Performance Evaluation of Production Lines in a Manufacturing Company Using Data Envelopment Analysis (DEA)

A brewery in Nigeria was considered as a case study with six (6) suggested most productive production lines selected. These lines were evaluated for efficiency using data envelopment analysis as a performance measure.

Relevant data were collected for the three most productive production lines based on the daily production records for one year. The production lines operate on the batch production system. Several products are produced from these lines with the same machines but different local contents. The data collected includes;

1. Available Production Time per month (in hours)
2. Total downtime per month (in hours)
3. Number of rejects and reworks
4. Total Number of Products per run
5. Total Number of runs per month
6. Number of personnel on each line per month

Choice of Inputs, Outputs and Decision-Making Units

Decision-making units (production lines) were chosen based on the perceived most efficient production lines in the system. Non-highly correlated variables were selected as inputs and outputs to forestall redundancy. Input variables which translated to measurable outputs were chosen for the study. These inputs and outputs include:

Input Variables

1. Actual Production Time per month (Hours) (APT)
2. Manpower (M)

Output Variable

1. Number of Products (Good + Rejects+ Reworks) (NP)

The virtual efficiencies were calculated with the aim of drawing the production possibility curve and determining the system benchmark. These metrics were computed using the Frontier Analyst Application tool and evaluated against the most efficient line chosen as the benchmark. The model developed using these inputs and outputs was also solved using the Frontier Analyst Software.

The DEA Model Formulation

The DEA model entails the identification of inputs into systems and the corresponding system output. The following notations hold for the mathematical model:

Notations

$j$ = Number of service units to be evaluated

$\theta$ = Efficiency rating of the service unit to be evaluated

$y_{p j}$ = Amount of output $p$  used by the service unit j

$x_{i j}$ = Amount of input $i$  used by service unit $j$

$p$ = Number of outputs generated by the service units

$m_p$ = Coefficient or weight assigned by DEA to output $p$

$n_i$ = Coefficient or weight assigned by DEA to input $i$

Model Assumptions

For the model development process, the following assumptions hold:

1. Efficiency relative to collected samples can be measured with DEA models.
2. A constant return to scale is assumed (i.e. increase in an input variable causes an increase in the output produced).

Benchmarking

Benchmark for a DEA model is obtained to evaluate the performance of other decision-making units (DMUs) against the selected benchmark. This enables decision-makers to know the extent to which other decision-making units fall short of the benchmark. To obtain the benchmark, equation 1 is used as the objective function. Aggregate inputs are used against the output. A production line with the highest efficiency is used as a benchmark. The production frontier curve is drawn using these aggregated inputs and output.

The Objective Function

An objective function was developed and applied to each of the six lines. The objective function of the model is aimed at maximizing the efficiency of each line in the system subject to the constraint. This is done such that none gives an efficiency greater than 100%. For instance, for any of the production lines, the objective function is derived using Eq. (1) subject to the service unit constraints (Eqns. (3)-(5))

Mathematically we have the following:

Maximize $\theta=\frac{m_1 y_1+m_2 y_2+m_3 y_3+\cdots+m_p y_p}{n_1 x_1+n_2 x_2+n_3 x_3+\cdots+n_i x_i}$     (1)

Generically written as:

Maximize $\theta=\frac{\sum_{i=1}^P m_i y_i}{\sum_{j=1}^P n_i x_i}$     (2)

Subject to:

Service Unit $1=\frac{m_1 y_1+m_2 y_2+m_3 y_3+\cdots+m_p y_p}{n_1 x_1+n_2 x_{21}+n_3 x_3+\cdots+n_i x_p} \leq 1$     (3)

Service Unit $2=\frac{m_1 y_{12}+m_2 y_{22}+m_3 y_{32}+\cdots+m_p y_{p 2}}{n_1 x_{12}+n_2 x_{22}+n_3 x_{32}+\cdots+n_i x_{i 2}} \leq 1$     (4)

Service Unit $p=\frac{m_1 y_{11}+m_2 y_{21}+m_3 y_{31}+\cdots+m_p y_{p 1}}{n_1 x_{11}+n_2 x_{21}+n_3 x_{31}+\cdots+n_i x_{i 1}} \leq 1$     (5)

$m_1, m_2 \ldots m_p>0 ; n_{1}, n_2, \ldots n_i \geq 0$

However, for the problem to be solved, the specific model becomes for each of the lines is derived thus using Eqns. (1)-(5);

Line 1

Maximize $\theta_1=\frac{m_1 \sum_{i=1}^{12} N P_{11}}{n_1 \sum_{i=1}^{12} A P T_{11}+n_2 \sum_{i=1}^{12} M_{21}}$     (6)

Subject to:

For Line $1 \Rightarrow \frac{m_1 \sum_{i=1}^{12} N P_{11}}{n_1 \sum_{i=1}^{12} A P T_{11}+n_2 \sum_{i=1}^{12} M P_{21}} \leq 1$     (7)

