- Email: [email protected]
Multiperiod production planning with demand and cost fluctuation
M&l.
Cornput. Modelling Vol. 28, No. 3, pp. 103-119, 1998 @ 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0895-7177/98 $19.00 + 0.00
Pergamon
PII:SOS95-7177(98)00102-2
Multiperiod Production Planning with Remand and Cost Fluctuation M.
EL
HAFSI
The A. Gary Anderson Graduate School of Management
The University of California, Riverside, CA 92521, U.S.A. S. X. BAI Department of Industrial and Systems Engineering Utility of Florida, Gainesville, FL 32611, U.S.A. (Received and accepted July 1997)
Abstract-We consider the production planning of a single product over a finite planning horizon. The latter is made up of time periods that may not necessarily be of the same length. In each time period, the product is subject to a constant demand and a finite production capacity. If demand is not satisfied immediately, it is backlogged. The cost rates of carrying inventory or bacldoggiw a unit of the product may also vary from one time period to the next. The objective is to determine an optimal production plan for the product so ss to minimize the total inventory and backlog costs incurred over the planning horizon. The problem is formulated ss an optimal control problem. First, the optimal solution of the onatimeperiod problem is determined. Then, based on the one-timeperiod optimal solution, the optimal solution for the whole planning horizon is obtained by solving a nonlinear progr ammiug problem. @ 1998 Etsevier Science Ltd. All rights reseti. Keywords-Production
planning, Optimal control, Nonlinear optimisation.
NOMENCLATURE time interval representing the planning horizon number of time periods in the planni~
horison
length of time period i, in time unikr starting time of time period i, in time units constant demaud rate in time period i, in units/time
unit
maximum production rate in time period i, in units/time
unit
inventory carrying cost in time period i, in $/[(unit)(cime
unit)]
backlog cost in time period i, in $/[(unit)(time
unit)]
starting surplus level in time period i, in units utilization factor of the production system in time period i demand rate at time t, in units,&me unit production rate at time t, in unite/time unit surplus level at time t, in unit8
1. INTRODUCTION In this paper, we consider the problem of ~1~~ the production of a single product over a finite planning horizon. We consider a model in which the planning horizon is divided i&o a finite number of time periods that may not necessarily be of the same length. Each time period
103
104
M. EL HAFSI AND S. X. BAI
is characterized by a constant demand rate for the product, and a finite production capacity, measured by a maximum production rate of the product. In certain time periods, the demand rate may exceed the maximum production rate. Backlog is permitted. Inventory and backlog costs are incurred each time a unit is held in inventory or backlogged. The costs may also differ from one time period to another. The objective is to determine a production plan for the product, consisting of a production rate function over time, so as to minimize the total inventory and backlog costs incurred over the planning horizon. The model can be applied to different practical situations. For instance, consider the following situation. Management would like to plan the production of a certain product over a 1Zmonth planning horizon. For this, the monthly product’s demand rate is forecast and a monthly maximum production capacity is allocated to the production of the product. The inventory carrying cost is adjusted monthly to reflect the interest rate associated with the capital tied up in inventory. The backlog cost rate reflects customer good will losses and other aspects associated with late delivery of products and are assumed to increase monthly at the same rate as the inventory cost rate. An optimal production plan, the one that minimizes the total cost incurred over the planning horizon, can be determined using the above model. Another application may be the case of a production system producing a single product and subject to periodic maintenance. Assume that the planning horizon is divided into a finite number of alternating up and down periods. Down periods correspond to periods where planned maintenance takes place during which the production capacity is zero (i.e., the maximum production rate is zero). Then, we can use the above model to determine the optimal production plan of the product in the presence of planned maintenance. Further, the model can be used to determine steady state conditions when the planning horizon becomes infinite. The model described above is closely related to dynamic lot-sizing models. The first dynamic lot-sizing model is the Wagner-Whitin model [l]. This model considers the problem of determining production lot sizes of a single product when demand is deterministic but varies with time. Time is discretized into periods of fixed length and material is treated as a continuous quantity. The objective to be minimized is the cost of holding inventory. The decision variables are the amount of the product produced in a period. The state variables are the amount of inventory left at the end of a period. Many variations of this basic model were proposed later. For a discussion of lot-sizing models, the reader is referred to the books by Johnson and Montgomery [2] and Nahmias [3] and the literature cited there. Most lot-sizing models are known to be hard to solve, in the sense that their computational effort grows at an exponential order with the size of the problem. In the early 1939s, influenced by the work of Kishel [4] and Olsder and Suri [5], Kimemia and Gershwin [6] developed a framework for the control of production in manufacturing systems. They treated part movement as a continuous flow rather than discrete. As a result, the complexity of production control models is greatly reduced. In addition, this allows the dynamics of production systems to be expressed in a form that is appropriate for control theory techniques. Within thii framework, different production systems were studied, ranging from singkproduct singlc+machine systems to multiproduct multimachme systems. A similar model to ours was proposed by Bai and Varanasi [7]. Our model is different from theirs in many aspects. They consider a quadratic penalty function that penalizes inventory and backlog the same way, while we penalize inventory and backlog in a linear fashion using different cost rates for each. They do not consider changes in capacity from one period to the next, while we do. Finally, we use a completely different approach that is baaed on the analytical solution of the one-timeperiod problem, while they use an algorithmic approach baaed on the clever notion of influencing intervals. Other models that are somewhat related to our model can be found in the literature. These models study a single machine single-product system that is subject to random failures and repairs j&12], deterministic up times and random repair times [13], or deterministic up and down times [14]. As described above, we consider the production planning of a product over an N-time-period planning horizon. The problem is formulated in Section 2. The solution approach is described
MultiperiodProductionPlanning
105
in Section 3. The analytical solution of the one-time-period problem is developed in Section 4. The optimal solution of the N-time-period problem is presented in Section 5. Two applications of the model are presented in Section 6. Section 7 concludes the paper.
2. PROBLEM FORMULATION Consider the production planning of a single product or part type over a fmite planning horizon. The latter consists of N time periods that may not be necessarily of the same length. In each period of time, the product is subject to a known constant demand rate and a finite production capacity. Any unmet demand is assumed to be completely backlogged. Further, inventory holding cost and backlog cost are incurred each time a unit of the product is held in inventory or backlogged. The production system is assumed to be perfectly reliable over the planning horizon. The objective is to determine a production plan for the product so as to minimize the total inventory and backlog costs incurred over the planning horizon. In order to specify the model, let the planning horizon [O,Z’]be divided into N time periods. Time period i starts at time ti, for i = 0, 1, . . . , N - 1. The ti’s satisfy the ordering relation to = 0 < tl < *** < tN_1 = T. Let z(t), u(t), d(t) denote the surplus level (the state variable), the production rate (the control variable), and the demand rate at time t, respectively. The surplus level z(t) is defined as the difference between the cumulative production and the cumulative demand up to time t. We assume x(t) E 82= (--00, oo), u(t) E 91i+= [0, co), t 3 0, and d(t) = 4, a positive constant for t E [ti, &+I). Surplus refers to inventory when x(t) 2 0 and backlog when x(t) < 0. The rate of change in surplus is given by the following system dynamics: 2(t) = u(t) - d(t),
z(0) = x0.
(1)
Here, x0 represents the surplus level at the beginning of the planning horizon. The surplus can be written as x(t) = x0 + ot[U(~) - d(7)] dr. J DEFINITION 1. A production control u(a) = (u(t), t 2 0) is admissible with respect to the initial state value x0 E 92 if u(t) is measurable in t, 0 5 u(t) < Vi, for t E [ti, ti+l), i = 0, 1, . . . , N - 1, and the corresponding state x(t) E 32, for al t 2 0. In the above definition, Vi represents the maximum production rate allowable in time period i. Here, we restrict ourselves to right-continuous production controls, i.e., at points of discontinuity, we have u(t) = lim,++ ~(7). DEFINITION 2. A function u = u(x) : W + R is an admissible feedback control if for any given initial state x0 E 8, the equation
k(t) = u(t) - d(t),
x(0) = x0,
has a unique solution, and u(x(.)) is admissible with respect to x0. The objective is to determine the production rate, u(t), over the planning horizon, [O,T], without over-producing or under-producing. Over-producing results in high levels of inventory, incurring large inventory costs. Under-producing results in large amount of backlogged demand which causes loss of customers’ good will and other undesirable effects. In other words, our objective is to keep the surplus level as close as possible to zero. Let ct be the dollar amount incurred for carrying a unit of the product for a unit of time in time period i, and c; be the dollar amount incurred for backlogging a unit of the product for a unit of time in time period i. We define the penalty function gi(x(*)) as follows:
Si(4t)) =
c:x(t), -c;(x)t),
if x(t) > 0,
0,
if x(t) = 0.
if x(t) < 0,
M. EL HAFSIAND S. X.
106
BAI
g&r(-)) represents the instantaneous cost incurred in time period i, when the surplus level at time t is z(t). For any admissible control u(s) and initial surplus q, define i+1
d+)) dt.
