Programming

Linear Programming

 

 Introduction

Supply chain management is a significant process of the business process. It is also known as a combination of the procurement, production, storage and delivery of the goods. The concept of the supply chain management can be seen in the each sector of management. In the business process, it can be also seen that companies or management face the problem in the regard of the different process such as procurement, production, storage and delivery of the goods. In order to solve these issues, management takes the help of the linear programming. It is a method to achieve the best outcome in the business process by the maximizing the profit profitability and minimizing the cost (Higle & Sen, 2013). Along with this, linear programming is a special case of mathematic programming. It is one of the simplest ways to perform optimization in the business activity. It is helpful to solve some very complex problem in the operation and provide the possible solution.

Answer 1

In the provided case study of Mecha Wire Works, it is found that the main problem of the company is to identify which order should procedure first and which order can be held. At the same time, it is found that the company is growing and its market demand is increase. But, at the same time, the company has not enough employees or labour so that company can complete its production process effectively

As the principle of the lineal programming that a company uses the linear programming for two purposes such as profit maximisation and cost reduction (Vanderbei, 2015). In the context of Mecha Wire Works, it has be assumed that objective of utilisation of the linear programming is to profit maximisation. In order to solve the problem of the given case study, the following assumptions are developed:

In order to profit maximisation, the production units of the Mecha Wire Works are:

Product W0075C = X1

Product W0033C = X2

Product W0005X = X3

Product W0007X = X4

At the same time, the given information also depict that the demand or order to the following products:

Order for Product W0075C = X1 = >= 150 Units

Order of units for Product W0007X = X4 = >= 600 Units

In the context of the plant capacity, following information is provide in the case study of the  Mecha Wire Works:

Plant name Capacity
Drawing<= 4000 hours
Extrusion<= 4200 hours
Winding<= 2000 hours
Packaging<= 2300 hours

The each product provides some profit to the company after the deduction of the cost. The profit from the each product is calculate in the below manner.

ProductmaterialLaborOverheadTotal expenseSelling priceProfit
W0075C$33$9.90$23.10$66$100$34
W0033C$25$7.50$17.50$50$80$30
W0005X$35$10.50$24.50$70$130$60
W0007X$75$11.25$64.75$150$175$25

 

 

Mecha Wire Works
W0075CW0033CW0005XWOOO7XLHSOPERATORRHSSLACK
Object function3430602559900
(Max. profit)drawing12012200<=40001800
Extrusion11411950<=42002250
Winding13001850<=2000150
Packaging10322300<=23000
Solution valuw11002500600

W0075CW0033CW0005XWOOO7X
Number of units1,100.00250.000.00600.00
Profit$34$30$60$25$59,900.0 
Constraints
Drawing time121$2,200.0<=4000
Extrusion time1141$1,950.0<=4200
Winding time13$1,850.0<=2000
Packaging time132$2,300.0<=2300
W75C orders1$1,100.0<=1400
W33C orders1$250.0<=250
W5X orders1$0.0<=1510
W7X orders1$600.0<=1116
Minimum W75C1$1,100.0>=150
Minimum W7X1$600.0>=600
LHSSignRHS

Total Maximum Profit
34X1 + 30X2 + 60X3 + 25X4Profit for product X1 per unit = 34
Equation Developed Using the ConstraintsProfit for product X2 per unit = 30
Drawing constraintProfit for product X3 per unit = 60
1X1 + 2X2+ 0X3+ 1X4 <= 4000Profit for product X4 per unit = 25
Extrusion constraint
1X1 +1X2 + 4X3 + 1X4 <=  4200Plant capacity (Hours)
Winding ConstraintDrawingExtrusionWindingPackaging
1X1 + 0X2 + 3X3 + 2X4 <= 23004000420020002300
Packaging ConstraintRequired hours/unit
1X1 + 0X2 + 3X3 + 2X3 + 2X4 <= 2300ProductDrawingPackagingExtrusionWinding
X1 >= 150;  X4 >= 600
X1 <= 1400W0075C1111
X2 <= 250W0033C2013
X3 <= 1510W0005X0340
X4 <= 1116W0007X1102

On the basis of the results, it can be recommended to Jon smith that it should produce the following quantity in order to achieve the optimum profit. It will be helpful to increase the profit that company wants to achieve (Plan & Vershynin, 2013). It will be also effective to satisfy all limitations:

W0075C = 1100 (in unites) (It is >= 150)

W0033C = 250 (in units)

W0005X = 0 (in units)

W0007X = 600 (in units) (It is >= 600)

Along with this, it is also found by the production such unites, the company will be able earn total net profit $59900.

In this, it is found that packaging hours will be used fully. But at the same time, the remaining hours will be unused including Drawing, extrusion.

Answer 2

Sensitivity analysis is a significant technique that is useful to demonstrate impact of an independent variable on the dependent variables in the given data set. This analysis technique is used in used in the some specific situations that depend on one or more input variable where requirement is regarding to measure the effect of change on other variables (Dantzig, 2016). For example, Roy is a sales manager in a company and it want to evaluate the impact of customer traffic on the sales of the company. Roy states that sale is an activity that contains that transaction of the price and product. In this, the price of product is $1000 and Roy team sole 100 unites. Hence, the total sales are $100000. Roy also states that a 10 percent increase leads to 5 percent in the total sales that allows Roy to develop a financial model and sensitivity analysis. In this, the question of the sensitivity analysis can be develop as that what happen sales if customer traffic increases by 20% 40% and 60%. Based on the assumption, it can be calculated and found that 20%, 40% and 60% increase in the customer traffic equates to an increase in transaction by 10%, 20% and 30%. The findings of the sensitivity analysis determine that sales are highly sensitive to changes in customer traffic.

Answer 3

On the basis of above discussion and finding, it is identified that there is no need of hiring the temporary in the drawing department. It is because hours for drawing are unemployed due to lack of winding and packaging hours. Along with this, Mecha Wire Works also has not the sufficient hours of packaging in order to precede the product after extrusion and drawing. Hence, the final goods will not become carrying out for delivery (Agrawal, et. al. 2014).

Conclusion

From above discussion, it is found the really linear programming is an effective technique that provides the solution of the operation process. It also enables the company to maximise profit and as well as also allows to reduce the cost. It brings effectiveness in the process at the plant.

 

 References

Agrawal, S., Wang, Z., & Ye, Y. (2014). A dynamic near-optimal algorithm for online linear programming. Operations Research62(4), 876-890. 

Dantzig, G. (2016). Linear programming and extensions. UK: Princeton university press.

Higle, J. L., & Sen, S. (2013). Stochastic decomposition: a statistical method for large scale stochastic linear programming. Germany: Springer Science & Business Media.

Plan, Y., & Vershynin, R. (2013). One‐Bit Compressed Sensing by Linear Programming. Communications on Pure and Applied Mathematics66(8), 1275-1297.

Vanderbei, R. J. (2015). Linear programming. Heidelberg: Springer.

 

 

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