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An optimal control method for aggregate production planning in large-scale manufacturing systems with capacity expansion and deterioration
Computersind. EngngVol. 28, No. 4, pp. 851-859, 1995
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Pergamon
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Copyright © 1995 Elsevier Science Printed in Great Britain. All rights 0360-8352/95 $9.50+ 0.00
AN OPTIMAL CONTROL METHOD FOR AGGREGATE PRODUCTION PLANNING IN LARGE-SCALE MANUFACTURING SYSTEMS WITH CAPACITY EXPANSION AND DETERIORATION KONSTANTIN KOGAN and EUGENE KHMELNITSKY Faculty of Engineering, Department of Industrial Engineering, Tel-Aviv University, Ramat-Aviv 69878, Israel
(Received January 1995)
Abstract--Anoptimal control approach to continuous-time aggregate production planning problems is presented. The proposed approach describes the production and capacity evolution (expansion, sell and deterioration) processes in the form of differential equations with regular production, subcontracting and capacity change rates controllable on one hierarchical level. In this way, the traditional disadvantages of the two-level problem consideration (one level for strategic capacity planning and the other for production smoothing) are avoided. Analytical properties for optimal production and capacity control regimes and conditions for their changeover are derived by the maximum principle. Based on these results, an insight into the optimal behaviour of the production system is gained and a fast numerical method is developed to identify and sequence the optimal regimes for arbitrary demand profiles. A computational example illustrates the effectiveness of the approach.
1. INTRODUCTION
There is considerable literature on aggregate production planning. It ranges from one-item continuous-time [1] and discrete [2] models to a more general analysis capturing different levels of aggregation in a hierarchical manner [3, 4]. Comprehensive reviews on models and achievements in this area can be found in the literature [5-7]. Production smoothing [8], hiring and firing [9], subcontracting [10] and capacity expansion [11] are the basic ways of responding to changes in demand. The objective of the aggregate production planning models is to minimize the total production cost (including inventory holding costs, costs of regular production, subcontracting, backlogging, capacity holding and so on). Forms of cost functions, used as a basis for comparing alternative production plans, are a decisive factor in the search for the optimal solution. For piecewise linear cost functions, linear programming formulations have been used in a straightforward manner [9, 12], while nonlinear, but convex cost assumptions lead either to an approximation [13] or a dynamic programming formulation [4]. In this paper, the focus is on an optimal control approach to aggregate production planning that allows production smoothing, subcontracting and capacity evolution (expansion, sell and deterioration) to be modeled on one hierarchical level. The approach presents the problem in the canonical form of optimal control and accounts for production constraints and arbitrary demand profiles. As a result, the maximum principle is applied to the problem and analytical properties of the optimal solution are derived. Based on these properties, an iterative algorithm is constructed to determine optimal production plans for different forms of cost functions. Due to the analytical origin, the algorithm possesses a fast convergence that is not available from linear or dynamic programming. Although single-item optimal production planning is considered in order to derive easy-to-use production smoothing and capacity planning regimes, the approach suggested here can be further extended for multiproduct analysis. 851
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Konstantin Kogan and Eugene Khmelnitsky 2. PROBLEM FORMULATION
Let a production system to be considered along a planning horizon [0, T] be characterized at each moment of time by inventory level X ( t ) and by the maximal production rate V(t). The maximal production rate can be expanded by the development of the capacity (equipment and manpower) and reduced by selling the capacity. At the same time, the capacity decreases (deteriorates) in an uncontrollable and random manner. Such a process is commonly described by a pure-death model [20]. Therefore, in large-scale manufacturing systems (which are under consideration) the rate of deterioration can be assumed to be deterministic and proportional to the capacity itself, flV(t). Also assume that the rates of the capacity expansion and sell, w(t) and y(t), respectively, are controllable and bounded from above. Furthermore, the upper bound of the selling rate is proportional to the capacity itself, 2V(t), thereby ensuring that the capacity is never exhausted. Thus, the dynamic change of the production capacity in the system is described as follows: l?(t) = w(t) - y ( t ) - flV(t), O <~w(t) <~ W,
V(O) = V o,
O <~y ( t ) <~2V(t),
(1) (2)
where W and 2V(t) are the maximal rates of the capacity expansion and sell, respectively. The difference between the total rate of controllable production u (t) and subcontracting s (t) and the rate of a given demand profile, d(t), determines inventory flow X(t). When X ( t ) is positive, it reflects a surplus, otherwise, it is a backlog: S((t) = u(t) + s(t) - d(t),
X(O) = Xo,
X ( T ) = Xv.
(3)
Both rates u(t) and s(t) are restricted by the maximal production rate V(t) and by the maximal subcontracting rate S(t) forecasted along the planning horizon [0, T], respectively: 0 <~u(t) <<.V(t);
0 <~s(t) <<.S(t).