(2)
Our goal is to choose an admissible production control u(e) so as to minimize the cost functional J(zo, u(m),to).
3. SOLUTION APPROACH The cost functional J(zc, u(e), to) can be written as J(zo, u(*), to) =
J
1
o
go(dt)) dt + Jh,
4.h td.
(3)
Now, assume that we know the optimal production control u*(t) for t E [tl,Z’], then the optimal production control u*(t) for t E [0, 2’1is obtained by solving the following optimal control problem: J*(zo,u*(.),
to) =
J
1
min go(+)) Olu(*)~cJo 0 z(to)=zo
dt + J’
(&u*(.),tl) .
Thii is a onetime-period optimal control problem with terminal cost J*(zi, u*(-), tl). inductive argument, we can write i=N-1
J* (x0, u*(q), to) =
c ito
Using an
ai+1
’ OSU%:vi Z(tc)=Zi
J ’-
gi(x(t))
dt.
(4
i
OBSERVATION 1. It is clear from (4) that the structure of the optimal solution will be the same in each time period of the planning horizon, since in each time period, we solve an optimal control problem that has exactly the same structure as the other time periods. OBSERVATION 2. Assume that we know the optimal ending surplus, $+i, in time period i. Then, we can transform the optimal control problem (4) into another in which the ending surplus, ski, of time period i is fixed and has to be reached by the trajectory z(t) at time ti+l which is also fixed. Of course, this is provided that we use the optimal production control in each time period.
Based on Observations 1 and 2, we can solve the optimal control problem in two steps. STEP 1. We assume that we are given the optimal starting and ending surpluses zi and Xi+1 for time period i. Then, we solve a one-timeperiod optimal control problem for which we determine the optimal production control u*(t) for t E [ti, &+I] analytically, as well ss the explicit expression of the cost incurred in time period i ss a function of 2i and zi+i, which we will denote by ui(zi, %+l)* STEP 2. Expressing the total cost J(zo,u(-), to) as a function of the q’s only (to keep the notation simple, we drop the arguments to and u(a)), and letting X = (zi,z~, . . . ,sN), we can write
Jbo, W = ~O(~Oco,~l)+4(x1,%2)
+*“+vN-l(ZN-l,xN),
(5)
which turns out to be a nonlinear function in the decision variables ~1, x2,. . . , XN. As will be shown later, J(zo,X) is a convex function in X and hence the optimal @,x$, . . . ,xk can be obtained by minimizing J(zo, X) using any classic nonlinear programming algorithm. Once the optimal xi, xi,. . . , Z> are obtained, the optimal surplus trajectory is constructed based The on the optimal production rates and the optimal intermediate surpluses xi, x&. . . ,x>. optimal production control together with the optimal surplus trajectory constitutes the complete production plan of the product over the planning horizon (0, T]. In the next section, we study the onetime-period analytically via the Pontriagin’s minimum principle.
optimal control problem which we solve
Multiperiod Production Planning
4. THE ONETIMEPERIOD
OPTIMAL
107
CONTROL PROBLEM
Since the problem is time invariant, we can use the interval [O,Ti] instead of the interval [ti, &+I] for time period i. Here, Ti denotes the length of time period i. We use Pi to denote the one-time-period optimal control problem, i.e.,
%(%Xi+1)= o<$)f
%+1,4*)).
ui(zi,si+r) denotes the minimum cost incurred, when the starting surplus is zi at time 0, and the ending surplus is zi+r at time Ti, in time period i. We solve problem Pi using Pontriagin’s minimum principle. The Hamiltonian of the system is defined as follows:
wm,
= g&m) + p(t)(4t)
4tMt))
- 419
for t E [0, Ti].
(6)
p(t) is the so-called costate function. Pontriagin’s minimum principle (see [15]) states that if u*(t) is the optimal production control and z*(t) is the optimal surplus trajectory, then there exists a continuous function p*(t) such that
k*@)
=
~(x*(t),u*(t),P*(t)); aP(t)
p*(t) = -
~(x*(t),u*(t),P*(t));
W) I
wx*(t)9~*(%P*(t))
'H(x*(t)&),P*(t)),
x(Ti) = xi+l,
x(0) = xi,
and
Ti is fixed,
for all admissible production controls u(t), t E [O,TJ. These conditions become 3*(t) = u*(t) - di;
(7)
P*(t)= Q{z(t)o}; Wx*(t),u*(t),p*(t))= o<$gui {g&*(t)) +P*(t)(u(t)- 4)), 3-l(x*(t),+),p*(t)) = g&*(t)> -p*(t)4 +.o,~i;,,~*(t)~(t), *and Ti is fixed. X(0) = Xi, x(Z) = xi+l,
(8) or
(9) (10)
Here, 11~~~1 is the indicator function which is defined as follows:
I{oeA}
=
1,
ifacA,
0,
otherwise.
Since the final time is fixed, and the Hamiltonian does not depend explicitly on the time, the former must be constant within the entire time interval [0, Ti]. Therefore, we can write ‘H(x*(t)+*(t)9P*(t))
= Qi,
for t E [0, Ti].
(11)
M. EL HAFSIAND S. X. BAI
108
Equation (9) implies that Vi,
ifp*(t) c 0,
a,
if p*(t) = 0,
( 0,
if p*(t) > 0,
z&*(t)=
(12)
where a E [0, Vi] is to be determined. According to equation (9), the optimal production control u*(t) is determined whenever p*(t) # 0. The fact that p*(t) = 0 provides no information about the relationship between u*(t), z*(t), and p*(t) indicates that the control problem is singular. In particular, a singular interval is an interval [TI, Q] c [O,Z”i]for which p*(t) = 0; and a stitching time is a time t,, not part of a singular interval, for which p*(td) = 0. To determine the optimal control u*(t) when p*(t) = 0, we need to investigate the possibility of the existence of singular intervals and switching times. The following proposition provides the answer. PROPOSITION
1. Assume that p*(t) = 0. If Vi 2 di, then the optimal control u*(t) is given by
u*(t) =
Vi,
if$(t)
< 0,
di,
if$(t)
= 0,
if$(t)
> 0,
{ 0,
and if Vi c di, then the optimal control u*(t) is given by u*(t) =
Uj, o { ,
if$(t)
< 0,
iff(t)
> 0.
PROOF. If p*(t) = 0 and T(t) < 0, t represents the time the costate function changes from nonnegative to negative. Since we consider right-continuous control, we have u*(t) = Vi. If p*(t)= 0,p*(t)= 0, and Vi 2 di, we have z*(t) = 0. Using equation (7) and the fact that u*(t) is right-continuous, we obtain u*(t) = di. Notice that t must lie inside a singular interval. Now, if Vi < di, u*(t) # 4 which implies that we cannot have singular intervals in this case. If p*(t) = 0 and f(t) > 0, t represents the time the costate function changes from negative or zero to positive. Since we consider right-continuous control, we have u*(t) = 0. Using Proposition 1, we can extend (12) to the following:
if Vi zdi,
if Vi < di,
then u*(t) =
then u*(t) =
Vi,
if p*(t) < 0,
Vi,
ifp*(t)=Oandp(t)
di,
if p*(t) = 0 and p*(t) = 0,
0,
if p*(t) = 0 and p’(t) > 0,
0,
ifp’(t)
Vi,
if p*(t) C 0,
Vi, o
if p*(t) = 0 and p*(t) < 0,
0:
(13)
> 0,
if p*(t) = 0 and p*(t) > 0,
(14
if p*(t) > 0.
As it is expected, the optimal production control u*(t) and the optimal trajectory x*(t) depend on the starting and ending surpluses Zi and xi+l. Based on the optima&y conditions (7)-(11) and (13),(14), the optimal solution of the onetime-period problem is readily obtained for both CW, Vi > di and Vi 5 di, 88 a function of xi and xi+l. The results are SUIIUIIM~ZHI in the next section. The optimal production control u*(t), the optimal surplus trajectory z*(t), as well as the optimal cost value Vi(Zi,Zi+l), as a function of zi and zi+l, are shown in each case. The optimal cost value vi(Zi, zi+l) is obtained by integrating the penalty function gi(x*(*)) over the interval [0, Ti] a
Multiperiod Production Planning
109
Optimal Solution for the Case Vi > 4 (a) zi I 0 and zi+l >_ 0. We distinguishtwo subcases. (a.1) Zi 5 diTi - (Ti/(l - Ti))Zi+l:
, -di(t
%(Xi,
Xi+l)
=
- Ti) +
ci’
a
[
xf +
ri
Xi+19
a
(1-i+l
I
’
(a.2) xi > 4Ti - (~i/(l - T~))x~+I:
0,
if t
Vi,
iftE
E
u*(t) =
0,2(x( * [ [
J$xi
xi - dd, x*(t)
%(Xici,Xi+l)
=
(1 - fi) ydi(t *
=
(b) xi < 0 and xi+1 2 0:
-
x~+~) + (1 - ri)Ti ,
- a+~) +
>
(1ift
-Ti)
-x*+1)2
r&G,Ti
E 0,$x* [ .
