Different stages of a Decision Making Unit (DMU) can be related to each other in series or parallel structures. These stages, sub processes, are very applicable in real world problems. Not considering these stages and their relations, while performance evaluation is being performed, would lead to wrong assessments which are far away from reality. For performance evaluation of an entity with its sub processes, utilizing those techniques which can be shown progress or regress can be helpful. One of these techniques is Malmquist Productivity Index (MIP). In this article considering DMUS each of which has sub process connected to each other in series, MPI technique has been developed in a way that for each sub process efficiency score and MIP index have been calculated. Moreover for each entity the aggregate efficiency score and the aggregate MIP index have also been introduced and calculated.

In some situations the producers desire to maximize total profit of Decision Making Units (DMUs) while the inputs and outputs prices of DMUs change from one time period to another. In this paper, the researchers develop productivity index when producers are going to maximize total profit when the price of inputs and outputs are known.The proposed method uses all price information about inputs and outputs for determining productivity index while previous methods use only cost of inputs to determine index of productivity. Therefore, the proposed Mulmquist productivity index in this paper is more precise than the other Mulmquist productivity index. Here, productivity change is decomposed into profit efficiency and profit technical change. Furthermore, profit efficiency change is decomposed into technical and allocative efficiency change and profit technical change into a part capturing shifts of input and output quantities and shifts of input and output prices. These decompositions provide a clearer picture of the root sources of productivity change.Finally, the proposed fractional programming problems are converted to the linear programming problems. By an illustrative example, we explain the proposed profit Malmquist productivity index.

Data Envelopment Analysis (DEA), a popular linear programming technique is useful to rate comparatively operational effiency of decision Making Unit (DMU) based on the their deterministic input output data. The Malmquist productivity index in DEA, calculable with the distance function, for measurement the productivity change among two variant time period or two variant group in the same time.This index is based on two factor of efficiency change index and a technological change index. In this paper, we operate on the collective Malmquist productivity index, which performs clustering operation DMUs with classification into different levels of efficient frontier, and then we discuss on the relation between Malmquist index on the efficiency layers and their attractiveness and progress.

Data envelopment analysis is a method for evaluating the relative efficiency of a collection of decision making units. The DEA classic models calculate each unit’ s efficiency in the best condition, meaning that finds a weight that the DMU is at its maximum efficiency. In this paper, utilizing the directional distance function model in the presence of undesirable outputs, the efficiency of each unit has been calculated in the best and worst condition and an efficiency interval for each DMU is designated and then with aid from these efficiency interval, we present an interval for each unit with a proportionate Malmquist productivity index, that these intervals indicate the progression or regression of each DMU.

In recent years, measuring and analyzing productivity changes is the main focus of various researches who study performance of organizations. All through widespread application of Malmquist Productivity Index, different types of data should be considered thoroughly, otherwise any defective study of the related data and deciding factors may yield poor results. Practical Malmquist Productivity Index (PMPI) models, presented in this research, are fundamentally capable of measuring the productivity of units in a competitive atmosphere, along with the hidden economic indexes such as time value of money, amortization and promoted skills of employees. Also these models would provide the productivity comparison over different periods of time. Moreover, these models are reliable as well as tangible for superior managers and it is noteworthy that they would offer significantly favorable conditions, lack of which may cause the unit under evaluation to face a great deal of regression.

The index is excellent by the Malmquist index as extended to productivity measurement. The index developed here is defined in terms of input cost rather than input quantity distance functions in supply chain. Therefore, we propose productivity change is decomposed into overall efficiency and cost technical change. These decompositions provide a clearer situation of the root sources of supply chain productivity change, so that illustrated here in a sample of supply chain; so that results are computed using non-parametric mathematical programming.

Data envelopment analysis (DEA) measures the relative efficiency of decision making units (DMUs) with multiple inputs and multiple outputs. DEA-based Malmquist productivity index measures the productivity change over time. We propose a dynamic DEA model involving network structure in each period within the framework a DEA. We have previously published the network DEA (NDEA) and the dynamic DEA (DDEA) models separately. Hence, this article is a composite of these two models. Vertically, we deal with multiple divisions connected by links of network structure within each period and, horizontally, we combine the network structure by means of carry-over activities between succeeding periods. We also introduce dynamic Malmquist index by which we can compare divisional performances over time.

The Malmquist productivity index evaluates the productivity change of a decision making unit (DMU) between two time periods. In this current study, a method is proposed to compute the Malmquist productivity index in several time periods (from the first to the last periods) in data envelopment analysis (DEA) and then, the obtained Malmquist productivity index is compared with Malmquist productivity index between two time periods (the first and the last time periods). The aim of this paper is to investigate progress and regress of decision making units (DMUs) in several time periods considering all time periods between the first and the last one. Consequently, when Malmquist productivity index is computed in several time periods, progress and regress of decision making units can be evaluated more carefully than before. At last, a numerical demonstration reveals the procedure of the proposed method then some conclusions are reached and directions for future research are suggested.