Optimizing Production of Chemicals Firm using Linear Programming Technique

doi:10.21203/rs.3.rs-3227561/v1

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Optimizing Production of Chemicals Firm using Linear Programming Technique

https://doi.org/10.21203/rs.3.rs-3227561/v1

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This paper is an insight Linear Programming problem for different products produced by PASVEP Chemicals Production Firm and its scare resource allocation. For the purpose, the linear program model was developed with the definite objective function and adequate constraints available to the Firm. The study conducted because the firm has optimization issues in production, and therefore needed optimal solution. The result from linear mathematical model using TORA software rightly identified the optimization problem arises due to inefficient allocation of scare resources. The application of sensitivity analysis, in this regard, provided solution where to resources divert to most needed raw material to get optimal level of production. Based on outcome, the study recommended to use data-driven decision-making methods like linear programming being first and successful application in chemical industry to optimize its resource allocation. The study also recommended that the firm should continue the production of all four products, however, with more focus on PVS solution to further enhance its output.

Chemicals

Linear Programming

Optimal Solution

Optimization

PASVEP

It is more efficient and effective for an organization to achieve its profit goals when scarce resources are allocated properly. Kilonzo [20] provided that the economy will clearly improve if sectors maximize productivity while using little resources. According to Akpan and Iwok [1], a firm's management decisions must result in higher output through cost minimization or output maximization, which increases the production, for an economy to grow. If costs are not kept to a minimum, it will be harder for the firm to maximize profit. However, it is important for a production manager to choose the best approach to distribute few resources in a way that would result in increased productivity and profit in order to reduce costs as presented by Esteban & Ray [10]. Differentiating between products that generate a profit and those that generate a loss is necessary. Since too much downtime on the production floor tends to waste resources there, plant managers must reduce downtime there.

Reducing cost and increasing profit is the concern of almost every firm, especially when the available resources are limited. For this purpose, many approaches are used, one of them being linear programming or mathematical programming [4, 27, 21]. It is used to identify inefficiency in the utilization of resources with the help of input and output analysis. David [8] pointed out that in the subjects of statistics and mathematics, linear programming is employed to optimize linear functions that are constrained. Therefore, it is made possible through linear programming to achieve the required objectives of profit maximization and cost minimization, and identifying the best possible solution, for the utilization of available resources. Linear programming in research may be considered as new development, but the objectives of concerns about increasing profit and reducing cost is not. Nearing & Tucker [25] demonstrated that most of the time, the management of manufacturing and production concerns, have been striving to maximize profit and reduce cost.

Application of Linear Programming can be seen in many sectors like Industrial, Government, Transportation, Military and civil engineering. Linear Programming in this regard is used as an essential part of a calculated plan, getting best possible solution for the problem of nonlinear programming, optimal control and making arrangements for emergency situation [22, 26]. The Science of Operations research may clarify and solve many sorts of problems, however numerous viable issues are considered as Linear Programming to get the best possible outcome with the given constraints. Fagoyinbo et al. [11] was on view that a significant method in operations research has an impact on many different areas of optimization, including duality, decomposition, and the significance of convexity and its generalizations. Linear Programming has a crucial role in marketing planning, financial planning, production scheduling, blending problems, portfolio selection, media selection etc. for the purpose to maximize the profit or to minimize the loss with the given available resources [6, 12].

The main objective of this research work is to determine the optimal level of production of four products manufactured by PASVEP chemicals production firm in province Khyber Pakhtunkhwa. For the purpose, Linear Programming system was applied for the first time by the said chemicals firm, so as to acquire the best possible outcome by using the available scarce resources. The paper also aims to identify the specific product(s) that should be given more devotion to optimize the profit for the firm with the help of linear programming technique.