(4)
In order to formulate the objective function, convex cost dependencies for the production process [see equation (3)], i.e. costs of production Cl (u), subcontracting C2(s) and inventory/shortage C3(X), were assumed [4, 10]. Although the remaining costs concerning capacity planning [see equation (1)], i.e. costs of capacity expansion Ca(w), sell Cs(y) and holding C6(V), can also be considered convex, the more traditional linear cost dependencies were adopted [19]. Thus, the aggregate production planning problem is as follows: min C =
j.T
e-P'{Ci [u(t)] + C2[s(t)] + C3[X(t)] + c4w(t) - csy(t) + c6 V(t)} dt,
(5)
0
subject to the constraints in equation (1)-(4), where p is the discount rate. Although a constant discount rate is adopted here to simplify the presentation, different rates for Ci, ci, i = 1. . . . . 6 can be easily accomodated by the solution approach. 3. ANALYSIS OF THE EXTREMAL BEHAVIOR OF THE PRODUCTION SYSTEM
Because the aggregate production planning problem is stated in the canonical form of an optimal control problem, the maximum principle can be used to study the extremal behaviour of the system [14]. The pioneering work by Hwang et al. [15] applies Pontryagin's maximum principle to the aggregate planning to solve a simple problem of unconstrained production with demand that is constant with respect to time. Due to the progress in the maximum principle theory [16], far more general problems with mixed constraints and arbitrary integrable functions of time (the demand profile in the current model) can be considered [17]. 3.1. The maximum principle
The maximum principle applied to the problem in equation (1)-(5) states that if a trajectory [X(t), V(t), u(t), s(t), w(t) and y(t)] is the optimal one, then there exist absolutely continuous
Optimal control method for aggregate production planning
853
functions @x(t) and @v(t) and integrable nonnegative functions au(t) and ay(t) such that the following is true:
~x(t) = e-PtC; [X(t)], ~v(t) = fl@v(t) + c6 e - p ' - a,(t) - 2ay(t), au(t) = ~ k x ( t ) - C~[u(t)]e P', if u(t) = V(t), ay(t)=-@v(t)+cse-"',
if y ( t ) = 2 V ( t ) ,
(6) ~bv(T)= 0, otherwise au(t) = 0 , otherwiseay(t)=0.
(7) (8) (9)
The Hamiltonian: H = -e
P'{C, [u(t)] + C2[s(t)] + C3[X(t)] + c4w(t )
--
csy(t) + c6 V(t)}
+ @x(t)[u(t) + s(t) - d(t)] + @v(t)[w(t) - y ( t )
- flV(t)],
is maximized for each t by the control variables u(t), s(t), w(t) and y ( t ) subject to the constraints in inequalities (2) and (4).
3.2. Optimal production control regimes With respect to the maximum principle, five optimal production control regimes can be derived. If the Hamiltonian as a function of u(t) and s(t) is maximized, then the following two subproblems arise for each t:
u(t)qJx(t) - Cj[u(t)]e-P'~max,
subject to 0 ~< u(t) <<.V(t),
s ( t ) ~ x ( t ) - C2[s(t)]e-P'~max,
subject to 0 ~
Presuming that C~ (0) > C~[ V(t)], i.e. the regular production is always cheaper than subcontracting, the solution for both problems is obtained immediately (see Fig. 1): If ~bx(t) < Ci(O)e P', then u(t) = s(t) = 0 [no production regime (NPR)]. If C~ (O)e p' ~< ~bx(t ) < C~ [V(t)]e -p', then u(t) is the root of the equation C~ [u(t)] = ~x(t)e pt and s(t) = 0 [underproduction regime (UPR)]. In order to be specific, when Cl (u) is of a quadratic form Cl (u) = la I u 2 + bj u + el, then u(t) = [~bx(t)ep' - bl ]/al. If C~[V(t)]e-P'<<.~kx(t)
MSR
C~(O)e-pt
C'l[V(t)]e-Pt
Ci(O)e-Pt NPR
Fig. 1. Optimal production control regimes.
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Konstantin Kogan and Eugene Khmelnitsky
3.3. Optimal capacity control regimes
Similar to the optimal production regimes, five optimal capacity control regimes can be derived when maximizing the Hamiltonian as a function of w(t) and y ( t ) (the third subproblem): [w(t) -y(t)]~bv(t) - [c4 w( t ) - csy( t )]e-P' ~ m a x , subject to O<<.w(t)<~W
and
O<~y(t)<~2V(t).
The solution of this problem is as follows (see Fig. 2): If ~bv(t)<<. c5 e -pt, then w ( t ) = 0 and y ( t ) = x V ( t ) [maximal sell regime (MSR)]. If ~bv(t) = c5 e -p~ holds on an interval of time, then w(t) = 0 and y ( t ) = - f l ( V ( t ) - [C~[X(t)] + p{c6 + c5(fl - p) + C'~[V(t)]}]/C'([V(t)] (see Lemma 1 in the Appendix), [partial sell regime (PSR)]. If c5 e -p' <<.qJv(t) < Ca e -pt, then w(t) = y ( t ) = 0 [no capacity expansion and sell (NES)]. If ~bv(t) = Ca e -pt holds on an interval of time, then y(t) = 0 and w(t) = flV(t) + [C~[X(t)] + p{c 6 + c4(fl + p) + C~[V(t)]}]/C'([V(t)] (see Lemma 2 in the Appendix), [partial expansion regime (PER)]. If c4e -pt <<.~Ov(t), then y ( t ) = 0 and w(t) = W [maximal capacity expansion regime (MCR)]. Note that because qJx(t) and ~bv(t) are continuous functions, then only adjacent regimes are eligible to follow each other (see Figs 1 and 2). This requirement is an important insight into optimal sequencing the regimes. Moreover, the solutions of the three subproblems yield that the greater ~x(t)[qlv(t)], the greater the production rate u ( t ) + v(t) [capacity expansion rate w(t) - y(t)] (see proofs of Lemmas 1 and 2 in the Appendix). This fact will significantly reduce the computational burden of the algorithm presented in Section 4. 4. OPTIMAL CONTROL ALGORITHM FOR PRODUCTION AND CAPACITY PLANNING
Five production and five capacity control regimes that make up various optimal control policies, each of which contains one production and one capacity control regime, were derived. Properties of the extremal behavior of the production system derived in Section 3 allow locating and sequencing of the optimal policies to be carried out numerically by a general algorithm. The algorithm is based on the well-known shooting method [18]. It searches for two missing left-hand boundary conditions ~bx(0) and ~kv(0) by means of iterative integration of differential eqs. (1), (3), (6) and (7). When the integration identifies a PSR or PER regime on which the solution is nonunique (see Corollaries of Lemmas 1 or 2), reshooting is required to locate the optimal moments for getting into and out of these regimes. The iterative procedure terminates when the Vv(t)
c4e-Pt
Cse-Pt PER PSR Fig. 2, Optimal capacity control regimes.