>
,
- x$+1) + (I-
+ xi+19 if t E 2(x* - $*+I) + (I-
ri)Ti > GGT,)
+2((1 -ri)x( +nxi+l)dil'i - (1 -rd&f]s
9 ,
M. EL HAFSIAND S. X. BAI
110
(c) xi 2 0 and xi+l < 0:
+
vi(xi, xi+d
-
= 3-x: 24
+ ci 2 ax’+‘.
(d) xi < 0 and xi+1 < 0. We distinguish two subcases. (d.1) xi 2 -((l
+ 4Ti):
- ri)/ri)(xi+l
,
Vi(XiyXi+l)
(d-2)Xi
<
=
ci 7- _ri)x:+x:+11 ri
(1
24
-((I - rd/rd)(xi+l +
vi,
iftc
0,
iftE
diTi):
o,-2i (Xi - xi+11+ r&C [ >
u+(t) =
xi +
x’(t) =
[
-2 ((xi -xi+11
(l-dgt, ri
-4(t
- Z) +x4+1,
+rA,Ti
, ,
>
if t E 0, -z(xi [ if t E
-2(x+
- x$+1) + rix) - zi+l)
,
+ r&,Ti
, >
vj(xi,xi+l) Fii
=
zi2i
[ribi
-
1 shows the optimal trajectory
X*+d2- S(rixi +
(1
-
ri)zi+l)drTi - (1 - r#Tf]
associated with each case.
.
Multiperiod Production Planning
111
Case (a.2).
Case(a.1).
Case (b). AX@
AX@
1
Cese (d.1).
Case (d.2).
Figure 1. Optimal surplus trajectory in the case Us 1 c&.
Optimal Solution for the Case Vi < 4 (a) zj 10 and q+t I 0:
0,
ifte
iTi,
iftE
xi -
x’(t)
= i
V&i,Zi+1)
=
0, [
u*(t) =
[
$Xi
d&,
(1 - ri) -d&-Ti)+zi+l, ri - zi+1)’
-
- Zi+1)
+ (1 - p’i)lli
%+1>
(1 -
+
, >
b)%i,
Ti
, >
if t E 0, J$cc - zi+l) + (1 - r,)E) [ iftE
+ 2((1 - r&i
[
2(3$ -
%+1)
+ qri+~)d&i
-I- (1 -
,
TI)Ti,Ti
, >
- (1 - ri)dfTf] .
M. EL HAFSI AND S. X. BAI
112
(b) zi 10 and Q+~ < 0:
u’(t)
=
0,
ift E
Vi,
if
0,
if t E [7-2,Ti),
[Wl),
E [71,72),
-dit + xi, (1 -dit
z’(t) =
Ti)
if t E [O,T~), + C,
if t E [71,72),
ri
-di(t - Ti) + Zi+l, 4 = a
Q(Q,Q+l)
ri(2i
_ q2
[
if
t E
[72,Ti),
1
_ kc2 (1 - Q)
+ diTi -
C)2 +
x~+I-
(xi+1 + diTi)2 - - ri (1
-
~2 Ti)
. I
Here,
(c)
71 =
:(Xi - C),
T2 =
z(Xi+l
-
C) + TiTi,
The constant C is determined by minimizing the cost function Vi(ziy zi+i) with respect to C. Notice here that when Vi = di, C becomes zero and therefore, we recover Case (c) of the previous section. Xi < 0 and Xi+1 < 0:
, x*(t)
=
3
fJi(Xi, Xi+l)
=
$i
[rib . -
xi+1)2 - 2(rixi + (1 - ri)zi+i)diTi
- (1 - ri)$Tf]
*
Figure 2 shows the optimal trajectory associated with each case. In the next section, we formulate the N-timeperiod problem as a nonlinear programming problem, which we solve using the standard gradient projection technique.
Multiperiod Production Planning
Case (a).
113
Case (b).
cese(c). Figure 2. Optimal surplus trajectory in the case Vi < 4.
5. THE N-TIMEPERIOD
OPTIMAL
CONTROL PROBLEM
Before we convert the N-time-period problem into a nonlinearprogrammingproblem, we notice that for the one-time-period optimal control to exist, we must impose the following constraint:
(1 - 5) Constraint (15) is nothing but a requirementfor xi+1 to be reached from xi under the associated optimal control. We illustratethis with Case (a) when Vi 14 and Case (b) when Vi < d+. For Case (a) when Vi > di, Figure 1 (Cases (a.1) and (a.2)) shows the time r where the surplus trajectory changes slope. The time r is given by 7=
-
%+I) + (1 -
ri)&Ti.
It is clear that T must lie inside the interval [O,Ti], otherwise xi+1 cannot be reached from xi. This translates to the following constraint:
which, when rearranged, is precisely constraint (15). For Case (b) when Vi < di, Figure 2 (Case (b)) shows ~1 and 7s as the times where the surplus trajectory changes slope. These are given by 71 =
$Xi - c)
and 72 = $y+i
- C) +riTi,
M. EL HAFSI AND S. X. BAI
114
where
For xi+1 to be reached from zi, we must have 0 5 71 5 72 < Ti. Now, consider the inequality 7s 5 Ti. The latter is equivalentto
From the i~~ecpality ~1 5 72, above inequality,we obtain
we
get xi - xi+1 5 &Ti. Using the fact that xi - xi+1 2 0 and the
(1 - Ti)
which is constraint (15). The remaining csses can be easily checked in the same fashion. Now, fori=0,1,2 ,..., N - 1, constraints (15) define a polyhedron in the N-dimensional space given by the set s = {x = (x1,22,. . . ) mvJT E fl I Ax 5 b} , (16) where A = {aij} is a 2N x N matrix given by
t&j
i = j,
i=l,2
3*=i+1,
i=N+1,...,2N,
j=l+l, i= A*
i=N+l,...,
1,
for
-1,
for
0,
otherwise,
=
,..., N,
i=l,2
,..., N,
di-S-1,
fori=1,2
,..., N,
(’ - T’-‘)di_lT’_l,
fori=N+1,...,2N.
I
(17)
2N,
and b is a 2N column vector, given by a, =
ri-1
(18)
It is clear that the set S is a convex set. The total cost incurred over the whole planning horizon is given by J(xo,X)
= V()(XO,Xl)
+v1(x1,x2)
(19)
+~~~+Wv-l(W-l,Q+
Notice that vi(xi, zi+i) can be written in a general format as foiiows: %(Xi, xi+l) = CriXf+
piXf+l + biXixi+l
+ &xi
+ rixi+l
+ Pi.
(20)
Substituting in J(xo, X), we obtain i==N-1 Jbo,
W
=
C
iSO
[aixf + &x:+1 +
oiXiXi+l + &Xi + %X4+1 + c(i] .
(21)
Multiperiod Production Planning
115
We now state the N-Cm&period problem. We use P to denote the N-timeperiod problem, i.e.,
optimal control
imN-1
minimize P:
J(20,X)
=
C
[a& + PiZf+, + U&Z*+1 + &Zi + r&+1 + hi] y
i=O
subject to
X E S,
/ J*(zrJ, X’) = &fs J(zo, X). Problem P is a nonlinear programming problem. The following proposition establishes the convexity of the objective function J(zo, X). PROPOSITION
2.
The objective function J(so, X) is convex in X E S.
PROOF. The proof is straightforward and is based on the Hessian matrix of the cost function J(zc, X), which we denote by H. The latter is given by
H=R+RT, where
0.
wO+~l) 71
R=
0
w1+
* 02)
72
.
0
0 .
* * ;
(PN-2
.
;aN-1) 7N-1
0
’
@N-l
The reader may verify that the coefficients ai and pi, for i = 0, 1,2,. . . , N - 1, are all strictly positive. Thus, the matrix R has all its eigenvaluee strictly positive, meaning that R is a positive deiinite matrix. This implies that XTHX
= 2XTRX > 0,
for X # 0 and X E S.
