Performance measurement of a production unit is crucial in determining whether it has achieved its objectives or not, improving production efficiency, and dealing with internal or external pressures by monitoring and benchmarking a company’s production [1] - [7] .

Malmquist productivity index (MPI) proposed is a management tool used to evaluate the productivity progress for multi-inputs and multi-outputs [8] - [12] . The MPI represents Total Factor Productivity (TFP) growth of a Decision Making Unit (DMU) and reflects the increase or decrease in efficiency with progress or regress of the frontier technology over time under multiple inputs and multiple outputs framework [13] - [15] . The TFP index can be used to estimate the productivity change, which is decomposed into efficiency change and technological change. The concept of productivity usually referred to labor productivity, this concept is very much related to TFP, defined as the product of efficiency change (catch-up) and technological change (frontier-shift). If TFP value is greater than one, this indicates a positive TFP growth from period (t) to period (t + 1), whereas a value less than one indicates a decrease in TFP growth or performance relative to the previous year. The framework employed in Malmquist can be illustrated in Figure 1, where a production frontier representing the efficient level of output (y) that can be produced from a given level of input (x) is constructed. The assumption made is that the frontier can shift over time. The frontier obtained in the current (t) and future (t + 1) time periods is labeled accordingly. When inefficiency exists, the relative movement of any given DMU over time will, therefore, depend on both its position relative to the corresponding frontier (technical efficiency) and the position of the frontier itself (technical change). If the inefficiency is ignored, then the productivity growth over time will be unable to distinguish between improvements that derive from a DMU catching up to its own frontier, or those that result from the frontier itself shifting up over time.

The input-based Malmquist productivity change index is formulated as [16] :

where M is the productivity of the most recent production point using the period t + 1 technology relative to the earlier production point using period t technology, D is input distance functions, the subscript I indicates CCR input-orientation. A value of greater than unity indicates a

positive total factor productivity growth between the two periods. Alternatively,

In other words, the Malmquist index is

The technical efficiency change (TEC) is given by:

where TE(t + 1) and TE(t) represent the technical efficiency at period (t + 1) and (t) respectively and can be calculated using the DEA model in Equation (5).

Subject to:

where represents the technical efficiency score of unit DMU_o and represents the dual variables that identify the benchmarks for inefficient units. Also, the technological change (TC) is formulated as:

where and represent the intertemporal efficiency indices between t + 1 and t and are calculated respectively as:

Subject to:

Also,

Subject to:

In Equation (2), the TFP index is the product of a measure of technical progress (Technological Change) as measured by shifts in the frontier measured at period t + 1 and period t (averaged geometrically) and a change in efficiency (Efficiency Change) over the same period. Technical efficiency refers to the ability to use a minimal amount of input to make a given level of output. If an organization fails to achieve an output combination on its production possibility frontier, and fails beneath this frontier, it can be said to be technically inefficient. Over time the level of output an organization is capable of producing will increase due to technological changes that affect the ability to optimally combine inputs and outputs. These technological changes cause the production possibility frontier to shift upward, as more outputs are obtainable from the same level of inputs. Thus, for any organization in an industry, productivity improvements over time (more outputs for the same or lower level of inputs) may be either technical efficiency improvements (catching up with their own frontier) or technological improvements (the frontier is shifting up over time) or both. Considering the Constant return to scale (CRS) and variable return to scale (VRS) models will result in four efficiencies: technical efficiency change (TEC), technological change (TC), pure technical efficiency change (PTEC), scale efficiency change (SEC), where all of which will be combined together in the total factor productivity change (TFPC).

Plastics are one of the most used materials on a volume basis in countries’ industrial and commercial life. Plastics are broadly integrated in today’s life style and make a major, irreplaceable contribution to virtually all product areas. A Jordanian company specialized in the production of plastic containers and covers used in food, oil and cosmetics production is interested to assess the performance of its production line during a given period of time. The company started its first production using one injection machine. In order to satisfy the growing demand, it has widened its production with more injection and blowing machines. Currently, the production plant has two types of machines: injection and five blowing machines. The company aims to measure and evaluate the productivity change of five production machines of the same type (blowing machines) M1, M2, M3, M4 and M5 in the company over the period from February to July 2014. This paper, therefore, utilizes Malmquist productivity index to assess the total factor productivity change of the five production machines. The results of this research provide valuable feedback to top managers regarding current improvement decisions and suggest guidelines to future planning. The remaining of this paper including the introduction is organized as follows. Section two describes the data collection and application of MPI. Section three conducts MPI analysis. Section four presents the results and discussions of MPI. The last section concludes the paper.

Figure 2 presents samples of the studied plastic products (covers, pets and containers). The production operation for producing several plastic products in a Jordanian company for plastic industries is depicted in Figure 3.

