# Appraisal of Optimal Production Quantity in Small and Medium Scale Industry

Ezeliora Chukwuemeka, Nwakobi John Oguejiofor, Sunday P Agu

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Ezeliora Chukwuemeka, Nwakobi John Oguejiofor, Sunday P Agu. Appraisal of Optimal Production Quantity in Small and Medium Scale Industry. International Journal of Advanced Engineering Research and Technology (IJAERT), 2017, 5 (1), pp.4-12. hal-05107690

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# Appraisal of Optimal Production Quantity in Small and Medium
Scale Industry

Ezeliora Chukwuemeka Daniel¹, Nwakobi John Oguejiofor², Agu Sunday P.³

Department of Mechanical Engineering, Nnamdi Azikiwe University, Awka, Anambra State, Nigeria¹

Department of Industrial and Production Engineering, Nnamdi Azikiwe University, Awka, Anambra State, Nigeria²,³

Emails: cezeliora@gmail.com¹, jnwakoby@hotmail.com² aguhsunny@yahoo.com³

## Abstract

The benefit and necessities of small and medium scale enterprise as means for developing a resilient growth in economic system has been established. This was revealed in the small and medium scale economic flexibility, economic building and excellent relationship and interactions with several other sectors of the economy. The appraisal of Productivity and its improvement in small and medium scale enterprise develops the economic conditions in the country and the industry on its side. In the cause of the research, the operational research techniques where employed to Golden plastic industry limited (GPIL). The tool employed is a linear programming technique that was used in maximizing profit. Productivity however, improve by applying the linear programming technique instead of applying trial and error technique which has been employed by the managers of the industry. Operation research techniques improve the productivity of the firm and thereby applying the techniques, the economics of developing countries like Nigeria can be enhanced. These techniques can be applied also to the crude oil problem in Nigeria and see an end to many industrial problems faced by small and medium scale enterprise.

**Key words:** Optimization, Bread, Profit, Productivity, Small and Medium Scale Enterprises, Response surface Modelling and Integer Programming

## 1.0 Background of Study

In modern days, industries are concerned with the productivity rate, which is concerned with the effective and the efficiency of the production system. However, in manufacturing industries all levels are faced with the challenges of producing goods of right quality and quantity and at right time and more especially at minimum cost (minimized cost) and maximum profit for their survival and growth. Productivity is defined as a measure of outcomes from production process and production planning, per unit of income in the system. Productivity could be seen as a ratio to measure how well an organization converts input resources (labour, materials, machines etc.) into goods and services.

Thus, this demands an increase in productive efficiency of the industry. In small and medium scale industries there exist lots and lots of business ventures that are being engaged by individuals, group of people or association, firms, industries and government with the main aim of maximizing profits. They range from small scale to medium and large scale. In the Nigerian economy, the small scale enterprises are the most common form of business. The aim of any economy (either industrialized or non-industrialized) depends largely on how well organized the small industries are, for instance, we look at the standard of practice of small scale industries in economically developed countries. The small scale enterprise in Nigeria seems too stagnant, less adventurous than developed countries. Meanwhile in economically developed countries
small scale business are better organized and coordinated than in the developing countries because the governments appreciate their significance to the national economy.

Most of the small businesses are solely operated which makes such companies sole proprietorship business ventures. The profit of the business is reaped by the owner; this includes people or individuals who want to invest in that type of business. It is also the easiest form of business ownership to organize because there is relatively small capital to use for its establishment. The owner also has managerial freedom and this will help the owner in employing his incentive to the maximum. Entrepreneurs are risk avoiders, not necessarily risk takers; but appear to be risk takers because they see market differently than the other forms of businesses. They can be referred to as risk eliminators because they seem to methodically eliminate all the factors that might prevent them from getting into a particular market.

There is no doubt that the relationship between the speeds with which businesses is created and sustained as well as the proximity of financing to such businesses is an essential element in the wealth creation process of developing countries.