t
Optimal control method for aggregate production planning
855
right-hand boundary conditions X ( T ) = Xr and ~Ov(T)= 0 are satisfied. The algorithm is constructed so that the desired values of Ox(O) and ~Ov(O)as well as the moments for getting into and out of the PSR and PER regimes be the roots of specially defined slack functions.
4. I. Definitions of the slack functions When @v(O) is given, the following six slack functions are calculated by integrating the prime and dual differential equations:
Function F~: ~Ox(O)~X(T) - Xr, i.e. the root of this function is the missing left-hand boundary condition, when neither a PER nor PSR exists on interval [0, 7];
Function Fx: $x(O)~$v(tin) - Cae -p'~", where ti. is defined from the equation ~bx(tio) - C~ [V(tin)]e -p'~" = -p[c6 + c4(fl + p)]e-a'% i.e. the root of this function is
Function F3: Function F4: Function Fs:
Function F~:
the missing left-hand boundary condition, when a PER exists and starts at moment t~o; tout~qJv(t~n) - Cae -p'", i.e. the root of this function is the moment for getting out of the current PER or PSR, when a subsequent PER exists and starts at moment ti,; tout~X(T) - Xr, i.e. the root of this function is the moment for getting out of the current PER or PSR, when neither a PER nor PSR exists on interval [tout, T]; ~ l x ( O ) " * ~ l v ( l i n ) - - C5 e -pt~n, where ti, is defined from the equation $x(ti,) - C'~ [V(tin)]e -p'~" = - p [c6 + cs(fl + p)]e -pt~", i.e. the root of this function is the missing left-hand boundary condition, when a PSR exists and starts at moment tin; tout---~v(tin) -- C5 e -pqn, i.e. the root of this function is the moment for getting out of the current PER or PSR, when a subsequent PSR exists and starts at moment tin.
Note, that all the slack functions are monotonic (see Lemma 3 in the Appendix) and, therefore, there is no need for the slow first-order shooting procedures to be adopted in the algorithm. Instead, fast and simple methods for finding the root of a monotonic function (e.g. the dichotomous, the golden-section or the Fibonacci search) can be applied.
4.2. The algorithm The algorithm consists of the following steps: Step 1. Step 2. Step 3. Step 4.
Step 5.
Step 6. Step 7. Step 8.
Assign any value to ~,v(0). Denote by I an empty set. Find the root of Function F t. If it exists, go to Step 10. Find all roots and corresponding moments tin for Functions F 2 and F 5. Denote by R the set of all moments t~n arranged in ascending order. If no root exists, go to Step 10. For every moment tin ~ R, define the interval [tin, tr] on which the conditions of the Corollary of Lemma 1 (for the roots of F2) or of Lemma 2 (for the roots of Fs) hold. Add the defined intervals to set L Find all roots tout of Functions F3, F 4 and F 6 that belong to the intervals of L and corresponding moments tin Functions F 3 and F 6. If no moment tin exists, then go to Step 6. Otherwise, add these moments to the set R and go to Step 4. Merge intervals [0, tint] and [ti.t, toutl], where t~,~ is a root of F2 or F5 and toutt E [tint, tr]. Assign j = 1. Merge intervals [to=j, tinj+~] and [t~j+l, to.tj+~], where t~nj+t is a root of F 3 o r F 6 and toutj+ I E [tinj+ I, if].