For Line $2 \Longrightarrow \frac{m_1 \sum_{i=1}^{12} N P_{11}}{n_1 \sum_{i=1}^{12} A P T_{12}+n_2 \sum_{i=1}^{12} M P_{22}} \leq 1$     (8)

For Line $3 \Longrightarrow \frac{m_1 \sum_{i=1}^{12} N P_{11}}{n_1 \sum_{i=1}^{12} O L E_{13}+n_2 \sum_{i=1}^{12} M P_{23}} \leq 1$     (9)

For Line $4 \Rightarrow \frac{m_1 \sum_{i=1}^{12} N P_{11}}{n_1 \sum_{i=1}^{12} A P T_{14}+n_2 \sum_{i=1}^{12} M P_{24}} \leq 1$     (10)

$m_1,>0 ; n_{1}, n_2 \geq 0$

Line 2

Maximize $\theta_2=\frac{m_1 \sum_{i=1}^{12} N P_{11}}{n_1 \sum_{i=1}^{12} A P T_{12}+n_2 \sum_{i=1}^{12} M P_{22}}$     (11)

Subject to:

For Line $1 \Longrightarrow \frac{m_1 \sum_{i=1}^{12} N P_{11}}{n_1 \sum_{i=1}^{12} A P T_{11}+n_2 \sum_{i=1}^{12} M P_{21}} \leq 1$     (12)

For Line $2 \Rightarrow \frac{m_1 \sum_{i=1}^{12} N P_{11}}{n_1 \sum_{i=1}^{12} A P T_{12}+n_2 \sum_{i=1}^{12} M P_{22}} \leq 1$     (13)

For Line $3 \Rightarrow \frac{m_1 \sum_{i=1}^{12} N P_{11}}{n_1 \sum_{i=1}^{12} A P T_{13}+n_2 \sum_{i=1}^{12} M P_{23}} \leq 1$     (14)

For Line $4 \Longrightarrow \frac{m_1 \sum_{i=1}^{12} N P_{11}}{n_1 \sum_{i=1}^{12} A P T_{14}+n_2 \sum_{i=1}^{12} M P_{24}} \leq 1$     (15)

$m_1,>0 ; n_{1}, n_2 \geq 0$

Line 3

Maximize $\theta_3=\frac{m_1 \sum_{i=1}^{12} N P_{11}}{n_1 \sum_{i=1}^{12} A P T_{13}+n_2 \sum_{i=1}^{12} M P_{23}}$     (16)

Subject to:

For Line $1 \Rightarrow \frac{m_1 \sum_{i=1}^{12} N P_{11}}{n_1 \sum_{i=1}^{12} A P T_{11}+n_2 \sum_{i=1}^{12} M P_{21}} \leq 1$     (17)

For Line $2 \Longrightarrow \frac{m_1 \sum_{i=1}^{12} N P_{11}}{n_1 \sum_{i=1}^{12} A P T_{12}+n_2 \sum_{i=1}^{12} M P_{22}} \leq 1$     (18)

For Line $3 \Longrightarrow \frac{m_1 \sum_{i=1}^{12} N P_{11}}{n_1 \sum_{i=1}^{12} A P T_{13}+n_2 \sum_{i=1}^{12} M P_{23}} \leq 1$     (19)

For Line $4 \Longrightarrow \frac{m_1 \sum_{i=1}^{12} N P_{11}}{n_1 \sum_{i=1}^{12} A P T_{14}+n_2 \sum_{i=1}^{12} M P_{24}} \leq 1$     (20)

$m_1,>0 ; n_{1}, n_2 \geq 0$

The effectiveness of the four systems was compared using the coefficients applied to their inputs and outputs.

Line 4

Maximize $\theta_4=\frac{m_1 \sum_{i=1}^{12} N P_{11}}{n_1 \sum_{i=1}^{12} A P T_{14}+n_2 \sum_{i=1}^{12} M P_{24}}$     (21)

Subject to:

For Line $1 \Longrightarrow \frac{m_1 \sum_{i=1}^{12} N P_{11}}{n_1 \sum_{i=1}^{12} A P T_{11}+n_2 \sum_{i=1}^{12} M P_{21}} \leq 1$     (22)

For Line $2 \Longrightarrow \frac{m_1 \sum_{i=1}^{12} N P_{11}}{n_1 \sum_{i=1}^{12} A P T_{12}+n_2 \sum_{i=1}^{12} M P_{22}} \leq 1$     (23)

For Line $3 \Longrightarrow \frac{m_1 \sum_{i=1}^{12} N P_{11}}{n_1 \sum_{i=1}^{12} A P T_{13}+n_2 \sum_{i=1}^{12} M P_{23}} \leq 1$     (24)

For Line $4 \Rightarrow \frac{m_1 \sum_{i=1}^{12} N P_{11}}{n_1 \sum_{i=1}^{12} A P T_{14}+n_2 \sum_{i=1}^{12} M P_{24}} \leq 1$     (25)

$m_1,>0 ; n_{1}, n_2 \geq 0$

The effectiveness of the four systems was compared using the coefficients applied to their inputs and outputs.