Small and Medium Enterprises (SME's) play crucial roles in any nations' economies as job creators, suppliers of basic inputs/parts for larger businesses, and key players in the primary sector of the economy. Increasing productivity of SME's with properly structured approaches and commitments could be profitable, and given the right focus, SME's provide the fillip required to pump-start economic development of any country.

Clifford (1980) emphasized that in view of the large increase in world population especially in Nigeria and many other developing countries, governments can certainly not afford to employ many more of its population. Moreover, the large-scale industries have dramatically failed to absorb more than a tiny fraction of the multitudes that need jobs. SME's are therefore possibly the only scope of employment creation; little wonders why policy makers are so eagerly attempting to promote and encourage them.

The path to economic recovery and growth may require increasing production inputs, land, labor, capital and technology and/or increasing their productivity.

Increasing productivity should be the focus because many other countries that have found themselves in the same predicaments have resolved them through productivity enhancement schemes. Given the importance of high productivity in boosting economic growth and the standards of living of the people, its improvement cannot but be of importance to both Managers and production engineers. Industries all over the world including Nigeria are continuously faced with shortages of production inputs which result in low capacity utilization and consequently low outputs. But an economy can grow if management decisions at the firm level result in boosted output through either cost minimization or output maximization culminating in increased production in the real sector. Thus firm managers are always seeking for the right decisions so as to meet their objectives which mainly revolve on how best to increase profit.

The growth in industries puts pressure on management in finding the optimal planning, organizing, leading and controlling levels of production in the various productive industries of the economy. As a result of this pressure, managerial theories of the firm are introduced to analyze business environments and to solve practical business problems such as operational problems emanating from within the industry, and environmental problems in which the industry operates. Both theoretical and quantitative techniques are developed to model and analyze these decision problems.
One decision area for the management of any firm is in determining the product mix of the firm. A product mix (also called product assortment), according to Kortler (1993) is the set of all product lines and items that a particular seller or producer offers for sale to buyers. Like most other developing countries of the world, Nigeria is faced with the general nature of scarcity of factors of production. As a result of the increasing cost of production, scarcity of factors and high customers' expectations, the management of industries can hardly do without a well-articulated course of action in their production decisions. This enables them to achieve their profit-maximizing objective.

The pronouncements of policy makers in Nigeria indicate that the aims of government economic policies are to increase the rate of growth of the economy and to increase the contribution (absolute and relative) of the industrial sector to GDP thus diversifying the economy. The contribution of the real sector to the rate of growth in the economy is very minimal. This is due to the sectorial dependence on crude-oil economy. The productive sector is neglected without a planned business environment to encourage national output, thus propping up unemployment, adverse exchange rate, rising factor costs, underutilization of installed plant capacity and rising prices of finished goods, among others. To stay this tide, researchers such as Sargeaunt (1985), Kurtz (1992), Lucey (2000) and Taha (2008) posit that the use of scientific methods, particularly linear programming, in the allocation of scarce resources is of vital importance to the manufacturing industries to boost output.

**The objective** of the study is the application of an optimal model to appraise the productivity of the small and medium scale industry.

## 2.0 Review of Literature

However according to Dantzig (1993) "For the purpose I had in mind, however, I saw that Leontief's model had to be generalized. His was a steady-state model and what the Air Force wanted was a highly dynamic model, one that could change over time. In Leontief's model there was a one-to-one correspondence between the production processes and the items being produced by these processes. What was needed was a model with alternative activities. Finally it had to be computable. Once the model was formulated, there had to be a practical way to compute what quantities of these activities to engage in that was consistent with their respective input-output characteristics and with given resources". The activity analysis model Dantzig formulated would be described today as a time-staged, dynamic linear program with a staircase matrix structure. Initially there was no objective function; broad goals were never stated explicitly in those days because practical planners simply had no way to implement such a concept. Non computability was the chief reason, I believe, for the total lack of interest in optimization prior to 1947.

