KLEIN

LP Modeling

Linear programming (LP) is a common method in which relationship complexities are illustrated through linear functions that find the greatest results, according to Evans (2013). In other words, it is simplifying real relationships to simplified linear relationships. In LP real-life problems are formulated into mathematical models, which includes the objective function, and subject to constraints. To express and solve LP models using the auxiliary variable cell method, an LP model must be set up, based on the instructions from Evans (2013).

To begin with, in the Klein Industries scenario, let S represent small compressors, let M represent medium compressors, and L represent large compressors. The non-negative value will be represented as follows: S, M, L ≥ 0 because negative amounts of units cannot be made. Next, the bending/forming time is equal to 23,400 minutes, in which total bending time cannot surpass the allotted minutes. It produces a constraint of 0.4S +0.7M +0.8L≤ 23400. The same time frame is allotted for welding, in which it also cannot surpass the allotted minutes. It produces a constraint of: 0.6S + 1.0M + 1.2L≤ 23400. Next, painting cannot surpass the 46,800 minutes allotted; It produces a constraint of: 1.4S + 2.6M + 3.1L≤ 46800. In addition, there is a minimum as well as a maximum bound of sales units that produce constraints of 1400 ≤ S ≤ 21000, 6200 ≤ M ≤ 12500, and 2600 ≤ L ≤ 4200. Finally, prices per unit in turn are $20.50, $34.00, and $42.00. To maximize the companies’ earnings, the objective function for maximization is 20.50S, + 34M + 42L. The linear programming model was set up to include the objective function and constraints. The model is as follows: Maximize 20.50S, + 34M + 42L. The constraints are: 0.4S + 0.7M + 0.8L ≤ 23400, 0.6S, + 1.0M + 1.2L ≤ 23400, 1.4S + 2.6M + 3.1L ≤ 46800, 14000 ≤ S ≤ 21000, 6200 ≤  M ≤ 12500, 2600 ≤ L ≤ 4200.

Further, a spreadsheet will need to be set up to include all available data, including the aux variables. List the number of compressors and the auxiliary for each type of compressor. The profit is equal to the sum of the small, medium and large product multiplied by their respective prices, $20.50, $34.00, and $42.00. Next set up formulas for multiplying each size compressor for each bending, welding, and painting time. Further, set up the formula for the auxiliary by doing the same.

Next, the solver is then set up to include all the values of each of the variables, the objectives, and the constraints of the linear optimization model. Click the solver and add in the necessary parameters. According to Evans (2013), solver helps you find the best possible solutions to an optimization problem. Since the general idea is to optimize profits, set the objective to represent the cell that the company will want to optimize while also ensuring the max button is checked. By changing variable cells, you can find the maximum unit price of each size compressor. Evans (2013) states those decision variables are what decide the output. In this model, the Constraints box will be the cells that represent the maximum and the minimum number of each size compressor; enter each constraint separately, check Make Unconstrained Variables Non-Negative since all of the decision variables will be greater than zero. Select the Simplex LP since this is a linear optimization problem. According to Evans (2013), when using the Simplex LP option, the solver will find the global best solution. Finally, click the Solve radial button. The solver box will provide a solution if one is found. In the solver results box, keep the solver solution, click on the reports you want to see, in this case, only the sensitivity report will be highlighted and click Ok. The Klein scenario LP analysis revealed a maximum profit of $651,221.42. However, to increase profits to $651,221.42, the company must only manufacture 16,157 of the smaller, 6,200 of the midsized, as well as 2,600 of the larger compressors.

Memo to Production Manager

In the analysis conducted, the solver identified $651,221.42 as the maximum profit that can be obtained if a maximum of 16157 of the smallest units is produced, 6200 of the middle sized units are produced as well as 2600 of the largest. The sensitivity report reveals the values of the midsized and larger units are $4.07, and $3.39 in turn. These are the maximum limits that the compressors can reach without reducing the ideal profit. The smaller units show that there can be a decrease by up to $1.53 also having no change in the profit. Constraints for the lower bound medium unit can be reduced by 2607 and for the lower bound large can be minimized by 2187 units. The report also suggests that the medium and larger units are not lucrative at maximum value, so only produce the minimum value of the medium and larger units in which the numbers have been provided above. In reference to painting, the shadow price shows 14.6429, indicating that an increase in available time to paint would improve profits by $14.64. The report also shows that painting time can be increased by an additional 6780 to reach maximum profit. 

Reduced Cost and Shadow Pricing

In solving the linear programming model without auxiliary variables, the same parameters are used as in the model with auxiliary variables, only do not include auxiliary cells. The sensitivity report shows a reduction in the cost of the medium and larger unit of $4.07 and $3.39 in turn. The lower bound shadow price is also identical. In addition, the shadow price in relation to painting is the exact same for the aux variables as well as the variables without the aux. In other words, for this scenario, whether you add an auxiliary cell or not, it does not change or have any effect on the results. Constraints increased by one on the right side means the shadow price will show the change in the objective function Evans (2013).

Conclusion

In setting up the LP model and using the Excel solver, Klein Industries learned what their maximum profit would be and how many to produce of each type of compressor. The Solver revealed $651,221.42 as the maximum profit. In order to reach that profit, the organization should create 16,157, 6,200, and 2.600 compressors to its respective size. In addition, when comparing an LP model with auxiliary and one without an auxiliary, it is evident that there is no effect on the results. The cost reduction and shadow prices were the same for the medium and larger units.

References

Evans, J. R. (2013). Statistics, Data Analysis, and Decision Modeling: International Edition (5th ed.). Retrieved from https://bookshelf.vitalsource.com/#/books/9780133071405/cfi/31!/4/4@0.00:0.00

Kashyap, S. (2017, February 2). Introductory guide on Linear Programming for (aspiring ) data scientists. Retrieved from https://www.analyticsvidhya.com/blog/2017/02/lintroductory-guide-on-linear- programming-explained-in-simple-english/