Step 9. If to,tj+t is a root of F4, then exit, otherwise j = j + 1 and go to Step 8. Step 10. Integrate the differential equations (1), (3), (6) and (7) on the time interval [0, T] to determine the prime and dual variables X(t), V(t), $x(t) and ~bv(t). For each moment of time, the control variables u(t), s(t), w(t) and y ( t ) are defined from the conditions derived in Section 3 (optimal production and capacity control regimes). Step 11. If the right-hand boundary condition $v(T) = 0 is satisfied with a given accuracy, then stop; the optimal solution has been found. Otherwise, correct the value of Sv(0) with respect to a root-finding method and go to Step 2. CAIE 2g/4~L
856
Konstantin Kogan and Eugene Khmelnitsky Table I. Second-order polynomial cost function parameters
a~ b, c~
C, (u)
C2(s)
C3(X)
0.00025 0.0 0.005
0.0007 0.003 0.0
0.000001/0.00002 0.0 0.0
5. N U M E R I C A L E X A M P L E
Numerical computations have been conducted to test the algorithm in a realistic production environment with complex demand profiles. An example from a soap manufacturing company was considered. The company utilizes a two-step process (three-level bill of materials) producing suds and several types of toilet soap which make up the aggregate product. At the beginning of the planning horizon 50 mixers produce the product at maximal rate V0 = 8 (product units per time unit). The liquid soaps obtained after mixing are pumped to an intermediate storage facility (aggregate buffer). The liquid is pumped from the buffer to a packing line of infinite capacity which releases the finished product. The cost dependencies were approximated by the quadratic polynomial functions of the form Ci(z) = ½ai22 + big + cl, i = 1, 2 and 3 (see Table 1, where t w o values for a 3 correspond to the costs incurred by surpluses and backlogs). For the linear costs, the parameters Ca, c5 and c6 are given as 0.05, 0.01 and 0.00001, respectively. The other parameters of the problem are set as follows: X0 = XT = 0 (product units), S ( t ) - 4 (product units per time unit), W = 0.1 (product units per square time units), p = 0.5% per time unit, ~ = 0.2% per time and 2 = 0.5% per time. In order to find values for the slack functions, the integration of the prime and dual systems was carried out by the Runge-Kutta method of the fourth-order with mesh points being equally distributed throughout the planning horizon [0, 50] and the step size being 0.25. The dichotomous search was adopted for solving root-finding problems, whereas, the accuracy of the search was chosen to be 0.001. The initial value for the dual variable assigned as Sv(0) = 0 was found on the optimal solution as Sv(0)= 0.0806. The results of the trajectory computations along with the given demand profile are presented in Fig. 3. The highest peak in the demand is compensated by both maximal capacity expansion (MCR) 17 13 9 5
i
,
1
v
I
i~ t
40 20 0
I 9
t
I
8 t
s(t) 4
0
P
w(t)
t
0.1
I 10
II
I
Ik
20
3o
I /40
It
-y(t)
t
Fig. 3. D e m a n d profile d(t), inventory levels X(t), capacity evolution V(t) and rates of production u(t), subcontracting s(t), capacity expansion w(t) and sell y(t).
Optimal control method for aggregate production planning
857
and partial subcontracting (PSR), while on the overall planning horizon, the regular production capacity is fully utilized. As a result, two curves u ( t ) and V ( t ) coincide in Fig. 3. Close to the end of the planning horizon, an extra capacity is sold (MSR). Formally, the optimal sequence of the production and capacity control policies computed for the given demand is as follows: PSR
-- MCR
=~ P S R
-- NES
=~ F P R
-
NES
=~ F P R
-
MSR
=~ P S R
-- MSR,
where the moments of changing over are equal to 13.5, 28.5, 38.5 and 40.5 time units. More than 100 runs of the algorithm have been carried out for different demands, cost functions and boundary conditions. The time required for an IBM PC-486 computer to compute the optimal solutions ranged from 0.6 to 9.0 min. 6. CONCLUSIONS
A continuous-time optimal control model has been stated to manage a large-scale manufacturing system where product capacity deteriorates with a rate proportional to the capacity itself. The system is controlled by production smoothing, subcontracting and capacity expansion and sell at one hierarchical level of production planning. Concurrent consideration of these means with respect to their linear and nonlinear costs allows correct production policies to be formulated in response to the arbitrary change in demand. The analysis of the maximum principle for the problem results in derivation of the optimal production and capacity control regimes and conditions for their changing over. A fast numerical algorithm which is based on finding the roots of specially constructed monotonic slack functions is suggested. The algorithm takes advantage of the analytical properties of the optimal policies for locating and sequencing them. REFERENCES 1. K. J. Arrow and S. Karlin. Smooth production plans. Studies in the Mathematical Theory oflnventory and Production, pp. 70-85. Stanford Universtiy Press, CA (1958). 2. C. C. Holt, F. Modigliani, J. F. Muth and H. A. Simon. Planning Production, Inventories and Work Force. Prentice-Hall, Englewood Cliffs, NJ (1960). 3. J. B. Lasserre. An integrated model for job-shop planning and scheduling. Mgt Sci. 38, 1201-1211 (1992). 