An optimisation approach for capacity planning: modelling insights and empirical findings from a tactical perspective

Andréa Nunes Carvalho ^**andrea.carvalho@int.gov.br, **lf.scavarda@puc-rio.br, ***fabricio.oliveira@puc-rio.br

Instituto Nacional de Tecnologia, Rio de Janeiro, RJ, Brasil

Luiz Felipe Scavarda

Pontifícia Universidade Católica do Rio de Janeiro, Rio de Janeiro, RJ, Brasil

Fabricio Oliveira

Pontifícia Universidade Católica do Rio de Janeiro, Rio de Janeiro, RJ, Brasil

RMIT, Melbourne, Australia

Group	Types of constraints	Description
Production flow related	Release date and deadline	To guarantee that activities are processed within their time windows.
	Non-interruption flow	To establish that once started, an activity must be processed in all subsequent periods until it is finished.
	Maximum and minimum intensities	To ensure that the intensity of an activity can never be less than the minimum intensity nor more than the maximum intensity permitted in a single time period.
	Cadenced flow	To guarantee the activities’ variable execution progress mode, their overlapping behaviour and the non-fixed precedence relationship among them.
Capacity-related	Regular, overtime and subcontracting	To guarantee the balance within an activity's processing time and its fractions in terms of regular, overtime and subcontracting hours.
	Maximum number of working hours	To establish the upper bound for internal capacity, which is available in terms of regular working hours and overtime working hours
	Availability of employees	To establish the upper bound of available employees, considering is the capacity changes (hiring and firing) along the time periods
	Committed workload	To ensure that the available capacity can never be more than the already committed workload and the incoming orders allocated in a single time period.
	Minimum employment period policy	To avoid employment instability by restricting the capacity changes within the time periods

Milestone	Cutting	Stamping	Machining	Assembly	Welding
0	0.010	0.003	0.002	0.004	0.003
1	0.060	0.018	0.009	0.026	0.019
2	0.220	0.066	0.032	0.097	0.070
3	0.242	0.090	0.050	0.105	0.078
4	0.289	0.140	0.070	0.124	0.096
5	0.346	0.200	0.095	0.148	0.118
6	0.394	0.250	0.115	0.167	0.136
7	0.451	0.310	0.140	0.191	0.157
8	0.498	0.360	0.160	0.210	0.175
9	0.555	0.420	0.184	0.234	0.197
10	0.603	0.470	0.205	0.253	0.215
11	0.650	0.520	0.225	0.273	0.233
12	0.691	0.549	0.299	0.327	0.283
13	0.729	0.577	0.369	0.371	0.327
14	0.773	0.609	0.449	0.421	0.377
15	0.812	0.637	0.519	0.465	0.421
16	0.830	0.648	0.552	0.484	0.438
17	0.850	0.674	0.602	0.521	0.472
18	0.890	0.720	0.692	0.589	0.533
19	0.910	0.745	0.742	0.626	0.567
20	0.930	0.771	0.792	0.664	0.601
21	0.960	0.844	0.875	0.778	0.740
22	0.980	0.935	0.951	0.929	0.916
23	1.000	1.000	1.000	1.000	1.000

Groups of decisions	Managerial decision	Objectives
Demand	Include new orders (or proposals) to generate a plan	Assess the impact of incoming orders on resource capacity
	Negotiate or adjust a project’s deadline	Determine a deadline or check on the feasibility of a customer’s deadline
	Modify the original already committed workload plan	Update the capacity plan with revised data relative to the already committed workload
Supply	Adjust release dates (material availability) based on information from suppliers and project engineers	Anticipate release dates of critical materials or postpone an incoming project to level demand
Supply	Allow the subcontracting of specific part components	Reduce the workload of critical work centres
Capacity	Alter the maximum working hours for specific work centres or authorize overtime for specific work centres	Adjust internal capacity of critical work centres
Flexibility	Change the employment period policy by extending the minimum employment period or expand the fixed capacity periods	Minimize capacity change costs
Flexibility	Change the employment period policy by reducing the minimum employment period or reduce the fixed capacity periods	Minimize idle capacity
Production flow	Adjust the cadence parameters or regulate the minimum and maximum intensity of the activities	Reduce a part component’s lead time

AP_ia	1 if activity a belongs to part i, 0 otherwise
AT_ba	1 if activity a is classified as activity type b, 0 otherwise
C_bg	accumulated intensity of activity type b in milestone g
CAP_w	maximum number of working hours per period at work centre w
CC	capacity change cost relative to hiring or firing one employee
CO_w	average salary per hour of an employee working an overtime hour at work centre w
CR_w	production processing hourly cost at work centre w
CS_w	average salary of an employee working at work centre w
DL_a	deadline of activity a
M	“big M”, a sufficiently large number
ME	minimum employment period
MN_a	minimum intensity of activity a in any time period
MX_a	maximum intensity of activity a in any time period
NE_w	number of employees initially allocated at work centre w
NP	number of periods in the planning horizon
OHE_w	number of overtime hours per employee per period at work centre w
PS_i	price of the subcontracted part i
Q_aw	processing time of activity a at work centre w
RD_a	release date of activity a
RHE_w	number of regular working hours per employee per period at work centre w
WH_wt	number of hours relative to the committed workload allocated to work centre w in period t
XS_i	1 if part i may be subcontracted, 0 otherwise

c_a	1 if activity a is processed in-house, 0 otherwise
d_i	1 if part i is processed in-house, 0 otherwise
e_at	1 if activity a has already started in period t or in an earlier period; 0 otherwise
ea_wt	number of employees allocated at work centre w in period t
ef_ltw	number of employees hired in period l and fired in period t at work centre w
eh_ltw	number of employees hired in period l and still working in time period t at work centre w
eo_wt	number of employees working overtime hours at work centre w in period t
er_wt	number of employees working regular hours at work centre w in period t
f_at	1 if activity a is entirely processed by period t; 0 otherwise
n_igt	1 if part i has completed at least milestone percentual g in period t; 0 otherwise
o_at	number of overtime hours processing activity a in period t
r_at	number of regular working hours processing activity a in period t
s_at	number of subcontracted hours processing activity a in period t
w_at	1 if activity a is processed in period t; 0 otherwise
wr_wt	number of regular hours relative to the committed workload allocated to work centre w in period t
wo_wt	number of overtime hours relative to the committed workload allocated to work centre w in period t
x_at	intensity of activity a in period t
z_at	accumulated intensity of activity a until period t

Brasil