This was the situation prior to 1946. In place of an explicit goal or objective function, there were a large number of ad hoc ground rules issued by those in authority to guide the selection. Without such rules, there would have been, in most cases, an astronomical number of feasible solutions to choose from. Incidentally, "Expert System" software which is very much in vogue today makes use of this ad hoc ground-rule approach. The foregoing about the early development of models took place before the advent of the computer. More precisely, in late 1946, it was obvious that the computer was going to exist.
Dantzig, (1963) formulates a model which satisfactorily represented the technological relations usually encountered in practice. He decided that the ad hoc ground rules had to be discarded and replaced by an explicit objective function. He formulated the planning problem in mathematical terms using a set of axioms. The axioms concerned the relations between two kinds of sets: the first was the set of items being produced or consumed and the second, the set of activities or production processes in which these items would be inputted or outputted in fixed proportions providing these proportions are non-negative multiples of each other. The resulting system to be solved was the minimization of a linear form subject to linear equations and inequalities. The use (at the time it was proposed) of a linear form as the objective function to be extremized was a novel feature of the model. There was also a linear objective to be extremized. Luckily the particular geometry used in his thesis was the one associated with the columns of the matrix instead of its rows. This column geometry gave him the insight which led him to believe that the simplex method would be a very efficient solution technique. He earlier had rejected the method when he viewed it in the row geometry because running around the outside edges seemed so unpromising. He proposed the simplex method in the summer 1947. But it took nearly a year before his colleagues and himself in the Pentagon realized just how powerful the method really was.

However, many authors has different views and to be in resonance with all these views we need to maintain and set up a standard optimization model for production processes that will be workable in bread industries.

Dibua (2004), explains that “Linear Programming model is best applied where a manufacturer wants to develop a production schedule / target and an inventory policy that will satisfy sales demand in future period. Ideally, the schedule and policy will enable the production company to satisfy demand and at the same time minimize the total production and inventory costs”.

Everette (2003) said that Linear programming could be used to provide uninterrupted production by optimizing production processes for efficiency.

## 3.0 Research Methods

The analysis of the data were based of the integer programme and response surface models applied to the data to obtain the maximum quantity and maximum profit in the bread production industry.

<table><caption>**Table 1:** Quantity of Long loaf (D1) Produced per month</caption><thead><tr><th>Year</th><th>Jan</th><th>Feb</th><th>March</th><th>April</th><th>May</th><th>June</th><th>July</th><th>Aug</th><th>Sep</th><th>Oct</th><th>Nov</th><th>Dec</th><th>total</th></tr></thead><tbody><tr><td>2010</td><td>8875</td><td>7944</td><td>9775</td><td>8225</td><td>7225</td><td>8050</td><td>8000</td><td>7625</td><td>6552</td><td>9800</td><td>7848</td><td>7725</td><td>97644</td></tr><tr><td>2011</td><td>7800</td><td>8376</td><td>6050</td><td>10250</td><td>6800</td><td>7800</td><td>10225</td><td>9550</td><td>7032</td><td>7025</td><td>7608</td><td>6400</td><td>94916</td></tr><tr><td>2012</td><td>11625</td><td>10344</td><td>8700</td><td>9775</td><td>9275</td><td>10325</td><td>8500</td><td>7100</td><td>6384</td><td>5850</td><td>7872</td><td>8225</td><td>103975</td></tr><tr><td>2013</td><td>10825</td><td>10248</td><td>9350</td><td>9300</td><td>8625</td><td>8350</td><td>7800</td><td>9575</td><td>9408</td><td>6325</td><td>6840</td><td>7800</td><td>104446</td></tr><tr><td>2014</td><td>6675</td><td>9336</td><td>10800</td><td>9775</td><td>9950</td><td>9775</td><td>9225</td><td>10200</td><td>7488</td><td>6100</td><td>7656</td><td>9350</td><td>106330</td></tr><tr><td>2015</td><td>8725</td><td>7416</td><td>11200</td><td>8200</td><td>9625</td><td>9450</td><td>10250</td><td>8725</td><td>9072</td><td>8725</td><td>7896</td><td>9250</td><td>108534</td></tr></tbody></table>
<table><caption>**Table 2:** Profit made in Long loaf (D1) per month</caption><thead><tr><th>Year</th><th>Jan</th><th>Feb</th><th>March</th><th>April</th><th>May</th><th>June</th><th>July</th><th>Aug</th><th>Sep</th><th>Oct</th><th>Nov</th><th>Dec</th><th>total</th></tr></thead><tbody><tr><td>2010</td><td>177500</td><td>158880</td><td>195500</td><td>164500</td><td>144500</td><td>161000</td><td>160000</td><td>152500</td><td>131040</td><td>196000</td><td>156960</td><td>154500</td><td>1952880</td></tr><tr><td>2011</td><td>156000</td><td>167520</td><td>121000</td><td>205000</td><td>136000</td><td>156000</td><td>204500</td><td>191000</td><td>140640</td><td>140500</td><td>152160</td><td>128000</td><td>1898320</td></tr><tr><td>2012</td><td>232500</td><td>206880</td><td>174000</td><td>195500</td><td>185500</td><td>206500</td><td>170000</td><td>142000</td><td>127680</td><td>117000</td><td>157440</td><td>164500</td><td>2079500</td></tr><tr><td>2013</td><td>216500</td><td>204960</td><td>187000</td><td>186000</td><td>172500</td><td>167000</td><td>156000</td><td>191500</td><td>188160</td><td>126500</td><td>136800</td><td>156000</td><td>2088920</td></tr><tr><td>2014</td><td>133500</td><td>186720</td><td>216000</td><td>195500</td><td>199000</td><td>195500</td><td>184500</td><td>204000</td><td>149760</td><td>122000</td><td>153120</td><td>187000</td><td>2126600</td></tr><tr><td>2015</td><td>174500</td><td>148320</td><td>224000</td><td>164000</td><td>192500</td><td>189000</td><td>205000</td><td>174500</td><td>181440</td><td>174500</td><td>157920</td><td>185000</td><td>2170680</td></tr></tbody></table>