4. S. P. Sethi, M. I. Taksar and Q. Zhang. Capacity and production decisions in stochastic manufacturing systems: an asymptotic optimal hierarchical approach. Product. Oper Mgt 1, 367-392 (1992). 5. S. C. Graves et al. (Eds). Handbooks in OR and MS. Elsevier, Amsterdam (1993). 6. A. C. Hax and D. Candea. Production and Inventory Management. Prentice-Hall, Englewood Cliffs, NJ (1984). 7. J. Bitran and A. Hax. On the design of hierarchical production planning systems. Decis. Sci. 8, 28-55 (1977). 8. J. O. McClain, L. J. Thomas and J. B. Mazzola. Operations Management: Production of Goods and Services. Prentice-Hall, Englewood Cliffs, NJ (1992). 9. A. C. Hax. Aggregate production planning. Handbook of Operations Research. Van Nostrand Reinhold, NY (1978). 10. M. I. Kamien and L. Li. Subcontracting, coordination, flexibility and production smoothing in aggregate planning. Mgt Sci. 36, 1352-1363 (1990). 11. J. C. Bean and R. L. Smith. Optimal capacity expansion over an infinite horizon. Mgt Sci. 31, 1523-1532 (1985). 12. S. A. Lipman, A. F. Rolfe, H. M. Wagner and J. S. C. Yuan. Algorithm for optimal production scheduling and employment smoothing. Opers Res. 15, 1011-1029 (1967). 13. W. H. Taubert. A search decision rule for the aggregate scheduling pattern. Mgt Sci. 14, B343-B359 (1968). 14. A. E. Bryson and Y.-C. Ho. Applied Optimal Control. Ginn & Co, Waltham (1969). 15. C. L. Hwang, L. T. Fan and L. I. Erickson. Optimum production planning by the maximum principles. Mgmt Sci. 13, 751-755 (1967). 16. A. Y. Dubovitsky and A. A. Milyutin. Theory of the maximum principle Methods of Theory of Extremum Problems in Economy, Nauka, Moscow (1981). 17. E. Khmelnitsky and K. Kogan. Necessary optimality conditions for a generalized problem of production scheduling. Optimal Control Applic. Meth. 15, 215-222 (1994). 18. R. L. Burden and J. D. Faires. Numerical Analysis. Boston, PWS-KENT (1989). 19. H. Luss. Operations research and capacity expansion problems: a survey. Opers Res. 30, 908-928 (1982). 20. H. M. Wagner. Principles of Operations Research with Applications to Managerial Decisions. Prentice-Hall, Englewood Cliffs, NJ (1969). APPENDIX
Lemma 1 Let the partial sell regime (PSR) be realized on a time interval [q, t2] then y ( t ) = - f l V ( t ) - [ C ~ [ X ( t ) ] + p{c 6+ c5([~ + p) + C~[V(t)}]/C~[V(t)].
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Konstantin Kogan and Eugene Khmelnitsky
/'roof 1. Differentiating the condition of the PSR, ~,v(t)= c5 e -p', and taking into the account the dual equation (7), the following equation results: [c6 + c5(fl + p)]e -pt = a~(t) + 2ar(t ).
(A1)
Because the PSR is under consideration, ay(t) equals 0 [see equation (9)]. Next, equation (A1) cannot be satified on the no production and underproduction regimes because a,(t) is equal to zero on these regimes [see equation (8)]. Thus, the FPR is the only regime that can occur along with the PSR [i.e. u(t) = V(t)] and a~(t) = Ox(t) - C~ [V(t)]e -p'. Consequently, equation (A1) takes the following form:
~kx(t ) = {c6 + c5(fl + p) + C~ [V(t)]}e -p'.
(A2)
2. Differentiating equation (A2) results in the following equation:
C][X(t)] = - p {c6 + cs(fl + p) + C~ [V(t)]} + CT[V(t)][-y(t) -/~V(t)].
(A3)
3. From equation (A3), it immediately follows that:
y(t) = - f l V ( t ) -
C~[X(t)] + p {c6 + c5(fl + p) + C'l [V(t)]}
c','[v(t)]
Corollary If, on an interval of time, the partial capacity sell regime is realized, then the following three conditions hold: (1) ~bv(t ) = c 5 e-pt;
(2) ~kx(t ) = {c6 + c5(fl + p) + C~[V(t)l}e-Pt;
C; [X(t)] + p {c6 + c s (fl + p) + C~[V(t)]} <~2V(t). C'([V(t)]
(3) 0 ~< --flV(t)
Proof The first condition holds by definition of the PSR. The second one was proved in Item I of Lemma 1. The third condition emanates from equation (2). Lemma 2 Let the PER be realized C~[V(t)]}]/C'~[V(t)].
on
a
time
interval
[tl,/2],
then
w(t)=flV(t)-[-[C~[X(t)]~-p{c6q-c4(fl+p)+
Proof 1. Differentiating the condition of the SPR, ~,v(t) = c4 e -p', and taking into account the dual equation (7), the following equation results:
[c6 + c4(fl + p)]e -pl = a~(t) + Aay(t).
(A4)
Because the PER is under consideration, %(t) equals 0 [see equation (9)]. Next, equation (A4) cannot be satisfied on the no production and underproduction regimes because a~(t) is equal to zero on these regimes [see equation (8)]. Thus, the F P R is the only regime that can occur along with the PER [i.e. u(t) = V(t)] and au(t ) = ~x(t) - C~ [V(t)]e -p'. Consequently, equation (A4) takes the following form:
~x(t) = {c6 + c,(fl + p) + C~[V(t)]}e -pt.
(A5)
2. Differentiating equation (A5) results in the following equation:
C'3[X(t)I = - 0 { c 6 + e4(/~ + p) + C~[V(t)]} + CT[V(t)][w(t) -/~V(t)l.
(A6)
3. From equation (A6), it immediately follows that:
w(t) = flV(t) +
C;[X(t)] + p{c6 + c4(/~ + p) + C~ [V(t)]}
c','[v(t)]
Corollary If, on an interval of time, the partial capacity expansion regime is realized, then the following three conditions hold: (1) ~kr(t) = c4 e-P'; (3) 0 ~
(2) ~bx(t ) --- {c6 + c4(fl + p) + C~[V(t)]}e-P';
C~[X(t)] + p {c, + c4([3 + p) + C~ [V(t)]} <<_w. C~[V(t)]
Proof The first condition holds by definition of the PER. The second one was proved in Item 1 of Lemma 2. The third condition emanates from equation (2). Lemma 3 Slack Functions F t , F 2, F3, F4, F5 and F 6 are nondecreasing functions of their arguments.