The application of using linear programming for measuring efficiency and regulate optimum production level is not new. From the literature, it is found that on several occasion, the approach is used on various sectors. Such as, Ekwonwune and Edebatu [9] used the simplex method in linear programming to optimize the output of brewed beverages for Nigeria's Golden Guinea Breweries. During the study, a linear programming problem-solving online tool was employed to access the company's profitability. The study shown that, when compared to the outcomes of manual computation, the linear programming approach yields a large return. Some products were found not contributing to the company’s total profit, therefore, it was suggested that these products should not be produced. While other products which are contributing to the cause of profit may continue. Similarly, to maximize the revenue of a small business concern i-e soap making factory, Igbinehi et al. [16] used linear programming model. The factory was making two different types of soap one being white and other being colored. According to the data, the manufacturer was spending more on colored soap, yet white soap was bringing in more money. Based on linear programming findings, the company adopted to producing white soap more, for the sake of profit maximization. The author concluded that linear programming can be used for improving decision making for business while operating. Another maximization of total profit case of an Ethiopian Chemical Company with headquarters there, Maurya et al. [24] developed a linear programming model that suggested a viable product-mix which ensure the optimum profit for Company. The collected data was transformed into Linear Programing model and the Simplex Algorithm and MS-Excel Solver were used to solve the issue in order to produce the best feasible product combination and maximize profit. The result shows that company produces 20 tons of Aluminum Sulphate per day and 6.98 tons of Sulphuric acid per day by getting the maximum profit of 107,353.17 per day for the company.

Iheagwara et al. [17] also focused on the maximization of profit by using the method of Simplex Algorithm of Linear Programming for the Niger Mills Company PLC Calabar, which is a Flour Mill (Production Industry) in Nigeria. The data obtained from the Account Department were converted into the Linear Mathematical form, thus the problem was solved by using available computer software i.e. Excel Solver and TORA. The result shows best available production program with the given constraints for the Niger Mills Company Calabar. Sensitivity Analysis was also applied on the said problem. Similar programming technique was used by Anieting et al. [5] for the purpose of achieving optimum capacity for Usmer Water Company, in Akwa-lbom State in Nigeria. They applied TORA software with big M-method. Results showed different outcome with different values. The results also talked about sensitivity analysis of the problem. Furthermore, in the context of economic crisis, Veselovska [28] discussed that managing the resources optimally is extremely important for industries to survive in economic crisis. For the purpose the author framed two models based on linear programming. One of the model will help in maximizing revenues while the second one will help in minimizing costs of a company.

Gupta and Griglani [13] applied linear programming on manufacturing concern to discuss maximization of profit. For the purpose the researchers selected water packaging industry in India. The constraints that were chosen included employee's time and water containers. Analysis of the data was carries out through MATLAB software and the researchers obtained optimal solution. With the help of linear programming technique, Fagoyinbo et al. [11] conducted research to achieve optimum profit maximization at GEEPEE Nigeria limited. For the purpose revised simplex method was used to solve the problem and acquire solution. The GEEPEE Nigeria Limited is a Tanks production firm, however in the research, as a case study, only four types of tanks were taken into consideration, i.e. Combo, Atlas, Rambo and Jumbo tanks of different sizes. The research conducted is to find out which tank should produce more, so as to maximize the total profit for the firm with the available limited resources of Rubber and Gas used for production of tanks. The result shows that Combo tanks contribute more and gave maximum profit.

Maryam et al. [23] discussed that the application of linear programming may be relatively new, however the approach does help in achieving optimum decisions while making business related strategies. It is particularly helpful in allocation of resources, inventory management, planning production and reducing cost. According to Joly [18], the basic objective of optimization in the oil industry is to push production processes or operations towards the maximum profit until they reach a point at which any further increases in profitability necessitate changes to the present system. Optimization is a key science for high-performance refineries. Al-Shayea [3] used similar approach to identify inefficiency in the use of available resources in King Khalid University Hospital. Similarly Haron and Chellakumar [15] used linear programming technique Data envelopment analysis for the performance of small, large and medium enterprises in Kenya. In the fields of oil exploration, the technique was used by Al-Najjar and Al-Jaybajy [2]. Furthermore, Linear programming techniques is one of the revolutionary method that has helped mankind to achieve highly desirable objectives with scarce resources was argued by Karagiannis and Apostolou [19]. They noted that very complex production problems can be solved with linear programming techniques very simply by developing inter-relationship of objective functions for the purpose of profit maximization.