Therefore, H is positive definite and J(zo, X) is convex in X E S. Proposition 2 implies that Problem P can be solved using any standard nonlinear programming algorithm. Once the optimal solution of Problem P is obtained, the optimal production control and the optimal surplus trajectory for the N-timeperiod problem can be easily constructed using the starting and ending surplus in each time period.
6. CASE STUDIES In this section, we use the N-time-period model to study two production planning problems: a finite horizon production planning problem and a periodic maintenance problem. In the latter, we study the transient and the steady state aspects. We use the Gradient Projection Algorithm of Rosen [16] to solve Problem P in each case. A Finite Horizon Production Planning Problem The data of this problem is shown in Table 1. The time periods correspond to the months of the year. The inventory carrying cost per unit per day is calculated on the basis of an annual interest rate of 10%. The backlog cost per unit per day is assumed to be five times the inventory carrying cost. The demand and maximum production rates are given in units per day. There is an initial inventory of 400 units of the product at the beginning of Period 1. The optimal surplus levels at the end of each time period were obtained by solving the twelvetimeperiod problem via the standard gradient projection technique of Rosen. Then, the optimal production plan was constructed using the optimal solution of the one-timeperiod problem.
M. EL HAFSI AND S. X. BAI
116
‘lbble 1. Date for the Unite horizon production planning problem.
Time Period
Ti
daye
+
4
vi
Ci
C;
units/day
units/day
$/[unit/day]
S/[unit/day]
1
31
34
35
15.00
75.00
2
28
38
25
15.11
75.60
3
31
22
28
15.24
76.20
4
30
10
26
15.36
76.80
5
31
49
40
15.48
77.40
6
30
63
35
15.61
78.10
7
31
38
45
15.73
78.70
8
31
31
20
15.86
79.30
9
30
12
25
15.98
79.90
10
31
20
25
16.11
80.60
11
30
27
30
16.24
81.20
12
31
30
35
16.37
81.90
likable2. Production plan. Number of Unite
Production Rate
Production Run Length
to be Produced
unite/day
(days)
1085
35
31
700
25
28
868
28
31
780
26
30
1240
40
31
1050
35
30
1395
45
31
620
20
31
750
25
30
775
25
31
QOO
30
30
385
35
11
700
30
20
800
DW Figure 3. Surplus trajectory.
Multiply
Production Planning
117
Table 2 shows the production plan for the entire horizon. Figure 3 shows the product’s surplus trajectory. Notice that the cumulative demand is 11,548 units and the cumulative production is 11,148 units over the entire horizon. Since there is an initial inventory of 400 units, the demand is completely satisfied and the surplus level is zero by the end of the planning horizon. The optimal plan is to produce at the maximum production rate in each period except the last period, where we have a production run of 11 days at the maximum production rate followed by a production run of 20 days at the demand rate. A Periodic Maintenance
Problem
Consider a manufacturing system subject to periodic maintenance. That is, the system goes through alternating up and down periods. Down periods correspond to maintenance. Each maintenance period takes up 2” time units and each up period t&es up TU time units. We assume that the number of time periods in the planning horizon, N, is even. In other words, there are N/2 up periods and N/2 down periods. The first period is assumed to be an up period. The product’s demand rate, inventory carrying cost rate, and backlog cost rate are assumed to be invariant throughout the planning horizon. Obviously, the maximum production rate must be zero during a down period. Thii implies that xi+1 = xi - di, for i = 1,3, . . . , N - 1, (22) To be able to use the N-time-period model developed above, all we have to do is to impose equations (22) as equality constraints in the nonlinear programming problem P, and use the following cost value functions in the one-time-period problem:
&
(xf*l - 2:) ,
ifx:i ZOandzi+r
20,
ifxiZOandxi+r
CO,
ifzi
(231
We use this modiied model to study the transient and steady state behavior of a manufacturing system producing a single product and subject to periodic planned maintenance. The plarming horizon consists of 24 time periods. The up period has a length of 20 time +.&is and the maintenance has a length of 5 time rkts. The maximum production rate is 15 un~ts/t~me unit during up periods and the demand rate is 10 units/time unit. The inventory carrying and backlog costs are $l.S/(zsnit/time unit] and $7.5/[unit/time unit], respectively. The system starts with a backlog of 200 units. As in the previous case, Problem P is first solved using the gradient projection technique, then the optimal production plan is constructed using the one-timeperiod problem optimal solution. The optimal surplus trajectory is shown in Figure 4. Notice that the optimal production plan is to eliminate the initial backlog as fast as possible, then as the system catches up, a certain positive surplus is built at the end of each of the following up periods. This positive surplus is usually referred to as the hedging level, since it hedges against the shortage in capacity brought about the maintenance period, Notice that the system reaches the steady state in Period 7 minoring the border effect of Period 24). At the steady state, the hedging level is 41.7 units.
7. CONCLUSION In this paper, we proposed a model for the plarming of the production of a single product over an N-time-period finite horison. The problem was formulated as an optimal control problem.
118
M. EL HAFSIAND S. X. BAI
-
1
1
r Figure 4. Surplus trajectory for the maintenance problem.
The optimal solution was obtained in two steps. First, the one-time-period problem was solved analytically. Then, based on the solution of the one-time-period problem, the N-time-period problem was formulated as a constrained nonlinear programming problem. The latter can be solved using any standard nonlinear programming algorithm. The model can be adapted to different situations. One of the adaptations of the model is to the case of a system with periodic maintenance which we presented in Section 6. Another adaptation could be to the case where customers may be impatient during certain periods and do not tolerate long delays in the delivery of the product. In this case, the backlog cost rate can be chosen such that it reflects customers’ tolerances for late delivery (high backlog cost rates may be assigned to periods of pressure from customers). The main disadvantage of the model is that it assumes that all the information is known with certainty and is static throughout the planning horizon. Thii is not always the case, especially when it comes to demand forecasts, since usually a demand realization is most likely to be different from the forecast of that demand. Nevertheless, the model can still be adapted to such dynamic changes using the idea of rolling horizons [3]. The latter consists of reevaluating the N-timeperiod production plan each period (as demand and capacity realizations occur) and implementing the production plan of the current period only.
REFERENCES 1. H.M. Wagner and T.M. Whitin, Dynamic version of the economic lot size model, Management Science 38 (6), 8Q-Q6 (1958). 2. L.A. Johnson and D.C. Montgomery, Editors, Opsmtions Research in Production Planning, Scheduling, and Inventory Control, Wiley & Sons, New York, (1974). 3. S. Nahmias, Editor, Production and Opemtions Analysis, Irwin, Homewood, IL, (1993). 4. Ft. Rlshel, Dynamic programming and minimum principles for systems with jump Markov diiurbances, SIAM Journal on Control 113 (2), 338-371 (1975). 5. G.L. Olsder and Ft. Suri, The optimaI control of parts routing in a manufacturing system with failure prone machines, Presented at the lgth IEEE Conjemnce on De&ion and Control, (1980). 6. J. Kimemia and S.B. Gershwin, An algorithm for the computer control of a flexible manufacturin.g system, IIE !&anaaci?ioM15 (4), 353-362 (1983). 7. S.X. Bai and S. Varanasi, An optimal production flow control problem with piecewise constant demand, Mathl. Comput. Modelling 34 (7), 87-107 (1996). a. It.. Akella and P.R. Kumar, Optimal control of production rate in a failure prone manufacturing system, IEEE Thneactions on Automatic Control AC-S1 (2), 116-126 (1986). 9. T. Bielecki and P.R. Kumar, Optimality of sero-inventory policies for unreliable manufacturing systems, Opemtions Remumh 36 (4), 532-541 (1988).
Multiperiod Production Planning
119
10. A. Sharifnie, Optimal production control of 8 manufacturing system with machine failures, IEEE Iftnneactions on Automatic Contnd AC-SS (7), 620-625 (1966). 11. E.K. Boukas and A. Haurie, Manufacturing flow control and preventive maintenance: A stoch8stic control 8ppro8ch, IEEE !hmsactiona on Automatic Contnd AC36 (9), 1024-1031 (1990). 12. M.C. C aramanis end G. Liberopolous, Perturbation analysis for the design of flexible m8nuf8cturing system flow controIIem, opecationa Reueorch 40 (6) (1992). 13. J.Q. Hu and D. Xiig, Optimal production control for failure prone manufecturing systems with deterministic up timea, Working p8per, Boeton University, (1992). 14. F.J. Kr&mer and S.X. Bai, A production flow control problem with periodic maintenance, Optimal Con-l Application and Methods 17, 281-307 (1996). 15. D.E. Kirk, OptimOr Control Theory, An Introduction. Network Serb, Prentice Hall, (1970). 16. J.B. Rosen, The gradient projection method for nonlinear progmmming. Part I. Linear constrainta, J. SIAM 8 (l), 181-217 (1960).