Line 5

Maximize $\theta_5=\frac{m_1 \sum_{i=1}^{12} N P_{11}}{n_1 \sum_{i=1}^{12} A P T_{15}+n_2 \sum_{i=1}^{12} M P_{25}}$     (26)

Subject to:

For Line $1 \Longrightarrow \frac{m_1 \sum_{i=1}^{12} N P_{11}}{n_1 \sum_{i=1}^{12} A P T_{11}+n_2 \sum_{i=1}^{12} M P_{21}} \leq 1$     (27)

For Line $2 \Rightarrow \frac{m_1 \sum_{i=1}^{12} N P_{11}}{n_1 \sum_{i=1}^{12} A P T_{12}+n_2 \sum_{i=1}^{12} M P_{22}} \leq 1$     (28)

For Line $3 \Longrightarrow \frac{m_1 \sum_{i=1}^{12} N P_{11}}{n_1 \sum_{i=1}^{12} A P T_{13}+n_2 \sum_{i=1}^{12} M P_{23}} \leq 1$     (29)

For Line $4 \Longrightarrow \frac{m_1 \sum_{i=1}^{12} N P_{11}}{n_1 \sum_{i=1}^{12} A P T_{14}+n_2 \sum_{i=1}^{12} M P_{24}} \leq 1$     (30)

$m_1,>0 ; n_{1}, n_2 \geq 0$

The effectiveness of the four systems was compared using the coefficients applied to their inputs and outputs.

Line 6

Maximize $\theta_6=\frac{m_1 \sum_{i=1}^{12} N P_{11}}{n_1 \sum_{i=1}^{12} A P T_{16}+n_2 \sum_{i=1}^{12} M P_{26}}$     (31)

Subject to:

For Line $1 \Longrightarrow \frac{m_1 \sum_{i=1}^{12} N P_{11}}{n_1 \sum_{i=1}^{12} A P T_{11}+n_2 \sum_{i=1}^{12} M P_{21}} \leq 1$     (32)

For Line $2 \Longrightarrow \frac{m_1 \sum_{i=1}^{12} N P_{11}}{n_1 \sum_{i=1}^{12} A P T_{12}+n_2 \sum_{i=1}^{12} M P_{22}} \leq 1$     (33)

For Line $3 \Rightarrow \frac{m_1 \sum_{i=1}^{12} N P_{11}}{n_1 \sum_{i=1}^{12} A P T_{13}+n_2 \sum_{i=1}^{12} M P_{23}} \leq 1$    (34)

For Line $4 \Rightarrow \frac{m_1 \sum_{i=1}^{12} N P_{11}}{n_1 \sum_{i=1}^{12} A P T_{14}+n_2 \sum_{i=1}^{12} M P_{24}} \leq 1$     (35)

$m_1,>0 ; n_{1}, n_2 \geq 0$

The effectiveness of the four systems was compared using the coefficients applied to their inputs and outputs.

Frontier Analyst Software

The computation was performed using the Frontier Analyst software. Model data in the form of inputs and outputs were extracted from the Operation Performance Indicator (OPI) worksheet. The constant rate of return model was selected (this reveals that an increase in the number of inputs causes a corresponding increase in the units of outputs produced). The output maximization model was selected from the model option panel. With these in place, the model was run and the results generated were reported in chapter four of this project.

Process Evaluation

Based on the results obtained, evaluation of the lines based on the slacks obtained from the model will be carried out on production lines with low overall performance efficiency ratings and low coefficients of inputs and outputs as obtained from the DEA model and necessary recommendations made.

Performance measurement of decision units in manufacturing industries is fast becoming a concern. In response to this, several techniques are being developed both parametric and non-parametric measures. These methods are being adopted by both researchers and management to evaluate production systems. Data envelopment analysis (DEA) is one of the non-parametric measures of a production system where dimensional inconsistency does not affect output.

Some production lines often appear productive to management, however, when subjected to specific analysis may prove less efficient compared to others in the same system. Two inputs and one output were considered in this analysis: the manpower and actual production time, and the number of products respectively. It was observed that some production lines have more of these inputs than required by the number of products produced per time. This can be seen in the case of product F, product B as well as product A production lines which have more manpower than required. In contrast, some other product lines are observed to be producing less than the most efficient production line. This is the case of product A, product B, product F and product E production line which produces 9.65%, 9.19%, 1.96% and 0.39% respectively less than the most efficient production line.

Additionally, waste in the aspect of manpower is predominant in the system analysed. There is more of the production workforce than necessary compared to the capacity of the plant. To address this, the following recommendations were made that can aid management in decision-making. Even though investment in the area of skills has been made in the workforce in this production system. It is therefore expedient that the excess workforce should be laid off or another product line is created to which they can be deployed based on the desired production level. If considering the financial status of the industry, the man-hour cost of the excess workforce cannot be afforded by the industry, laying off the excess workforce could be a decision that can only be guided through continuous evaluation for performance improvement.