## 4.0 Response Optimization of Long Loaf Quantity

### Parameters

Goal Lower Target Upper Weight Import

Yield Maximum 20000 65000 65000 1 1

### Global Solution

X1 = 11625

X2 = 12096

X3 = 11200

X4 = 10725

X5 = 9950

X6 = 10550

### Predicted Responses

Yield = 66146, desirability = 1.000000

Composite Desirability = 1.000000

<figure><img src="image_2.png" /><figcaption>**Figure 1:** Contour Plots of Yield of Long Loaf Quantity</figcaption></figure>
<figure><img src="image_3.png" /><figcaption>Figure 2: Surface Plots of Yield of Long Loaf Quantity</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Figure 3: Optimization Plot of Long Loaf Quantity</figcaption></figure>

## INTEGER PROGRAMMING B&B TREE

Title: Quantity of Long Bread per month

Subproblem N10

Defining branches:

Objective value, z= 69298.00 Branching variable: x?
The results were developed from the Response surface modelling and Integer programming modelling. The models develop the optimum production quantity of the long loaf of bread in the case study company. The application of response surface model reveals that the optimum production quantity of the long loaf bread is 66,146 quantity of the loaf in monthly bases. However, the application of the integer programming model shows the optimum monthly production quantity of 69, 298.00 units of the long loaf product.

## 5.0 Response Optimization of Long Loaf Profitability

### Parameters

Goal Lower Target Upper Weight Import
Monthly Prof Maximum 80000 124000 124000 1 1

### Global Solution

X1 = 232500

X2 = 241920

X3 = 224000

X4 = 214500

X5 = 199000

X6 = 211000

### Predicted Responses

Monthly Prof = 1322920, desirability = 1.000000

Composite Desirability = 1.000000

<figure><img src="image_5.png" /><figcaption>Figure 4: Optimization Plot</figcaption></figure>

## INTEGER PROGRAMMING B&B TREE

Title: Profit of Long Bread per month
Subproblem N10
Defining branches:

Objective value, z= 1443737.00 Branching variable: x?

The analysis and evaluation of the long bread product was evaluate using response surface algorithm and Integer programming algorithm. The results reveals that the optimum profit made in long bread over any given month using response surface algorithm is ₦1,322,920, while the optimum profit made in long bread over any given month using integer programming algorithm is ₦1,443,737.00. The response surface plot also shows the optimum plot and the current independent variable used to establish the optimum profit of the profit made in the long bread over any given month. It also develops the design new points for the independent variables.