Cornput. Modelling Vol. 28, No. 3, pp. 103-119, 1998 @ 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0895-7177/98 $19.00 + 0.00
Pergamon
PII:SOS95-7177(98)00102-2
Multiperiod Production Planning with Remand and Cost Fluctuation M.
EL
HAFSI
The A. Gary Anderson Graduate School of Management
The University of California, Riverside, CA 92521, U.S.A. S. X. BAI Department of Industrial and Systems Engineering Utility of Florida, Gainesville, FL 32611, U.S.A. (Received and accepted July 1997)
Abstract-We consider the production planning of a single product over a finite planning horizon. The latter is made up of time periods that may not necessarily be of the same length. In each time period, the product is subject to a constant demand and a finite production capacity. If demand is not satisfied immediately, it is backlogged. The cost rates of carrying inventory or bacldoggiw a unit of the product may also vary from one time period to the next. The objective is to determine an optimal production plan for the product so ss to minimize the total inventory and backlog costs incurred over the planning horizon. The problem is formulated ss an optimal control problem. First, the optimal solution of the onatimeperiod problem is determined. Then, based on the one-timeperiod optimal solution, the optimal solution for the whole planning horizon is obtained by solving a nonlinear progr ammiug problem. @ 1998 Etsevier Science Ltd. All rights reseti. Keywords-Production
planning, Optimal control, Nonlinear optimisation.
NOMENCLATURE time interval representing the planning horizon number of time periods in the planni~
horison
length of time period i, in time unikr starting time of time period i, in time units constant demaud rate in time period i, in units/time
unit
maximum production rate in time period i, in units/time
unit
inventory carrying cost in time period i, in $/[(unit)(cime
unit)]
backlog cost in time period i, in $/[(unit)(time
unit)]
starting surplus level in time period i, in units utilization factor of the production system in time period i demand rate at time t, in units,&me unit production rate at time t, in unite/time unit surplus level at time t, in unit8
1. INTRODUCTION In this paper, we consider the problem of ~1~~ the production of a single product over a finite planning horizon. We consider a model in which the planning horizon is divided i&o a finite number of time periods that may not necessarily be of the same length. Each time period
103
104
M. EL HAFSI AND S. X. BAI
is characterized by a constant demand rate for the product, and a finite production capacity, measured by a maximum production rate of the product. In certain time periods, the demand rate may exceed the maximum production rate. Backlog is permitted. Inventory and backlog costs are incurred each time a unit is held in inventory or backlogged. The costs may also differ from one time period to another. The objective is to determine a production plan for the product, consisting of a production rate function over time, so as to minimize the total inventory and backlog costs incurred over the planning horizon. The model can be applied to different practical situations. For instance, consider the following situation. Management would like to plan the production of a certain product over a 1Zmonth planning horizon. For this, the monthly product’s demand rate is forecast and a monthly maximum production capacity is allocated to the production of the product. The inventory carrying cost is adjusted monthly to reflect the interest rate associated with the capital tied up in inventory. The backlog cost rate reflects customer good will losses and other aspects associated with late delivery of products and are assumed to increase monthly at the same rate as the inventory cost rate. An optimal production plan, the one that minimizes the total cost incurred over the planning horizon, can be determined using the above model. Another application may be the case of a production system producing a single product and subject to periodic maintenance. Assume that the planning horizon is divided into a finite number of alternating up and down periods. Down periods correspond to periods where planned maintenance takes place during which the production capacity is zero (i.e., the maximum production rate is zero). Then, we can use the above model to determine the optimal production plan of the product in the presence of planned maintenance. Further, the model can be used to determine steady state conditions when the planning horizon becomes infinite. The model described above is closely related to dynamic lot-sizing models. The first dynamic lot-sizing model is the Wagner-Whitin model [l]. This model considers the problem of determining production lot sizes of a single product when demand is deterministic but varies with time. Time is discretized into periods of fixed length and material is treated as a continuous quantity. The objective to be minimized is the cost of holding inventory. The decision variables are the amount of the product produced in a period. The state variables are the amount of inventory left at the end of a period. Many variations of this basic model were proposed later. For a discussion of lot-sizing models, the reader is referred to the books by Johnson and Montgomery [2] and Nahmias [3] and the literature cited there. Most lot-sizing models are known to be hard to solve, in the sense that their computational effort grows at an exponential order with the size of the problem. In the early 1939s, influenced by the work of Kishel [4] and Olsder and Suri [5], Kimemia and Gershwin [6] developed a framework for the control of production in manufacturing systems. They treated part movement as a continuous flow rather than discrete. As a result, the complexity of production control models is greatly reduced. In addition, this allows the dynamics of production systems to be expressed in a form that is appropriate for control theory techniques. Within thii framework, different production systems were studied, ranging from singkproduct singlc+machine systems to multiproduct multimachme systems. A similar model to ours was proposed by Bai and Varanasi [7]. Our model is different from theirs in many aspects. They consider a quadratic penalty function that penalizes inventory and backlog the same way, while we penalize inventory and backlog in a linear fashion using different cost rates for each. They do not consider changes in capacity from one period to the next, while we do. Finally, we use a completely different approach that is baaed on the analytical solution of the one-timeperiod problem, while they use an algorithmic approach baaed on the clever notion of influencing intervals. Other models that are somewhat related to our model can be found in the literature. These models study a single machine single-product system that is subject to random failures and repairs j&12], deterministic up times and random repair times [13], or deterministic up and down times [14]. As described above, we consider the production planning of a product over an N-time-period planning horizon. The problem is formulated in Section 2. The solution approach is described
MultiperiodProductionPlanning
105
in Section 3. The analytical solution of the one-time-period problem is developed in Section 4. The optimal solution of the N-time-period problem is presented in Section 5. Two applications of the model are presented in Section 6. Section 7 concludes the paper.
2. PROBLEM FORMULATION Consider the production planning of a single product or part type over a fmite planning horizon. The latter consists of N time periods that may not be necessarily of the same length. In each period of time, the product is subject to a known constant demand rate and a finite production capacity. Any unmet demand is assumed to be completely backlogged. Further, inventory holding cost and backlog cost are incurred each time a unit of the product is held in inventory or backlogged. The production system is assumed to be perfectly reliable over the planning horizon. The objective is to determine a production plan for the product so as to minimize the total inventory and backlog costs incurred over the planning horizon. In order to specify the model, let the planning horizon [O,Z’]be divided into N time periods. Time period i starts at time ti, for i = 0, 1, . . . , N - 1. The ti’s satisfy the ordering relation to = 0 < tl < *** < tN_1 = T. Let z(t), u(t), d(t) denote the surplus level (the state variable), the production rate (the control variable), and the demand rate at time t, respectively. The surplus level z(t) is defined as the difference between the cumulative production and the cumulative demand up to time t. We assume x(t) E 82= (--00, oo), u(t) E 91i+= [0, co), t 3 0, and d(t) = 4, a positive constant for t E [ti, &+I). Surplus refers to inventory when x(t) 2 0 and backlog when x(t) < 0. The rate of change in surplus is given by the following system dynamics: 2(t) = u(t) - d(t),
z(0) = x0.
(1)
Here, x0 represents the surplus level at the beginning of the planning horizon. The surplus can be written as x(t) = x0 + ot[U(~) - d(7)] dr. J DEFINITION 1. A production control u(a) = (u(t), t 2 0) is admissible with respect to the initial state value x0 E 92 if u(t) is measurable in t, 0 5 u(t) < Vi, for t E [ti, ti+l), i = 0, 1, . . . , N - 1, and the corresponding state x(t) E 32, for al t 2 0. In the above definition, Vi represents the maximum production rate allowable in time period i. Here, we restrict ourselves to right-continuous production controls, i.e., at points of discontinuity, we have u(t) = lim,++ ~(7). DEFINITION 2. A function u = u(x) : W + R is an admissible feedback control if for any given initial state x0 E 8, the equation
k(t) = u(t) - d(t),
x(0) = x0,
has a unique solution, and u(x(.)) is admissible with respect to x0. The objective is to determine the production rate, u(t), over the planning horizon, [O,T], without over-producing or under-producing. Over-producing results in high levels of inventory, incurring large inventory costs. Under-producing results in large amount of backlogged demand which causes loss of customers’ good will and other undesirable effects. In other words, our objective is to keep the surplus level as close as possible to zero. Let ct be the dollar amount incurred for carrying a unit of the product for a unit of time in time period i, and c; be the dollar amount incurred for backlogging a unit of the product for a unit of time in time period i. We define the penalty function gi(x(*)) as follows:
Si(4t)) =
c:x(t), -c;(x)t),
if x(t) > 0,
0,
if x(t) = 0.
if x(t) < 0,
M. EL HAFSIAND S. X.
106
BAI
g&r(-)) represents the instantaneous cost incurred in time period i, when the surplus level at time t is z(t). For any admissible control u(s) and initial surplus q, define i+1
d+)) dt.
