ARMA processes are processes in which a random vector Y_t in stage t depends on previously observed values Y_t–k, k > 0, along with some independent, identically distributed process v_t–k, k ³ 0. The ARMA(p,q) process [1] can be written mathematically as

,

where M_i, i=1,…,p, and N_j, j=1,…,q, are fixed matrices.

The use of such processes within the SMPS format is illustrated by a multi-stage production/inventory problem. The decision maker must choose the optimal levels for production and inventory levels, in order to satisfy a random demand. Capacity constraints limit the levels of production and inventory. Mathematically this problem can be formulated as follows.

where

c_it is the cost of producing one unit of product i in stage t

x_its is the amount of product i produced in stage t under scenario s

h_it is the cost of holding one unit of product i from stage t to stage t+1

y_its is the amount of product i held from stage t to stage t+1 under scenario s

a_ij is the amount of resource j used in producing one unit of product i

f_ik is the amount of resource k used in storing one unit of product i from one stage to the next

S is the set of scenarios

S(t) is the set of scenarios active in stage t; S(0) is a singleton, and

b_jt is the amount of resource b available in stage t

g_kt is the amount of resource k available in stage t

d_its is the amount of product i demanded in stage t under scenario s

a(s) is the ancestor of scenario s

p_ts is the probability of reaching scenario s in stage t

Time file

Core file

Stoch file

Reference

1. Wolfram Research, Inc., “Multivariate ARMA Models”, world-wide web page http://documents.wolfram.com/applications/timeseries/UsersGuidetoTimeSeries/1.2.5.html