Discussion

The discussion is based on the data collected, analysis of the data, the results developed, the tables and the charts developed in the research work. The research models applied in the cause of this research were Integer programming algorithm, Response Surface Algorithm and the Linear Programming Algorithm. The various optimum values of the optimization models for the aforementioned company were summarized in tables 4.1 and 4.2. In this research work, the researcher develops two types of optimization method. Firstly is the optimization of the production quantity over any given month, while the second method is the optimization of the profitability in the product over any given month. The researcher considers three major products, namely; long loaf, giant loaf and small loaf of bread in the case study company.
In Long loaf product, the results were developed from the Response surface modelling and Integer programming modelling. The models develops the optimum production quantity of the long loaf of bread in the case study company.
The application of response surface model reveals that the optimum production quantity of the long loaf bread is 66,146 (units) quantity of the loaf in monthly bases. However, the application of the integer programming model shows the optimum monthly production quantity of 69, 298.00 units of the long loaf product. The contour plot and the surface plot in figure 4.1 and figure 4.2 respectively showed the impact of the monthly variations in the variables for the six years of study. The response surface algorithm however develops new design points for the dependent variables and also develops the analysis of variance (ANOVA) for the dependent and independent variables.

However, in profitability analysis of the long loaf product, the results revealed the optimum profit made in long bread over any given month using response surface algorithm is ₦1,322,920, while the optimum profit made in long bread over any given month using integer programming algorithm is ₦1,443,737.00. The response surface plot also shows the optimum plot and the current independent variable used to establish the optimum profit of the profit made in the long bread over any given month. It also develops the design new points for the independent variables. However, response surface also observed the analysis of variance (ANOVA) in both dependent and independent variables.

Finally, the results of the optimization techniques applied in the case company has been revealed and the researcher was able to develop the results of the optimum monthly production quantity and optimum monthly profitability of each of the products under investigation. The results developed has yield a great influence to the aforementioned company. Based on
the discussions, the researcher has made a tremendous contribution to academics and especially to the case study company.

**5.1 Conclusion**

The application of the optimization models for optimum production quantity and optimum profitability of the product in the small-scale industry for productive and competitive production is very essential. Having observed that the profitability of the long loaf is ₦1,322,920.00, using response surface algorithm. Using integer programming, the optimum profit of the long loaf is ₦1,443,737.00. However, it was observed that the optimum quantity of the long loaf is 66,146.00 units using response surface algorithm. Using integer programming, the optimum profit of the long loaf is 69,298.00 units. The results developed have given an edge and has tremendously establish a standard for the aforementioned company and especially for optimization of productivity and profitability in the research under investigation.

**References**

[1] Dantzig, G., 1993, "Computational Algorithm of the Revised Simplex Methods", RAND Memorandum RM-1266

[2] Dantzig, G.B.(1963)."Linear Programming and Extensions", Princeton university Press, Princeton, New York.

[3] Dibua E. C. (2004). "Quantitative Approach to Decision making in Organizations", Paulic Publications Ltd. Enugu, Nigeria.

[4] Everette Adams (2003). "Production and Operation Management," Prentice Hall of India.

[5] Kotler, P., 1993, "Marketing Management: Analysis, Planning, Implementation and Control", Prentice-Hall of India Private Limited, New Delhi

[6] Kurtz, D et al, 1992, "Principles of Management", McGraw-Hill Inc., USA

[7] Lucey, T., 2002, "Quantitative Techniques", Book power, London

[8] Murugan, N and S. Manivel, 2009, "Profit Planning of an NGO run Enterprises using Linear Programming Approach", *International Research Journal of Finance and Economics*, Issue 23, 443-454

[9] Sargeaunt, M, 1965, "Operation Research – An introduction", Macmillan New York

[10] Taha, H., 2008, "Operations Research: An Introduction", 110001 PHI Learning Private Limited, New Delhi.