For many years, the planning and scheduling of discrete manufacturing
systems have mostly focused on the management of machines and the decision
making process at the shop-floor level has been addressed by the complete
idea in optimizing the usage of machines. Some of the present researchers
claim to also manage workforce, but accomplish most of the time a local
allocation of operators to machines (Grabot and Letouzey, 2000). For the
corporations to efficiently adapt to new technologies, the interactions
between tasks and human ability must be simultaneously matched. Therefore,
the Just-In-Time (JIT) philosophy (Xiaobo and Ohno, 1997; Hasgül,
2005) is necessarily considered to promote a better balance between the
machine and workforce management into this study. However, this is a combinatorial
problem. Hence, the workforce is defined as the working hours so that
it is introduced to be a continuous function during production into this
study. Besides, the cost of labor is relatively small to the operational
cost of machines; hence, an upper integer of the optimum workforce will
reasonably represent the optimum manpower allocation.
A production line is generally configured by a sequence of workstations
and each workstation consists of one or more parallel machines of the
same type (Lan and Lan, 2000). As modern Computer Numerical Controlled
(CNC) machines are widely used in the computer-based manufacturing systems,
the workforce management is then condensed to the viewpoint of material
handling rather than machine operating in shifting from workshops to Flexible
Manufacturing Systems (FMS) (Wang and Luh, 1996; Kim and Moon, 2007).
Therefore, to appropriately compose the manpower scheme for material handling
among a group of parallel machines surely becomes consequential for operation.
Practically, the loading-unloading workforce on the machine is normally
one of the operator`s jobs and it is considered fixed. Contrarily, the
workforce for raw material and finished parts handling in between the
machines and storage varies with the production rate. Thus, the workforce
for material and part handling among a group of parallel machines is then
mathematically contemplated as a function of production rate into this
Many simple models of workforce allocation have been solved with Kuhn`s
Hungarian Algorithm (Bazaraa et al., 1990). However, they all boiled
down to assigning a number of workers to a number of jobs. Konno and Ishii
(1995) have established a fuzzy analysis to obtain the optimal solution
for a complex workforce scheduling problem. But, without the specificity
of individual abilities from each worker, the mathematical model will
never be adaptable. A modification of genetic annealing was developed
through solving the real size manpower allocation problem (Abbound et
al., 1998; Yong and Yong-quan, 2007); nevertheless, the corresponding
performance of the proposed model was merely verified for small size versions
and the optimal solution might not be obtainable from the real size of
human resources. Thus, the average manpower performance is practically
presented to extend the applicability in this study.
Production control is often modeled as optimization problems for constructing
optimal profits. As the marginal operation cost is a linear increasing
function of productivity (Kamien and Schwartz, 1991), the marginal operation
cost of the machine is also considered to be a linear increasing function
of production rate in this study. It is that the higher production rate
results higher operational cost such as machine maintenance and depreciation.
For most researches with this viewpoint, the production rate is fixed
because of the difficulty in controlling the variable production rate.
Nevertheless, through the modern computer-integrated interface to program
the feed rate with fixed cutting speed and depth of cut on Computer Numerical
Controlled (CNC) machines (Balazinski and Songmene, 1995; Wang et al.,
2007), the production rate is suited of being dynamically controlled.
In addition, while the machines are idle or breakdown, the operation cost
is negligible (Lan and Lan, 2000; Chen and Lan, 2001). This is because
that the consumption of input resources does not exist and electricity
fees of idle machines are relatively small comparing with those of the
whole system. Although several models engaged in a profit function were
described by Kalir and Arzi (1998), none is related to the workforce.
Actually, the overall profit and the productivity are both the most concerned
problems confronting manufacturing industry.
The optimal production and workforce control to balance machine productivity
and human ability in computer-based manufacturing systems is rebellious
and crucial to industrial management. In addition, to complete an order
earlier will freeze the capital, raise the inventory cost and indicate
the sub-optimal resource utilization. On the other hand, an order accomplished
later than the production deadline may lose customers. Therefore, meeting
the production deadline is also the most desirable objective of management
(Soroush, 1999; Qiu et al., 2007). With the reasons above, it is
necessary to not only economically solve the workforce scheduling problem
for material handling among all parallel machines, but also competently
optimize the production rate for a deterministic production quantity and
punctually match the production deadline to reach the maximum profit.