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Proof Let the system of ordinary differential equations (i), (3), (6) and (7) be given, and [Xi (t), VI (t), ~bxl (t), ~bvl (t)] and IX2 (t),
V2(t), tpx2(t), ~ ( t ) ] be two solution of the system with the same initial conditions except ~xl (0)> ~x2(0): 1. Consider a small interval of time [0,t~] on which ~x~(t)> ~x2(t).
2. From maximizing the Hamiltonian (see Section 3, optimal production control regimes) it follows that u(t)+ s(t) is a nondecreasing function of ~x(t), i.e. ul (t) + sL(t) >/u 2(t) + s2(t), t ~ [0, tj ]. 3. From equation (3), it immediately follows that ~l(t)/> )(2(t) on the same interval of time. 4. Because Xl(0) = :(2(0) and )(l(t) >t )C2(t), then Xl(t) >tX2(t). 5. Because C3(X) is a convex differentiable function, then C~(X) is a nondecreasing function. From equation (6) the following results: ~xl (t) 1> ~x2(t). 6. Because ~xl (0) > ~:a (0) and ~xl (t)/> ~x2(t), then ~xl (tl) - Ip~ (tl) >~~xl (0) - ~x~(0). 7. It has been proven that when ~bxl(0) > ~ ( 0 ) , then Xl(tl)>I Xz(tl) and ~xl(tl)- ~bx2(tl)>~~x~(0)- ¢'x2(0)> 0. Therefore, the proof can be continued in the same way on subsequent time intervals each of which is no shorter than [0, tl] until the planning horizon [0, T] is covered. Thus, Xt(T) >1X2(T) and Ft is a nondecreasing function.
The proof for the remaining slack functions is analogous and, therefore, is omitted here.
~
Pergamon
0360-8352(95)00010-0
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Copyright © 1995 Elsevier Science Printed in Great Britain. All rights 0360-8352/95 $9.50+ 0.00
AN OPTIMAL CONTROL METHOD FOR AGGREGATE PRODUCTION PLANNING IN LARGE-SCALE MANUFACTURING SYSTEMS WITH CAPACITY EXPANSION AND DETERIORATION KONSTANTIN KOGAN and EUGENE KHMELNITSKY Faculty of Engineering, Department of Industrial Engineering, Tel-Aviv University, Ramat-Aviv 69878, Israel
(Received January 1995)
Abstract--Anoptimal control approach to continuous-time aggregate production planning problems is presented. The proposed approach describes the production and capacity evolution (expansion, sell and deterioration) processes in the form of differential equations with regular production, subcontracting and capacity change rates controllable on one hierarchical level. In this way, the traditional disadvantages of the two-level problem consideration (one level for strategic capacity planning and the other for production smoothing) are avoided. Analytical properties for optimal production and capacity control regimes and conditions for their changeover are derived by the maximum principle. Based on these results, an insight into the optimal behaviour of the production system is gained and a fast numerical method is developed to identify and sequence the optimal regimes for arbitrary demand profiles. A computational example illustrates the effectiveness of the approach.
1. INTRODUCTION
There is considerable literature on aggregate production planning. It ranges from one-item continuous-time [1] and discrete [2] models to a more general analysis capturing different levels of aggregation in a hierarchical manner [3, 4]. Comprehensive reviews on models and achievements in this area can be found in the literature [5-7]. Production smoothing [8], hiring and firing [9], subcontracting [10] and capacity expansion [11] are the basic ways of responding to changes in demand. The objective of the aggregate production planning models is to minimize the total production cost (including inventory holding costs, costs of regular production, subcontracting, backlogging, capacity holding and so on). Forms of cost functions, used as a basis for comparing alternative production plans, are a decisive factor in the search for the optimal solution. For piecewise linear cost functions, linear programming formulations have been used in a straightforward manner [9, 12], while nonlinear, but convex cost assumptions lead either to an approximation [13] or a dynamic programming formulation [4]. In this paper, the focus is on an optimal control approach to aggregate production planning that allows production smoothing, subcontracting and capacity evolution (expansion, sell and deterioration) to be modeled on one hierarchical level. The approach presents the problem in the canonical form of optimal control and accounts for production constraints and arbitrary demand profiles. As a result, the maximum principle is applied to the problem and analytical properties of the optimal solution are derived. Based on these properties, an iterative algorithm is constructed to determine optimal production plans for different forms of cost functions. Due to the analytical origin, the algorithm possesses a fast convergence that is not available from linear or dynamic programming. Although single-item optimal production planning is considered in order to derive easy-to-use production smoothing and capacity planning regimes, the approach suggested here can be further extended for multiproduct analysis. 851
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Konstantin Kogan and Eugene Khmelnitsky 2. PROBLEM FORMULATION
Let a production system to be considered along a planning horizon [0, T] be characterized at each moment of time by inventory level X ( t ) and by the maximal production rate V(t). The maximal production rate can be expanded by the development of the capacity (equipment and manpower) and reduced by selling the capacity. At the same time, the capacity decreases (deteriorates) in an uncontrollable and random manner. Such a process is commonly described by a pure-death model [20]. Therefore, in large-scale manufacturing systems (which are under consideration) the rate of deterioration can be assumed to be deterministic and proportional to the capacity itself, flV(t). Also assume that the rates of the capacity expansion and sell, w(t) and y(t), respectively, are controllable and bounded from above. Furthermore, the upper bound of the selling rate is proportional to the capacity itself, 2V(t), thereby ensuring that the capacity is never exhausted. Thus, the dynamic change of the production capacity in the system is described as follows: l?(t) = w(t) - y ( t ) - flV(t), O <~w(t) <~ W,
V(O) = V o,
O <~y ( t ) <~2V(t),
(1) (2)
where W and 2V(t) are the maximal rates of the capacity expansion and sell, respectively. The difference between the total rate of controllable production u (t) and subcontracting s (t) and the rate of a given demand profile, d(t), determines inventory flow X(t). When X ( t ) is positive, it reflects a surplus, otherwise, it is a backlog: S((t) = u(t) + s(t) - d(t),
X(O) = Xo,
X ( T ) = Xv.
