value for Ingredient A is 1.216, means if minimum requirement is increased by one unit of A, the objective function value will increase by $1.216. If the minimum requirement is decreased by one unit, the objective function value will decrease by $1.216 but within the allowable range of resources.
Constraint
Equation
Minimum requirement of A
0.8 S + 0.2 E >= 3
Minimum requirement of B
1.0 S + 1.5 E + 3.0 A >= 6
Minimum requirement of C
0.1 S + 0.6 E + 2.0 A >= 4
Maximum weight constraint
S + E + A <= 6
Nonnegative constraint
S, E, A >= 0
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