With the model proposed in this study, this issue becomes realistically
and concretely solvable.
ASSUMPTIONS AND NOTATIONS
Assumptions: Before formulating this study, several conditions
are assumed. They are described as follows:
||The production project is a continuous machining operation
with no breakdown. And, the order quantity is equivalently assigned
to each parallel machine.
||No delay or scrapping of parts occurs during the machining process.
||The raw materials are shipped just in time from inventory to manufacturing
for all parallel machines.
||All manufactured parts are collected and held in the shop until
the whole production quantity is done.
||The marginal operation cost of machines is a linear increasing function
of the production rate (Kamien and Schwartz, 1991).
||The operational cost for idle or breakdown machines is negligible
(Lan and Lan, 2000; Chen and Lan, 2001). That is, the operation cost
is considered only from the parallel machines assigned for production.
||The workforce for material handling is defined as the working hours
and considered as a continuous function of production time.
||All products are shipped and sold at a given price immediately at
the time the order quantity is done.
Parameters and Notations: Throughout the study, the parameters
and notations are used. They are defined and listed as follows:
||Maximum production rate of the machine, which is limited
by the maximum machining conditions
||Marginal operation cost for each machine at the production rate
x`(t), where b is a constant
||Operational cost for each machine at time t
||Product holding cost for unit part per unit time
||Average labor cost of material handling per unit workforce
||Average material handling ability per unit workforce, which denotes
the average number of parts that is capable to be handled by unit
||Number of parallel machines assigned for production
||Sales price per unit product
||Order quantity of the production project
||Production deadline that is given by the customer or production
||Workforce scheduling for material handling at time t,
which is defined as working hours for material handling at time t
||Cumulative parts manufactured per unit machine during time interval
||Production rate per unit machine at time t, which means number of
parts manufactured per unit time
The production quantity distributed to each parallel machine, Q/m, is
not particular to be an integer. Therefore, to meet the production quantity
of the machining project, a larger quantity Q + m is necessarily considered
for production. In order to practically introduce the JIT philosophy in
balancing the productivity and workforce, the constraint is also applied
In this study, pQ describes the total revenue of the production quantity
and represents the workforce cost during the production period [0, T].
In addition, express the operation cost and part holding cost during the
production period [0,T], respectively. Thus, the mathematical model in
achieving the maximum profit and its constraints are then formulated as
Set (x*, k*) to be the optimal solution of the mathematical model. And,
assume that the time interval is
the maximum subinterval of [0, T] to satisfy Euler Equation (Kamien and
Schwartz, 1991; Chiang, 1992). There are two feasible situations to be
discussed in this study.
Situation 1: x`(t) will never reach the maximum limit B before
The optimum solution for situation 1 is shown as follows:
The detail is described in Appendix A.
With Eq. 2 and 3, it is found that the optimal production rate x*`(t)
and the optimal workforce scheme are both linear increasing functions
of t before touching the maximum limit.
Before discussing the other situation, one PROPERTY is proposed and described
PROPERTY: If the line y = x*`(t) touches the line y = B, two lines
should overlap to be y = B from the touch point to the end point T.
The proof of PROPERTY is discussed in Appendix B.
Situation 2: x`(t) will reach the maximum limit B before T.
The detail is described in Appendix C.
Decision criteria: With Eq. 4, two possible decision criteria
are classified as follows:
it means .
This contradicts the assumption of Situation 2. It is that the optimal
production rate x*`(t) will not touch the maximum limit B within
the production deadline T. The optimal solution is situation 1.
This is that the optimal production rate x*`(t) will reach the maximum
limit B within the production deadline T. The optimal solution is
The sensitivity analyses for situation 1: It is considered that
x*(t), x*`(t) and k*(t) are the decision functions in this case. Q, m,
T and L are the relevant parameters in the analysis.
From Eq. 1 and 2, it is claimed that the optimal cumulative products
per unit machine x*(t) and the optimal production rate per unit machine
x*`(t) are both in inverse proportion to the production deadline T and
are both in direct proportion to the order quantity Q. In addition, it
is derived from Eq. 3 that the optimal workforce scheduling for material
handling k*(t) is an increasing function of both the order quantity Q
and the number of parallel machines m. On the other hand, k*(t) is a decreasing
function of the production deadline T and the average workforce ability
The sensitivity analysis on x*(t), x*`(t) and k*(t) with respect to q,
m, T and L for situation 1 is shown in Table 1.