(3)
Both rates u(t) and s(t) are restricted by the maximal production rate V(t) and by the maximal subcontracting rate S(t) forecasted along the planning horizon [0, T], respectively: 0 <~u(t) <<.V(t);
0 <~s(t) <<.S(t).
(4)
In order to formulate the objective function, convex cost dependencies for the production process [see equation (3)], i.e. costs of production Cl (u), subcontracting C2(s) and inventory/shortage C3(X), were assumed [4, 10]. Although the remaining costs concerning capacity planning [see equation (1)], i.e. costs of capacity expansion Ca(w), sell Cs(y) and holding C6(V), can also be considered convex, the more traditional linear cost dependencies were adopted [19]. Thus, the aggregate production planning problem is as follows: min C =
j.T
e-P'{Ci [u(t)] + C2[s(t)] + C3[X(t)] + c4w(t) - csy(t) + c6 V(t)} dt,
(5)
0
subject to the constraints in equation (1)-(4), where p is the discount rate. Although a constant discount rate is adopted here to simplify the presentation, different rates for Ci, ci, i = 1. . . . . 6 can be easily accomodated by the solution approach. 3. ANALYSIS OF THE EXTREMAL BEHAVIOR OF THE PRODUCTION SYSTEM
Because the aggregate production planning problem is stated in the canonical form of an optimal control problem, the maximum principle can be used to study the extremal behaviour of the system [14]. The pioneering work by Hwang et al. [15] applies Pontryagin's maximum principle to the aggregate planning to solve a simple problem of unconstrained production with demand that is constant with respect to time. Due to the progress in the maximum principle theory [16], far more general problems with mixed constraints and arbitrary integrable functions of time (the demand profile in the current model) can be considered [17]. 3.1. The maximum principle
The maximum principle applied to the problem in equation (1)-(5) states that if a trajectory [X(t), V(t), u(t), s(t), w(t) and y(t)] is the optimal one, then there exist absolutely continuous
Optimal control method for aggregate production planning
853
functions @x(t) and @v(t) and integrable nonnegative functions au(t) and ay(t) such that the following is true:
~x(t) = e-PtC; [X(t)], ~v(t) = fl@v(t) + c6 e - p ' - a,(t) - 2ay(t), au(t) = ~ k x ( t ) - C~[u(t)]e P', if u(t) = V(t), ay(t)=-@v(t)+cse-"',
if y ( t ) = 2 V ( t ) ,
(6) ~bv(T)= 0, otherwise au(t) = 0 , otherwiseay(t)=0.
(7) (8) (9)
The Hamiltonian: H = -e
P'{C, [u(t)] + C2[s(t)] + C3[X(t)] + c4w(t )
--
csy(t) + c6 V(t)}
+ @x(t)[u(t) + s(t) - d(t)] + @v(t)[w(t) - y ( t )
- flV(t)],
is maximized for each t by the control variables u(t), s(t), w(t) and y ( t ) subject to the constraints in inequalities (2) and (4).
3.2. Optimal production control regimes With respect to the maximum principle, five optimal production control regimes can be derived. If the Hamiltonian as a function of u(t) and s(t) is maximized, then the following two subproblems arise for each t:
u(t)qJx(t) - Cj[u(t)]e-P'~max,
subject to 0 ~< u(t) <<.V(t),
s ( t ) ~ x ( t ) - C2[s(t)]e-P'~max,
subject to 0 ~
Presuming that C~ (0) > C~[ V(t)], i.e. the regular production is always cheaper than subcontracting, the solution for both problems is obtained immediately (see Fig. 1): If ~bx(t) < Ci(O)e P', then u(t) = s(t) = 0 [no production regime (NPR)]. If C~ (O)e p' ~< ~bx(t ) < C~ [V(t)]e -p', then u(t) is the root of the equation C~ [u(t)] = ~x(t)e pt and s(t) = 0 [underproduction regime (UPR)]. In order to be specific, when Cl (u) is of a quadratic form Cl (u) = la I u 2 + bj u + el, then u(t) = [~bx(t)ep' - bl ]/al. If C~[V(t)]e-P'<<.~kx(t)
MSR
C~(O)e-pt
C'l[V(t)]e-Pt
Ci(O)e-Pt NPR
Fig. 1. Optimal production control regimes.