In conclusion, the literature evaluation has, offered a thorough assessment of the present status of research. The literature does, however, point up significant study gaps that should be filled. For the optimal production of various goods in the chemical sector, linear programming to be implemented and proven effective. By bridging the gap, linear programming and its ramifications will be better understood overall.

The very basic situation of mathematics has unitary functional objective either maximization of profit or minimization of loss or cost, which are subject to constraints that includes resources, procedures, time etc. The function and objectives in linear programming are in linear form.

Linear Objective Function (Maximize profit or Minimize cost):

$$Z=C1X1+C2X2+C3X3+\dots +CnXn$$

Linear Constraints:

$$a11X1+a12X2+a13X3+\dots \dots \dots +anXn \left(\ge \text{o}\text{r} \le \right) RHS1$$

$$a21X1+a22X2+a23X3+\dots \dots \dots +anXn \left(\ge \text{o}\text{r} \le \right) RHS2$$

$$a31Xm1+a32Xm2+a33Xm3+\dots \dots \dots +anXn \left(\ge \text{o}\text{r} \le \right) RHS3$$

Non-Negative Constraint:

$$X1, X2, X3>0$$

Any problem whose mathematical formulation fit this general model is linear program problem. The technique of linear programming is used widely to solve problems and thousands of variables can be accommodated. However, the use of variables is limited and represent constraints. It is important that objective function and constraints are related linearly viewed by Gupta & Hira [14]

For the purpose of application of linear programming, it is important to define the objective of the firm. And in most cases, it is either maximizing the profit or minimizing the cost. Therefore, with the available resources and existing outputs, and objective function is drawn.

In the above model, 2 to 4 are Constraints. These expressions specify elements of problem such as lack of available resources, or challenges such as meeting certain demand.

The xi variables in the above given model are decision variables; that is, they are the variables whose value is determined when the Linear Programming model is solved. Their values provided the answers that are being sought in the Linear Programming analysis. Such model needs data to establish the values of the decision variables.

The input data constants are often referred to as Parameters. The “a” and “c” in the above given model are all parameters of the model. While, condition 5 is called non-negativity conditions [7].

Following assumptions are made, while drawing on the objective function.

That the available resources needed for production are limited.
Optimum allocation of available resources will help in achieving optimum results i-e maximizing profit and minimizing cost.
The qualities of the raw material that is used, is assumed to be standard, and not inferior.

3.1. Problem Formulation of PASVEP Chemicals Production Firm

PASVEP chemicals Production Company produce three different types of chemical products i.e. Varnish Paint, PVC Solution and Acrylic and import one product i.e. Enamel Paint. These products produce from the mixture of different types of raw materials and different phases of the machine.

The accounts department of the company, provided that to produce 36 kg of Varnish Paint requires 12 liters of kerosene oil, 24 kg of varnish resin, 200 gm of Petrosin and 10 gm of Cobalt, while to produce 80 kg of PVC Solution requires 65 kg of cyclohexanone, 11 kg of PVC resin, 10 kg of Ethyl acetate and 500 gm of color powder. Furthermore, to produce 100 kg of Acrylic requires 100 liters of filter water and 10 kg of Polyvinyl. The monthly availability of the raw material is given in the below table:

Table 1

Monthly availability of Raw Material
Raw Materials	Monthly Availability
Kerosene Oil	100 liter
Cyclohexanone	950 kg
PVC resin	198 kg
Ethyl acetate or MEK	180 kg
Color powder	10 kg
Filter Water	600 liter
Polyvinyl	60 kg
Varnish resin	74 kg
Petrosin	2 kg
Cobalt	2 kg