(2)
Our goal is to choose an admissible production control u(e) so as to minimize the cost functional J(zo, u(m),to).
3. SOLUTION APPROACH The cost functional J(zc, u(e), to) can be written as J(zo, u(*), to) =
J
1
o
go(dt)) dt + Jh,
4.h td.
(3)
Now, assume that we know the optimal production control u*(t) for t E [tl,Z’], then the optimal production control u*(t) for t E [0, 2’1is obtained by solving the following optimal control problem: J*(zo,u*(.),
to) =
J
1
min go(+)) Olu(*)~cJo 0 z(to)=zo
dt + J’
(&u*(.),tl) .
Thii is a onetime-period optimal control problem with terminal cost J*(zi, u*(-), tl). inductive argument, we can write i=N-1
J* (x0, u*(q), to) =
c ito
Using an
ai+1
’ OSU%:vi Z(tc)=Zi
J ’-
gi(x(t))
dt.
(4
i
OBSERVATION 1. It is clear from (4) that the structure of the optimal solution will be the same in each time period of the planning horizon, since in each time period, we solve an optimal control problem that has exactly the same structure as the other time periods. OBSERVATION 2. Assume that we know the optimal ending surplus, $+i, in time period i. Then, we can transform the optimal control problem (4) into another in which the ending surplus, ski, of time period i is fixed and has to be reached by the trajectory z(t) at time ti+l which is also fixed. Of course, this is provided that we use the optimal production control in each time period.
Based on Observations 1 and 2, we can solve the optimal control problem in two steps. STEP 1. We assume that we are given the optimal starting and ending surpluses zi and Xi+1 for time period i. Then, we solve a one-timeperiod optimal control problem for which we determine the optimal production control u*(t) for t E [ti, &+I] analytically, as well ss the explicit expression of the cost incurred in time period i ss a function of 2i and zi+i, which we will denote by ui(zi, %+l)* STEP 2. Expressing the total cost J(zo,u(-), to) as a function of the q’s only (to keep the notation simple, we drop the arguments to and u(a)), and letting X = (zi,z~, . . . ,sN), we can write
Jbo, W = ~O(~Oco,~l)+4(x1,%2)
+*“+vN-l(ZN-l,xN),
(5)
which turns out to be a nonlinear function in the decision variables ~1, x2,. . . , XN. As will be shown later, J(zo,X) is a convex function in X and hence the optimal @,x$, . . . ,xk can be obtained by minimizing J(zo, X) using any classic nonlinear programming algorithm. Once the optimal xi, xi,. . . , Z> are obtained, the optimal surplus trajectory is constructed based The on the optimal production rates and the optimal intermediate surpluses xi, x&. . . ,x>. optimal production control together with the optimal surplus trajectory constitutes the complete production plan of the product over the planning horizon (0, T]. In the next section, we study the onetime-period analytically via the Pontriagin’s minimum principle.
optimal control problem which we solve
Multiperiod Production Planning
4. THE ONETIMEPERIOD
OPTIMAL
107
CONTROL PROBLEM
Since the problem is time invariant, we can use the interval [O,Ti] instead of the interval [ti, &+I] for time period i. Here, Ti denotes the length of time period i. We use Pi to denote the one-time-period optimal control problem, i.e.,
%(%Xi+1)= o<$)f
%+1,4*)).
ui(zi,si+r) denotes the minimum cost incurred, when the starting surplus is zi at time 0, and the ending surplus is zi+r at time Ti, in time period i. We solve problem Pi using Pontriagin’s minimum principle. The Hamiltonian of the system is defined as follows:
wm,
= g&m) + p(t)(4t)
4tMt))
- 419
for t E [0, Ti].
(6)
p(t) is the so-called costate function. Pontriagin’s minimum principle (see [15]) states that if u*(t) is the optimal production control and z*(t) is the optimal surplus trajectory, then there exists a continuous function p*(t) such that
k*@)
=
~(x*(t),u*(t),P*(t)); aP(t)
p*(t) = -
~(x*(t),u*(t),P*(t));
W) I
wx*(t)9~*(%P*(t))
'H(x*(t)&),P*(t)),
x(Ti) = xi+l,
x(0) = xi,
and
Ti is fixed,
for all admissible production controls u(t), t E [O,TJ. These conditions become 3*(t) = u*(t) - di;
(7)
P*(t)= Q{z(t)
(8) or
(9) (10)
Here, 11~~~1 is the indicator function which is defined as follows:
I{oeA}
=
1,
ifacA,
0,
otherwise.
Since the final time is fixed, and the Hamiltonian does not depend explicitly on the time, the former must be constant within the entire time interval [0, Ti]. Therefore, we can write ‘H(x*(t)+*(t)9P*(t))
= Qi,
for t E [0, Ti].
(11)
M. EL HAFSIAND S. X. BAI
108
Equation (9) implies that Vi,
ifp*(t) c 0,
a,
if p*(t) = 0,
( 0,
if p*(t) > 0,
z&*(t)=
(12)
where a E [0, Vi] is to be determined. According to equation (9), the optimal production control u*(t) is determined whenever p*(t) # 0. The fact that p*(t) = 0 provides no information about the relationship between u*(t), z*(t), and p*(t) indicates that the control problem is singular. In particular, a singular interval is an interval [TI, Q] c [O,Z”i]for which p*(t) = 0; and a stitching time is a time t,, not part of a singular interval, for which p*(td) = 0. To determine the optimal control u*(t) when p*(t) = 0, we need to investigate the possibility of the existence of singular intervals and switching times. The following proposition provides the answer. PROPOSITION
1. Assume that p*(t) = 0. If Vi 2 di, then the optimal control u*(t) is given by
u*(t) =
Vi,
if$(t)
< 0,
di,
if$(t)
= 0,
if$(t)
> 0,
{ 0,
and if Vi c di, then the optimal control u*(t) is given by u*(t) =
Uj, o { ,
if$(t)
< 0,
iff(t)
> 0.
PROOF. If p*(t) = 0 and T(t) < 0, t represents the time the costate function changes from nonnegative to negative. Since we consider right-continuous control, we have u*(t) = Vi. If p*(t)= 0,p*(t)= 0, and Vi 2 di, we have z*(t) = 0. Using equation (7) and the fact that u*(t) is right-continuous, we obtain u*(t) = di. Notice that t must lie inside a singular interval. Now, if Vi < di, u*(t) # 4 which implies that we cannot have singular intervals in this case. If p*(t) = 0 and f(t) > 0, t represents the time the costate function changes from negative or zero to positive. Since we consider right-continuous control, we have u*(t) = 0. Using Proposition 1, we can extend (12) to the following:
if Vi zdi,
if Vi < di,
then u*(t) =
then u*(t) =
Vi,
if p*(t) < 0,
Vi,
ifp*(t)=Oandp(t)
di,
if p*(t) = 0 and p*(t) = 0,
0,
if p*(t) = 0 and p’(t) > 0,
0,
ifp’(t)
Vi,
if p*(t) C 0,
Vi, o
if p*(t) = 0 and p*(t) < 0,
0:
(13)
> 0,
if p*(t) = 0 and p*(t) > 0,
(14
if p*(t) > 0.
As it is expected, the optimal production control u*(t) and the optimal trajectory x*(t) depend on the starting and ending surpluses Zi and xi+l. Based on the optima&y conditions (7)-(11) and (13),(14), the optimal solution of the onetime-period problem is readily obtained for both CW, Vi > di and Vi 5 di, 88 a function of xi and xi+l. The results are SUIIUIIM~ZHI in the next section. The optimal production control u*(t), the optimal surplus trajectory z*(t), as well as the optimal cost value Vi(Zi,Zi+l), as a function of zi and zi+l, are shown in each case. The optimal cost value vi(Zi, zi+l) is obtained by integrating the penalty function gi(x*(*)) over the interval [0, Ti] a
Multiperiod Production Planning
109
Optimal Solution for the Case Vi > 4 (a) zi I 0 and zi+l >_ 0. We distinguishtwo subcases. (a.1) Zi 5 diTi - (Ti/(l - Ti))Zi+l:
, -di(t
%(Xi,
Xi+l)
=
- Ti) +
ci’
a
[
xf +
ri
Xi+19
a
(1-i+l
I
’
(a.2) xi > 4Ti - (~i/(l - T~))x~+I:
0,
if t
Vi,
iftE
E
u*(t) =
0,2(x( * [ [
J$xi
xi - dd, x*(t)
%(Xici,Xi+l)
=
(1 - fi) ydi(t *
=
(b) xi < 0 and xi+1 2 0:
-
x~+~) + (1 - ri)Ti ,
- a+~) +
>
(1ift
-Ti)
-x*+1)2
r&G,Ti
E 0,$x* [ .
