where, as we will see, the input and output units are number of parts per monetary unit.
For manufacturers of discrete products a system’s output is the number of parts produced over a certain time frame. The system inputs are those resources needed to acquire and convert raw material into a finished product over that same time frame. Typical resource input comprises labor, capital, material, and energy. Even though each of these inputs is vastly different, they can be expressed in monetary terms. Thus, productivity will be expressed in terms of the number of parts produced per dollar of input (# of parts/$ input). This is a simple and effective means of accessing a manufacturing operation, machine, process, system or facility’s performance.
Obviously, the time frame over which a form of input is measured should be the same as the output that results. For products manufactured in high quantities, the time unit is usually hours; however, time units of days, weeks, or years can be used as well.
A partial productivity (PP) calculation considers only one input (such as labor). It is defined as:
PP = PO/PI,
where
| P O | = | number of parts output from a process in a specified time frame (# of parts/hr); this is often termed the production rate of the process |
| P I | = | amount of money input into the process over the same time frame ($/hr). |
A combined productivity (PC) calculation considers two or more inputs. Accordingly, combined productivity is given by the following equation:
PC = PO/SPI
where
| SP I | = | monetary sum of partial productivity measures input over a given time frame ($/hr). |
Thus,
SPI = PI labor + PI cap + PI mat + PI energy
The following examples demonstrate calculation of partial and combined productivity.
Example 2.1
A manufacturing process can produce 120 parts per hour. The process requires two laborers, each earning $18/hour. What is labor partial productivity of the process?
Solution
The governing equation is:
PP labor = PO /PI labor
The output in parts per hour is
PO = 120 parts/hr
Since there are 2 laborers, the labor input is
PI labor = 2 laborers × ($18/hr)/laborer = $36/hr.
Thus,
PP labor = PO/PI labor = (120 parts/hr)/($36/hr) = 3.33 parts/$.
So here the process can produce 3.33 parts for every dollar of labor input.
Example 2.2
The manufacturing process described in Example 2.1 uses a machine with capital cost of $45/hr. The machine requires 75 kW of power to operate. Cost of electricity is $0.057/kW-hour (kWh). The machine processes 150 lb of material per hour. The material costs $0.45/lb. Using the labor input costs found in Example 2.1, calculate the combined productivity of the process.
Solution
The governing equations are
PC = PO/SPI
SPI = PI labor + PI cap + PI mat + PI energy.
The output in terms of parts per hour was given as
PO = 120 parts/hr.
The labor partial productivity input was determined in Example 2.1 to be
PI labor = $36/hr.
The cost of capital to run the machine is given as
PI cap = $45/hr.
The cost of energy input per hour is found by converting power into energy. For 1 hour of operation the energy used will be:
energy = (power)(time) = (75 kW)(1 hr) = 75 kWh.
Thus, the machine will use 75 kWh for every hour of use. Therefore, the cost of the energy input into the process will be
PI energy = (energy use/hr)(electricity cost) = (75 kWh/hr)($0.057/kWh) = $4.28/hr.
The material cost input into the process is determined by multiplying the amount of material used per hour by the cost of the material:
PI mat = (material use per hr)(material cost) = (150 lb/hr)($0.45/lb) = $67.50/hr.
Therefore,
| SP I | = | PI labor + PI cap + PI mat + PI energy |
| = | $36/hr + $45/hr + $67.50/hr + $4.28/hr = $152.78/hr. |
Correspondingly,
PC = PO/SPI = (120 parts/hr)/($152.75/hr) = 0.79 part/$.
When all the inputs to the system are considered, the process produces less than one part (0.79) for every dollar of input.
In the examples listed above, the process output in parts/hr was provided. Typically this information is determined through calculations. The following section provides mathematical concepts to quantify production and thereby provide a method of calculating output in parts/hr. Subsequent sections demonstrate how to develop input capital costs of automated machines.
2.3 Process Outputs and Mathematical Concepts for Quantifying Production
In order to determine if an automation strategy selected will provide the desired productivity improvements we must first quantify the current and proposed manufacturing process. Or, in other words, we must measure and document each process’s performance. This serves as the output for the productivity calculations. The performance of the automation can then be quantified, and productivity calculations as well as direct comparison of the automation to the existing process can be made.
There is one measure, as observed in the last section, which is of prime importance. That measure is called the production rate (PO) of the process. It is a measure of how many parts are produced over a specific time period, typically expressed in parts per hour. This measures the output of the process. By combining this measure with other factors, several other mathematical quantifying concepts, in addition to productivity, can be examined.