The product Varnish Paint is produces and came to final shape through different machines processes, i.e. Drilling/Mixing, Filling and Packing/scaling. To produce 36 kg of varnish paint requires 40 minutes for Drilling/Mixing. After Drilling/Mixing process, the same quantity requires 50 minutes for Filling and at the end, the same quantity requires 20 minutes for Packing and scaling. To produce 80 kg of PVC Solution it requires 70 minutes for Drilling/Mixing and the next stage is filling, which requires 90 minutes for filling and 30 minutes for packing and scaling. Furthermore, to produce 100 kg of Acrylic requires 100 minutes for Drilling/Mixing and 70 minutes for filing and at the end it requires 40 minutes for packing and scaling.

To estimate the limits of time availability of all three machines are as follows:

5 hours (per day) * 60 minutes = 300 minutes (per day), we get, 300 minutes * 22 days = 6600 Minutes available per month.

The company also imports enamel Paint and mixes some chemicals into it and sale the high quality Enamel Paint. For 16 kg of High Quality Enamel Paint requires 14 kg of enamel paint with monthly availability of 368 kg, 2 liter of Kerosene oil with monthly availability of 100 Liters and 2 kg of varnish with monthly availability of 46 kg. On machine, Drilling and mixing require 20 minutes, while in next stage, filling requires 40 minutes and at the end Packing/scaling requires 20 minutes to get 16 kg of High-Quality Enamel Paint.

The Profit generated from the sale of Varnish Paint, PVC Solution, Acrylic and High-Quality Enamel Paint per kg is Rs. 235, 540, 61 and 300, respectively.

3.2. Formulation of Linear Programming Model

For the purpose of formulation of linear programming model, we let the following;

$X1$ : Quantity of Varnish Paint.

$X2$ : Quantity of PVC solution.

$X3$ : Quantity of Acrylic.

$X4$ : Quantity of High Enamel Paint.

The mathematical form of the problem is given below, to determine which product the company should produce more so as to maximize the total profit.

The Linear Objective Function:

$$Maximize=235X1+540x2+61x3+300x4$$

Subject to:

Raw Materials Constraints:

Kerosene Oil:

$$12X1+2X4 \le 100$$

Varnish Paint:

$$24X1 \le 74$$

Petrosin:

$$.20X1 \le 3$$

Cobalt:

$$.10X1 \le 2$$

Cyclohexanone:

$$65X2 \le 950$$

PVC Resin:

$$11X2 \le 198$$

Ethyl Acetate:

$$10X2 \le 180$$

Color Powder:

$$.5X2 \le 10$$

Filter Water:

$$100X3 \le 600$$

Polyvinyl:

$$10X3 \le 60$$

Enamel Paint:

$$14X4 \le 368$$

Varnish:

$$2X4 \le 46$$

Machine Time Constraints:

Drilling/Mixing: $40X1+70X2+100X3+20X4\le 6600$ (19)

Filling: $50X1+90X2+70X3+40X4\le 6600$ (20)

Scaling/Packing: $20X1+30X2+40X3+20X4\le 6600$ (21)

Non-negative Constraints:

$$X1, X2, X3, X4 \ge 0$$

In general, a linear programming model has linear objective function having equations that has inequalities. The selected problem, therefore, has all the characteristics that are needed by linear programming model, making this case a linear programming problem.