>
,
- x$+1) + (I-
+ xi+19 if t E 2(x* - $*+I) + (I-
ri)Ti > GGT,)
+2((1 -ri)x( +nxi+l)dil'i - (1 -rd&f]s
9 ,
M. EL HAFSIAND S. X. BAI
110
(c) xi 2 0 and xi+l < 0:
+
vi(xi, xi+d
-
= 3-x: 24
+ ci 2 ax’+‘.
(d) xi < 0 and xi+1 < 0. We distinguish two subcases. (d.1) xi 2 -((l
+ 4Ti):
- ri)/ri)(xi+l
,
Vi(XiyXi+l)
(d-2)Xi
<
=
ci 7- _ri)x:+x:+11 ri
(1
24
-((I - rd/rd)(xi+l +
vi,
iftc
0,
iftE
diTi):
o,-2i (Xi - xi+11+ r&C [ >
u+(t) =
xi +
x’(t) =
[
-2 ((xi -xi+11
(l-dgt, ri
-4(t
- Z) +x4+1,
+rA,Ti
, ,
>
if t E 0, -z(xi [ if t E
-2(x+
- x$+1) + rix) - zi+l)
,
+ r&,Ti
, >
vj(xi,xi+l) Fii
=
zi2i
[ribi
-
1 shows the optimal trajectory
X*+d2- S(rixi +
(1
-
ri)zi+l)drTi - (1 - r#Tf]
associated with each case.
.
Multiperiod Production Planning
111
Case (a.2).
Case(a.1).
Case (b). AX@
AX@
1
Cese (d.1).
Case (d.2).
Figure 1. Optimal surplus trajectory in the case Us 1 c&.
Optimal Solution for the Case Vi < 4 (a) zj 10 and q+t I 0:
0,
ifte
iTi,
iftE
xi -
x’(t)
= i
V&i,Zi+1)
=
0, [
u*(t) =
[
$Xi
d&,
(1 - ri) -d&-Ti)+zi+l, ri - zi+1)’
-
- Zi+1)
+ (1 - p’i)lli
%+1>
(1 -
+
, >
b)%i,
Ti
, >
if t E 0, J$cc - zi+l) + (1 - r,)E) [ iftE
+ 2((1 - r&i
[
2(3$ -
%+1)
+ qri+~)d&i
-I- (1 -
,
TI)Ti,Ti
, >
- (1 - ri)dfTf] .
M. EL HAFSI AND S. X. BAI
112
(b) zi 10 and Q+~ < 0:
u’(t)
=
0,
ift E
Vi,
if
0,
if t E [7-2,Ti),
[Wl),
E [71,72),
-dit + xi, (1 -dit
z’(t) =
Ti)
if t E [O,T~), + C,
if t E [71,72),
ri
-di(t - Ti) + Zi+l, 4 = a
Q(Q,Q+l)
ri(2i
_ q2
[
if
t E
[72,Ti),
1
_ kc2 (1 - Q)
+ diTi -
C)2 +
x~+I-
(xi+1 + diTi)2 - - ri (1
-
~2 Ti)
. I
Here,
(c)
71 =
:(Xi - C),
T2 =
z(Xi+l
-
C) + TiTi,
The constant C is determined by minimizing the cost function Vi(ziy zi+i) with respect to C. Notice here that when Vi = di, C becomes zero and therefore, we recover Case (c) of the previous section. Xi < 0 and Xi+1 < 0:
, x*(t)
=
3
fJi(Xi, Xi+l)
=
$i
[rib . -
xi+1)2 - 2(rixi + (1 - ri)zi+i)diTi
- (1 - ri)$Tf]
*
Figure 2 shows the optimal trajectory associated with each case. In the next section, we formulate the N-timeperiod problem as a nonlinear programming problem, which we solve using the standard gradient projection technique.
Multiperiod Production Planning
Case (a).
113
Case (b).
cese(c). Figure 2. Optimal surplus trajectory in the case Vi < 4.
5. THE N-TIMEPERIOD
OPTIMAL
CONTROL PROBLEM
Before we convert the N-time-period problem into a nonlinearprogrammingproblem, we notice that for the one-time-period optimal control to exist, we must impose the following constraint:
(1 - 5) Constraint (15) is nothing but a requirementfor xi+1 to be reached from xi under the associated optimal control. We illustratethis with Case (a) when Vi 14 and Case (b) when Vi < d+. For Case (a) when Vi > di, Figure 1 (Cases (a.1) and (a.2)) shows the time r where the surplus trajectory changes slope. The time r is given by 7=
-
%+I) + (1 -
ri)&Ti.
It is clear that T must lie inside the interval [O,Ti], otherwise xi+1 cannot be reached from xi. This translates to the following constraint:
which, when rearranged, is precisely constraint (15). For Case (b) when Vi < di, Figure 2 (Case (b)) shows ~1 and 7s as the times where the surplus trajectory changes slope. These are given by 71 =
$Xi - c)
and 72 = $y+i
- C) +riTi,
M. EL HAFSI AND S. X. BAI
114
where
For xi+1 to be reached from zi, we must have 0 5 71 5 72 < Ti. Now, consider the inequality 7s 5 Ti. The latter is equivalentto
From the i~~ecpality ~1 5 72, above inequality,we obtain
we
get xi - xi+1 5 &Ti. Using the fact that xi - xi+1 2 0 and the
(1 - Ti)
which is constraint (15). The remaining csses can be easily checked in the same fashion. Now, fori=0,1,2 ,..., N - 1, constraints (15) define a polyhedron in the N-dimensional space given by the set s = {x = (x1,22,. . . ) mvJT E fl I Ax 5 b} , (16) where A = {aij} is a 2N x N matrix given by
t&j
i = j,
i=l,2
3*=i+1,
i=N+1,...,2N,
j=l+l, i= A*
i=N+l,...,
1,
for
-1,
for
0,
otherwise,
=
,..., N,
i=l,2
,..., N,
di-S-1,
fori=1,2
,..., N,
(’ - T’-‘)di_lT’_l,
fori=N+1,...,2N.
I
(17)
2N,
and b is a 2N column vector, given by a, =
ri-1
(18)
It is clear that the set S is a convex set. The total cost incurred over the whole planning horizon is given by J(xo,X)
= V()(XO,Xl)
+v1(x1,x2)
(19)
+~~~+Wv-l(W-l,Q+
Notice that vi(xi, zi+i) can be written in a general format as foiiows: %(Xi, xi+l) = CriXf+
piXf+l + biXixi+l
+ &xi
+ rixi+l
+ Pi.
(20)
Substituting in J(xo, X), we obtain i==N-1 Jbo,
W
=
C
iSO
[aixf + &x:+1 +
oiXiXi+l + &Xi + %X4+1 + c(i] .
(21)
Multiperiod Production Planning
115
We now state the N-Cm&period problem. We use P to denote the N-timeperiod problem, i.e.,
optimal control
imN-1
minimize P:
J(20,X)
=
C
[a& + PiZf+, + U&Z*+1 + &Zi + r&+1 + hi] y
i=O
subject to
X E S,
/ J*(zrJ, X’) = &fs J(zo, X). Problem P is a nonlinear programming problem. The following proposition establishes the convexity of the objective function J(zo, X). PROPOSITION
2.
The objective function J(so, X) is convex in X E S.
PROOF. The proof is straightforward and is based on the Hessian matrix of the cost function J(zc, X), which we denote by H. The latter is given by
H=R+RT, where
0.
wO+~l) 71
R=
0
w1+
* 02)
72
.
0
0 .
* * ;
(PN-2
.
;aN-1) 7N-1
0
’
@N-l
The reader may verify that the coefficients ai and pi, for i = 0, 1,2,. . . , N - 1, are all strictly positive. Thus, the matrix R has all its eigenvaluee strictly positive, meaning that R is a positive deiinite matrix. This implies that XTHX
= 2XTRX > 0,
for X # 0 and X E S.
Therefore, H is positive definite and J(zo, X) is convex in X E S. Proposition 2 implies that Problem P can be solved using any standard nonlinear programming algorithm. Once the optimal solution of Problem P is obtained, the optimal production control and the optimal surplus trajectory for the N-timeperiod problem can be easily constructed using the starting and ending surplus in each time period.