The sensitivity analyses for situation 2: It is considered that
x*(±), x*`(t) and k*(t) are the key variables in this case. p,
b, B, Q, T and L are the relevant parameters in the analysis. From Eq.
4, it is claimed that is
directly proportional to the marginal operation constant b, the maximum
productivity B and the production deadline T and it is inversely proportional
to the sales price per unit product p and the average material handling
ability per unit workforce L.
In addition, it is asserted from Eq. 5 and 6 that the optimal cumulative
products per unit machine x*(t) and the optimal production rate per unit
machine x*`(t) are both increasing functions of the order quantity Q and
the maximum production rate B. Moreover, through Eq. 7, the optimal workforce
scheduling for material handling k*(t) is a decreasing function of the
production deadline T and the average workforce ability L and it is an
increasing function of maximum productivity B and the order quantity Q.
||The sensitivity analyses for situation 1
+: Decision variable is an increasing function of
the parameter, –: Decision variable is a decreasing function
of the parameter, #: Decision variable depends on the changes of
other relevant parameters
||The sensitivity analyses for situation
+: Decision variable is an increasing function of the
parameter, –: Decision variable is a decreasing function of
the parameter, #: Decision variable depends on the changes of other
The sensitivity analysis on ,
x*(±), x*`(t) and k*(t) with respect to p, b, B, Q, T and L for
Situation 2 are shown in Table 2.
The interest of productivity and workforce management grows up in manufacturing
systems with the necessity of being more and more flexible. The production
quantity, number of parallel machines, maximum production rate, operation
cost, product holding cost and the average ability of workforce are considered
simultaneously in balancing humans and machines to optimize the profit
for a manufacturing project. This is an extremely hard-solving and complicated
issue and the existing researches are far from giving the satisfactory
answer to this viewpoint. However, through our proposed model, the problem
becomes practically and concretely solvable.
In addition, with the optimum solution and the decision criteria, the
operational scheme on both production control and workforce scheduling
is then precisely determined. Therefore, the production planning, production
cost estimating and even the contract negotiation can be further approached
through this study. With this viewpoint, the applicability of the proposed
model is significantly extended.
The production rate and the workforce are both important control factors
of a machining project. Besides, the control of machine productivity and
the manpower scheduling are also critical for production planners. This
study surely generates the idea of managing both machines and humans and
also contributes the solution to optimize the overall profit matter.
Future researches for managing the machines and the humans under floating
sales prices and variation of production quantities or orders are encouraged
in this approach. In sum, this study surely generates a reliable and applicable
idea of optimal control to the techniques and also provides a better and
practical solution to this field.
Appendix A: The optimum solution for Situation 1.
With the boundary condition
and the constraint
the objective function is then rearranged as
From Euler Equation (Kamien and Schwartz, 1991; Chiang, 1992), it is
There exists a constant k1 satisfying
Integrating Eq. A1 with t, then
Using the boundary conditions, x(0) = 0 and
into Eq. A2 separately; we have
Applying Eq. A3 and A4 into Eq. A1 and A2, x*(t) and x*`(t) are thus
Substituting x*(t) into the constraint then
k*(t) is found.
Appendix B: The proof of PROPERTY.
From Eq. 2, x*`(t) is a strictly increasing linear function of t. And
it holds for any subinterval during [0, T] satisfying 0 ≤x*`(t)≤B.
Therefore, x*≤(t) in the time interval (shown
in Fig. 1) cannot exist because it contradicts to be
a decreasing linear function of t, the PROPERTY of is thus verified.
||Possible condition of line y = B and y = x*`(t)
Appendix C: The optimum solution for Situation 2.
Before reaching the maximum limit B, Eq. 1, 2 and 3 are satisfied for
this situation either. By applying the PROPERTY into the objective function,
it is then modified as:
From the transverality condition of salvage value for free end value
and Schwartz, 1991; Chiang, 1992), it is obtained that
Eq. C1, the optimal time to
reach the maximum limit is obtained.
the PROPERTY into the Eq. 1 and 2, the optimal solution x*(t) and x*`(t)
are then found.
Substituting x*(t) into the constraint
then k*(t) is found.