In this study, a developed linear program was used to successfully handle and optimize the production issues of a chemical firm using TORA software. The study uncovered the firm's most successful product, outpacing all others. Additionally, outcome of sensitivity analysis offered insightful information that helped determine the best quantity of each raw material, enabling the achievement of ideal production levels. The following table provides a summary of the research's findings and results:

Table 2

TORA Output Solution Summery for PASVEP Chemicals Firm
Variables	X1		X2	X3	X4		Total
Solution	3.08		14.62	6.00	23.00		15882.89
Raw Materials		Monthly Availability				Slack- / Surplus+
Kerosene Oil		100				17-
Cyclohexanone		950				0
PVC resin		198				2.38-
Ethyl acetate		180				1.69-
Color powder		10				0
Filter Water		600				37.23-
Polyvinyl		60				33.85-
Varnish resin		74				2.69-
Petrosin		3				0
Cobalt		2				0
Enamel Paint		368				-46
Varnish		46				0
Drilling/Mixing		6600				4393.59-
Filling		6600				3790.45-
Scaling/Packing		6600				5399.87-

The model was analyzed by using TORA Computer Software for which the result in Table 2 shows the optimum solution for X1 is 724.58 by producing 3.08 quantity of X1. The optimum solution for X2 is 7892.31 by producing 14.62 quantity of X2. The best solution for X3 is 366 with the given quantity of 6.00 and the optimum solution for X4 is 6900 with the given quantity of 23.00. By combining the values of all four products, the PASVEP Chemicals Production firm gets the maximum optimal profit of Rs. 15882.89 as given above.

Similarly, the study witnessed that the available raw materials, such as, Cyclohexanone, Color Powder, Petrosin, Cobalt and Varnish were fully utilized during the production process. While, 17 kg Kerosene Oil, 2.38 kg PVC resin, 1.69 kg Ethyl acetate, 37.23 kg filter water, 33.85 kg Polyvinyl, 2.69 kg Varnish resin and 46 kg Enamel Paint were left un-used. Furthermore, TORA result of Constraints and Machine time are also provided in Table 2. The outcome of which demonstrated that 4393.59 minutes for Drilling/Mixing, 3790.45 minutes for Filling and 5399.87 minutes for Scaling/Packing were allocated more than required.

The above-mentioned linear program's results showed that some of the raw materials used in production were left over. This clearly demonstrated poor management in the PASVEP chemical firm's production department resulted in a probable loss for the company. The scarce resources than were actually needed were spent on the procurement of raw materials, which could have been put to use in other productive endeavors un-necessary. According to the findings, the aforementioned chemical company may use funds from excess raw materials identified by the study to buy raw materials that were fully used for maximum output. The outcome also showed that the time apportioned for the various production stages was severely mishandled by allotting additional time th-at was not required. The allotted time for manufacturing by the firm may be cut in half to minimize costs.

Table 3

Sensitivity Analysis result to Optimize Output
Products	Current Obj. Coefficient	Min Obj. Coefficient	Max Obj. Coefficient	Reduced Cost
Varnish Paint	235	0	Infinity	0
PVC Solution	540	0	Infinity	0
Acrylic	61	0	Infinity	0
High Enamel Paint	300	0	Infinity	0
Materials	Current RHS	Min RHS	Max RHS	Dual Price
Kerosene Oil	100	83	Infinity	0
Varnish resin	74	0	108	9.79
Petrosin	3	0.62	Infinity	0
Cobalt	2	0.31	Infinity	0
Cyclohexanone	950	0	1170	8.31
PVC resin	198	160.77	Infinity	0
Ethyl acetate	180	146.15	Infinity	0
Color powder	10	7.31	Infinity	0
Filter Water	600	0	600	0.61
Polyvinyl	60	60	Infinity	0
Enamel Paint	368	322	Infinity	0
Varnish	46	0	52.57	150
Drilling/Mixing	6600	2206.41	Infinity	0
Filling	6600	2809.55	Infinity	0
Scaling/Packing	6600	1200.13	infinity	0