The data were obtained from the production report over a period of six months (February-2014 to July-2014) for both day and night shifts for the five blowing machine; (M1-M5). Data includes the planned production in units (PP), defect quantity in units (DQ), and idle time in units (IT) and are selected as inputs, whereas the actual production quantity in units (PQ) is set output for each period. Each month was divided into two periods; each period consists of two weeks where (H1) represents the first half of the month and (H2) represents the second half of the month. Inputs and outputs data are represented in Tables 1-5 for M1 to M5, respectively. Table 6 lists the descriptive statistics of the inputs and the output for both day and night shifts.

The input-based Malmquist productivity change index described in Equation (1) is used to analyze the perfor-

mance of the five blowing machines over the period from February to July 2014 in both day and night shift. Firstly, the TE are calculated and presented for the five blowing machines in Table 7.

In Table 7 the TE value (= 0.9771) of the first half of February (Feb. H1) for M1 at day work (M1d1) is calculated using Equation (5) as follows:

Subject to:

Similarly, the TE value (= 0.6394) for M1 in Feb. H2 day working (M1d2) is estimated as follows:

Subject to:

The TE values of the other periods for M1 at day and night shifts are estimated similarly. The TE values at day and night shifts for the other blowing machines (M2-M5) over the period from February-July 2014 are calculated in a similar manner and are also presented in Table 7.

Secondly, the and for the five blowing machines are calculated and then the results are presented in Table 8. For example, the (=1.0346) and (=0.6588) for M1d are calculated using the input and output data shown in Table 9. Mathematically,

Subject to:

Subject to:

The remaining and for M1 and the other blowing machines (M2-M5) over the period from February-July 2014 are calculated in a similar manner and then presented in Table 8 for both day and night shifts.

Utilizing the results displayed in Table 7 and Table 8, the MPI values are calculated as follows. At beginning, the values of technical efficiency change (TEC), which measures the change in efficiency between current (t) and next (t + 1) periods, are estimated.

For example, the TEC value between the first half of February (Feb. H1) and the second half of February (Feb. H2) for M1d is calculated by applying Equation (4) as follows:

The TEC values of M1 for the other periods and the blowing machines from M1-M5 are calculated then presented in Table 10. Next, the technological change (TC) values are calculated for all periods. The TC is the development of new products or the development of new technologies that allows methods of production to improve and results in the shifting upwards of the production frontier as more outputs are obtainable from the same level of inputs. The TC includes new production processes, called process innovation, and the discovery of new

products called product innovation. For illustration, the TC value (= 0.9864) between the first half of February (Feb. H1) and the second half of February (Feb. H2) for M1d is calculated by applying Equation (6) as the following:

The calculated TC values for the blowing machines from M1-M5 are calculated in a similar manner and are also presented in Table 10. Finally, the MPI is used to measure the productivity change of a DMU over time and is calculated by the multiplication of TEC and TC of the same period. For example, the MPI value (= 0.6544) between the first half of February (Feb. H1) and the second half of February (Feb. H2) for M1d is calculated using Equation (2) as follows:

The MPI values for all of the blowing machines at day shift from (M1-M5) are calculated in a similar manner and are displayed in Table 10. Moreover, the TC, TEC and MPI values for all of the blowing machines (M1-M5) are calculated using the data of night shift and then shown in Table 11.

This paper assesses the performance of blowing process for plastic industries using Malmquist Index approach during the period from February to July 2014 for both day and night shifts. Five blowing machines are studied. Two primary issues are addressed in the computation of Malmquist indices of productivity growth. The changes in productivity are divided into technical efficiency change (TEC) and technological change (TC). Data include the planned production in units (PP), defect quantity in units (DQ), and idle time in units (IT) and are selected as inputs, whereas the actual production quantity in units (PQ). Inefficiency is observed in each period. The percentage of input utilization is determined in all periods. Then, the Malmquist productivity index (MPI) values are calculated for all periods. Finally, comparisons of TEC, TC and MPI are conducted among the five machines and between the day and night shifts for each machine. It is concluded that: (1) to improve the technical efficiency, a need for internal training, effective operating procedures, and enhancing quality is required, (2) to increase technological change, figuring out more efficient ways of making existing products or using new technologies allowing output to grow at a faster rate than economic inputs is needed, and (3) with the Malmquist productivity index analysis the company is now able to assess the productivity change of the production machines over time. The results of this research will also help decision makers identify the possible causes of decline in productivity within each production machine and guide them in appropriate proactive/corrective plans.

AbbasAl-Refaie,Mohammad D.Al-Tahat,RubaNajdawi, (2015) Using Malmquist Index Approach to Measure Productivity Change of a Jordanian Company for Plastic Industries. American Journal of Operations Research,05,384-400. doi: 10.4236/ajor.2015.55032