After a false retirement Bill and Jeanne Quain realized their destined action in the plant and shrub business. The need for a high-quality commercial fertilizer prompted the innovation of a blended fertilizer called “Quain-Grow”. Working with chemists at Rutgers University, a mixture was constructed from four compounds, C-30, C-92, D-21 and E-11.
Specifications (i.e constraints) for the mixture demanded that Chemical E-11 must constitute for at least 15% of the blend, C-92 and C-30 must together constitute at least 45% of the blend, and D-21 and C-92 can together constitute no more than 30% of the blend. Lastly, Quain-Grow is packaged and sold in 50-pound bags.
The objective of this analysis is to determine what blend of the four chemicals will allow Quain to minimize the cost of a 50-lb bag of the fertilizer. To do this we have used Linear Programming (LP) – a technique specifically designed to help managers make decisions relative to the allocation of resources. In this case, C-30 = , C-92 = , D-21 = , and E-11 = . The constraints for this case were translated into linear equations (i.e. inequalities) to mathematically express their meaning. The first constraint that C-11 must constitute for at least 15% of the blend can be expressed as: . The second constraint that C-92 and C-30 must together constitute at least 45% of the blend can be expressed as: . The third constraint that D-21 and C-92 can together constitute no more than 30% of the blend can be expressed as: . Lastly, the fourth constraint is that Quain-Grow is packaged and sold in 50-lb bags can be expressed as: . These equations were obtained and entered into a POM LP as a minimizing function. The objective function of this case was calculated and expressed as .
These results show that we can recommend the following ratios of C-30, C-92, D-21 and E-11 respectively so that the cost of a 50-lb bag of fertilizer is minimized: 7.5 lbs, 15 lbs, 0 lbs and 27.5 lbs. In checking to see if these align with the given restraints we found the following to be true; ; ; and . The actually cost result of this minimization analysis was calculated to be $3.35 per 50 lb bag of fertilizer. The equation for this result is as follows: . Additionally, we performed a sensitivity analysis to project how much our recommendation may change if there are changes in the variables or input data. This type of analysis is also called postoptimality analysis. There are two approaches to determining just how sensitive an optimal solution is to changes: (1) a trial-and-error approach and (2) the analytic postoptimality method. In this case analysis we used the analytic postoptimality method.
After we had solved the LP problem, we used the POM software to determine a range of changes in problem parameters that would not affect the optimal solution or change the variables in the solution. While using the information in the sensitivity report, it is pertinent to assume the consideration of a change to only a single input data value at a time. This is because the sensitivity information does not generally apply to simultaneous changes in several input data values. Our main objective when performing this analysis was to obtain a shadow price (or dual value) – the value of one additional unit of a scarce resource in LP. In any scenario, the shadow price is valid as long as the right-hand side of the constraint stays in a range within which all current corner points continue to exist.
The information to compute the upper and lower limits of this range is given by the entries labeled Allowable Increase and Allowable Decrease in the sensitivity report. Our results from the sensitivity analysis were produced in two parts. The first shows the impact of changing the objective function coefficients on the optimal solution and gives the range of values (lower and upper bound) for which the optimal solution remains unchanged. The second part of the report shows the impact of changing the R.H.S of the constraints of the objective function value, with the help of Dual Value (Shadow Price), with the lower and upper bounds for which the shadow price is valid.
Lastly, these results explain that the price of C-30 can vary within the range of .09 to Infinity without affecting the optimal solution. Likewise the range for C-92 is between –Infinity and .12, the range for D-21 is between 15 and 42.5, and the range for E-11 is between 30 and Infinity. The second part of this sensitivity analysis show the ranges for which the shadow prices are valid. Constraint 1 has a dual value of 0 and is valid between –Infinity and 27.5. Constraint 2 has a dual value of -.08 and is valid between 15 and 42.5. Constraint 3 has a dual value of .03 and is valid between 0 and 22.5. Finally, Constraint 4 has a dual value of -.04 and is valid between 30 and Infinity.