The output of Sensitivity Analysis for Objective function is given in above table revealed that if objective coefficient for product X1 increases to infinity or decrease to zero, the optimal linear Programming solution will remain unchanged. Similar result can be drawn for other products, i.e. X2, X3 and X4. However, the result of Sensitivity Analysis for Constraints revealed that Kerosene oil decreases to 83, Petrosin decreases to 0.62, Cobalt decreases to 0.31, PVC Resin decreases to 160.77, Ethyl Acetate decreases to 164.15, Color Powder decreases to 7.31, Polyvinyl decreases to 60, Enamel Paint decreases to 322 and all these given materials increases to infinity would have no positive or negative effect on the Objective function value. While, if the quantity of Varnish Resin increase from 74 kg to 108 kg, the total profit will increase by 9.79 per unit. The increase in the quantity of Cyclohexanone from 950 kg to 1170 kg, will also increase the total profit but by 8.31 per unit. Similarly, increase in Filter water to 600 kg, the revenue will improve by 0.61 per unit and increase in Varnish to 52.57 kg, will raise the total profit by 150 per unit.

Contrary to above, allowable increase or decrease in the machine time (Drilling, Filling and Packing) would have no positive or negative effect on the objective function value, as given in Table 3.

The result of sensitivity analysis identified the problem of inefficiency in PASVEP chemical firm. The ineffectiveness is witnessed by the fact that the firm has shortage of raw materials, such as, varnish resin, Cyclohexanone, filter water and varnish. While, the remaining raw materials were available in excess shows mishandling within firm. The result of sensitivity analysis in regard to varnish resin and filter water is in contradiction with the outcome of linear program given in Table 1. However, the shortage of raw materials for production, such as, Cyclohexanone and varnish were identified in both Table 1 and 2. The outcome of the study represented that the firm must allocate the resources in such a way that optimal production level is attained.

In the current study, production and mixing problems of a chemical firm were solved, so as to get the optimal output. For the purpose, a mathematical linear program was developed based on the data obtained from PASVEP chemical firm and a linear objective function was set with the given linear constraints. The mathematical form model of chemical firm data was then solved by using TORA computer software. From the result, the research was able to determine that the firm's main issue was wasteful allocation of resources towards the procurement of specific raw materials. As a result, the company is currently experiencing a lack of several essential raw materials. The firm's profits would have increased if these raw materials had been bought in a timely manner. The study used sensitivity analysis to determine the best potential solution by identifying the ideal quantities for each raw material in order to reach optimal production levels. The analysis also showed that the PVC solution, as a company product, makes a larger profit contribution to the company than any other product in their line-up.

The model, which demonstrated the enormous potential of this mathematical technique in optimizing production level and boosting overall efficiency, was a successful application of the operations research technique in the PASVEP chemicals production firm. The results of this study show how using linear programming in production planning and scheduling has considerable advantages. The current case study provides a useful example for chemical industry by illustrating the effects of using linear programming in the real world for resource allocation and decision-making.

From the result, the study suggested the following recommendations for the PASVEP Chemicals Production Firm to optimize its resource allocation, mitigate shortages and surpluses.

PASVEP Chemicals Production Firm should produce all the products, as all products contribute to the total optimal profit. But the product, PVC Solution contribute more than any other product and generate high profit for the firm. Therefore, it is recommended that the firm should produce more of PVC Solution to further optimize the revenue.
To optimize resource allocation, use data-driven decision-making methods like linear programming. The company can avoid extreme surpluses and shortages by spotting possible imbalances and modifying purchasing methods accordingly.
Put in place real-time production progress and raw material level monitoring. As a result, the company will be able to react swiftly to changes in demand and make wise choices to deal with any inconsistencies in the allocation of resources.

Funding No external funding.

Data Availability Not applicable.

Code Availability Not applicable.

Declarations

Ethics Approval Not applicable.

Conflict of Interest The authors declare no conflicts of interest associated with this manuscript.

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No competing interests reported.

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Optimizing Production of Chemicals Firm using Linear Programming Technique

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Abstract

1. Introduction

2. Literature Review

3. Methodology and Data

3.1. Problem Formulation of PASVEP Chemicals Production Firm

3.2. Formulation of Linear Programming Model

4. Results & Discussion

5. Conclusions

Declarations

References

Additional Declarations

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