6. CASE STUDIES In this section, we use the N-time-period model to study two production planning problems: a finite horizon production planning problem and a periodic maintenance problem. In the latter, we study the transient and the steady state aspects. We use the Gradient Projection Algorithm of Rosen [16] to solve Problem P in each case. A Finite Horizon Production Planning Problem The data of this problem is shown in Table 1. The time periods correspond to the months of the year. The inventory carrying cost per unit per day is calculated on the basis of an annual interest rate of 10%. The backlog cost per unit per day is assumed to be five times the inventory carrying cost. The demand and maximum production rates are given in units per day. There is an initial inventory of 400 units of the product at the beginning of Period 1. The optimal surplus levels at the end of each time period were obtained by solving the twelvetimeperiod problem via the standard gradient projection technique of Rosen. Then, the optimal production plan was constructed using the optimal solution of the one-timeperiod problem.
M. EL HAFSI AND S. X. BAI
116
‘lbble 1. Date for the Unite horizon production planning problem.
Time Period
Ti
daye
+
4
vi
Ci
C;
units/day
units/day
$/[unit/day]
S/[unit/day]
1
31
34
35
15.00
75.00
2
28
38
25
15.11
75.60
3
31
22
28
15.24
76.20
4
30
10
26
15.36
76.80
5
31
49
40
15.48
77.40
6
30
63
35
15.61
78.10
7
31
38
45
15.73
78.70
8
31
31
20
15.86
79.30
9
30
12
25
15.98
79.90
10
31
20
25
16.11
80.60
11
30
27
30
16.24
81.20
12
31
30
35
16.37
81.90
likable2. Production plan. Number of Unite
Production Rate
Production Run Length
to be Produced
unite/day
(days)
1085
35
31
700
25
28
868
28
31
780
26
30
1240
40
31
1050
35
30
1395
45
31
620
20
31
750
25
30
775
25
31
QOO
30
30
385
35
11
700
30
20
800
DW Figure 3. Surplus trajectory.
Multiply
Production Planning
117
Table 2 shows the production plan for the entire horizon. Figure 3 shows the product’s surplus trajectory. Notice that the cumulative demand is 11,548 units and the cumulative production is 11,148 units over the entire horizon. Since there is an initial inventory of 400 units, the demand is completely satisfied and the surplus level is zero by the end of the planning horizon. The optimal plan is to produce at the maximum production rate in each period except the last period, where we have a production run of 11 days at the maximum production rate followed by a production run of 20 days at the demand rate. A Periodic Maintenance
Problem
Consider a manufacturing system subject to periodic maintenance. That is, the system goes through alternating up and down periods. Down periods correspond to maintenance. Each maintenance period takes up 2” time units and each up period t&es up TU time units. We assume that the number of time periods in the planning horizon, N, is even. In other words, there are N/2 up periods and N/2 down periods. The first period is assumed to be an up period. The product’s demand rate, inventory carrying cost rate, and backlog cost rate are assumed to be invariant throughout the planning horizon. Obviously, the maximum production rate must be zero during a down period. Thii implies that xi+1 = xi - di, for i = 1,3, . . . , N - 1, (22) To be able to use the N-time-period model developed above, all we have to do is to impose equations (22) as equality constraints in the nonlinear programming problem P, and use the following cost value functions in the one-time-period problem:
&
(xf*l - 2:) ,
ifx:i ZOandzi+r
20,
ifxiZOandxi+r
CO,
ifzi
(231
We use this modiied model to study the transient and steady state behavior of a manufacturing system producing a single product and subject to periodic planned maintenance. The plarming horizon consists of 24 time periods. The up period has a length of 20 time +.&is and the maintenance has a length of 5 time rkts. The maximum production rate is 15 un~ts/t~me unit during up periods and the demand rate is 10 units/time unit. The inventory carrying and backlog costs are $l.S/(zsnit/time unit] and $7.5/[unit/time unit], respectively. The system starts with a backlog of 200 units. As in the previous case, Problem P is first solved using the gradient projection technique, then the optimal production plan is constructed using the one-timeperiod problem optimal solution. The optimal surplus trajectory is shown in Figure 4. Notice that the optimal production plan is to eliminate the initial backlog as fast as possible, then as the system catches up, a certain positive surplus is built at the end of each of the following up periods. This positive surplus is usually referred to as the hedging level, since it hedges against the shortage in capacity brought about the maintenance period, Notice that the system reaches the steady state in Period 7 minoring the border effect of Period 24). At the steady state, the hedging level is 41.7 units.
7. CONCLUSION In this paper, we proposed a model for the plarming of the production of a single product over an N-time-period finite horison. The problem was formulated as an optimal control problem.
118
M. EL HAFSIAND S. X. BAI
-
1
1
r Figure 4. Surplus trajectory for the maintenance problem.
The optimal solution was obtained in two steps. First, the one-time-period problem was solved analytically. Then, based on the solution of the one-time-period problem, the N-time-period problem was formulated as a constrained nonlinear programming problem. The latter can be solved using any standard nonlinear programming algorithm. The model can be adapted to different situations. One of the adaptations of the model is to the case of a system with periodic maintenance which we presented in Section 6. Another adaptation could be to the case where customers may be impatient during certain periods and do not tolerate long delays in the delivery of the product. In this case, the backlog cost rate can be chosen such that it reflects customers’ tolerances for late delivery (high backlog cost rates may be assigned to periods of pressure from customers). The main disadvantage of the model is that it assumes that all the information is known with certainty and is static throughout the planning horizon. Thii is not always the case, especially when it comes to demand forecasts, since usually a demand realization is most likely to be different from the forecast of that demand. Nevertheless, the model can still be adapted to such dynamic changes using the idea of rolling horizons [3]. The latter consists of reevaluating the N-timeperiod production plan each period (as demand and capacity realizations occur) and implementing the production plan of the current period only.
REFERENCES 1. H.M. Wagner and T.M. Whitin, Dynamic version of the economic lot size model, Management Science 38 (6), 8Q-Q6 (1958). 2. L.A. Johnson and D.C. Montgomery, Editors, Opsmtions Research in Production Planning, Scheduling, and Inventory Control, Wiley & Sons, New York, (1974). 3. S. Nahmias, Editor, Production and Opemtions Analysis, Irwin, Homewood, IL, (1993). 4. Ft. Rlshel, Dynamic programming and minimum principles for systems with jump Markov diiurbances, SIAM Journal on Control 113 (2), 338-371 (1975). 5. G.L. Olsder and Ft. Suri, The optimaI control of parts routing in a manufacturing system with failure prone machines, Presented at the lgth IEEE Conjemnce on De&ion and Control, (1980). 6. J. Kimemia and S.B. Gershwin, An algorithm for the computer control of a flexible manufacturin.g system, IIE !&anaaci?ioM15 (4), 353-362 (1983). 7. S.X. Bai and S. Varanasi, An optimal production flow control problem with piecewise constant demand, Mathl. Comput. Modelling 34 (7), 87-107 (1996). a. It.. Akella and P.R. Kumar, Optimal control of production rate in a failure prone manufacturing system, IEEE Thneactions on Automatic Control AC-S1 (2), 116-126 (1986). 9. T. Bielecki and P.R. Kumar, Optimality of sero-inventory policies for unreliable manufacturing systems, Opemtions Remumh 36 (4), 532-541 (1988).
Multiperiod Production Planning
119
10. A. Sharifnie, Optimal production control of 8 manufacturing system with machine failures, IEEE Iftnneactions on Automatic Contnd AC-SS (7), 620-625 (1966). 11. E.K. Boukas and A. Haurie, Manufacturing flow control and preventive maintenance: A stoch8stic control 8ppro8ch, IEEE !hmsactiona on Automatic Contnd AC36 (9), 1024-1031 (1990). 12. M.C. C aramanis end G. Liberopolous, Perturbation analysis for the design of flexible m8nuf8cturing system flow controIIem, opecationa Reueorch 40 (6) (1992). 13. J.Q. Hu and D. Xiig, Optimal production control for failure prone manufecturing systems with deterministic up timea, Working p8per, Boeton University, (1992). 14. F.J. Kr&mer and S.X. Bai, A production flow control problem with periodic maintenance, Optimal Con-l Application and Methods 17, 281-307 (1996). 15. D.E. Kirk, OptimOr Control Theory, An Introduction. Network Serb, Prentice Hall, (1970). 16. J.B. Rosen, The gradient projection method for nonlinear progmmming. Part I. Linear constrainta, J. SIAM 8 (l), 181-217 (1960).