854
Konstantin Kogan and Eugene Khmelnitsky
3.3. Optimal capacity control regimes
Similar to the optimal production regimes, five optimal capacity control regimes can be derived when maximizing the Hamiltonian as a function of w(t) and y ( t ) (the third subproblem): [w(t) -y(t)]~bv(t) - [c4 w( t ) - csy( t )]e-P' ~ m a x , subject to O<<.w(t)<~W
and
O<~y(t)<~2V(t).
The solution of this problem is as follows (see Fig. 2): If ~bv(t)<<. c5 e -pt, then w ( t ) = 0 and y ( t ) = x V ( t ) [maximal sell regime (MSR)]. If ~bv(t) = c5 e -p~ holds on an interval of time, then w(t) = 0 and y ( t ) = - f l ( V ( t ) - [C~[X(t)] + p{c6 + c5(fl - p) + C'~[V(t)]}]/C'([V(t)] (see Lemma 1 in the Appendix), [partial sell regime (PSR)]. If c5 e -p' <<.qJv(t) < Ca e -pt, then w(t) = y ( t ) = 0 [no capacity expansion and sell (NES)]. If ~bv(t) = Ca e -pt holds on an interval of time, then y(t) = 0 and w(t) = flV(t) + [C~[X(t)] + p{c 6 + c4(fl + p) + C~[V(t)]}]/C'([V(t)] (see Lemma 2 in the Appendix), [partial expansion regime (PER)]. If c4e -pt <<.~Ov(t), then y ( t ) = 0 and w(t) = W [maximal capacity expansion regime (MCR)]. Note that because qJx(t) and ~bv(t) are continuous functions, then only adjacent regimes are eligible to follow each other (see Figs 1 and 2). This requirement is an important insight into optimal sequencing the regimes. Moreover, the solutions of the three subproblems yield that the greater ~x(t)[qlv(t)], the greater the production rate u ( t ) + v(t) [capacity expansion rate w(t) - y(t)] (see proofs of Lemmas 1 and 2 in the Appendix). This fact will significantly reduce the computational burden of the algorithm presented in Section 4. 4. OPTIMAL CONTROL ALGORITHM FOR PRODUCTION AND CAPACITY PLANNING
Five production and five capacity control regimes that make up various optimal control policies, each of which contains one production and one capacity control regime, were derived. Properties of the extremal behavior of the production system derived in Section 3 allow locating and sequencing of the optimal policies to be carried out numerically by a general algorithm. The algorithm is based on the well-known shooting method [18]. It searches for two missing left-hand boundary conditions ~bx(0) and ~kv(0) by means of iterative integration of differential eqs. (1), (3), (6) and (7). When the integration identifies a PSR or PER regime on which the solution is nonunique (see Corollaries of Lemmas 1 or 2), reshooting is required to locate the optimal moments for getting into and out of these regimes. The iterative procedure terminates when the Vv(t)
c4e-Pt
Cse-Pt PER PSR Fig. 2, Optimal capacity control regimes.
t
Optimal control method for aggregate production planning
855
right-hand boundary conditions X ( T ) = Xr and ~Ov(T)= 0 are satisfied. The algorithm is constructed so that the desired values of Ox(O) and ~Ov(O)as well as the moments for getting into and out of the PSR and PER regimes be the roots of specially defined slack functions.
4. I. Definitions of the slack functions When @v(O) is given, the following six slack functions are calculated by integrating the prime and dual differential equations:
Function F~: ~Ox(O)~X(T) - Xr, i.e. the root of this function is the missing left-hand boundary condition, when neither a PER nor PSR exists on interval [0, 7];
Function Fx: $x(O)~$v(tin) - Cae -p'~", where ti. is defined from the equation ~bx(tio) - C~ [V(tin)]e -p'~" = -p[c6 + c4(fl + p)]e-a'% i.e. the root of this function is
Function F3: Function F4: Function Fs:
Function F~:
the missing left-hand boundary condition, when a PER exists and starts at moment t~o; tout~qJv(t~n) - Cae -p'", i.e. the root of this function is the moment for getting out of the current PER or PSR, when a subsequent PER exists and starts at moment ti,; tout~X(T) - Xr, i.e. the root of this function is the moment for getting out of the current PER or PSR, when neither a PER nor PSR exists on interval [tout, T]; ~ l x ( O ) " * ~ l v ( l i n ) - - C5 e -pt~n, where ti, is defined from the equation $x(ti,) - C'~ [V(tin)]e -p'~" = - p [c6 + cs(fl + p)]e -pt~", i.e. the root of this function is the missing left-hand boundary condition, when a PSR exists and starts at moment tin; tout---~v(tin) -- C5 e -pqn, i.e. the root of this function is the moment for getting out of the current PER or PSR, when a subsequent PSR exists and starts at moment tin.
Note, that all the slack functions are monotonic (see Lemma 3 in the Appendix) and, therefore, there is no need for the slow first-order shooting procedures to be adopted in the algorithm. Instead, fast and simple methods for finding the root of a monotonic function (e.g. the dichotomous, the golden-section or the Fibonacci search) can be applied.
4.2. The algorithm The algorithm consists of the following steps: Step 1. Step 2. Step 3. Step 4.
Step 5.
Step 6. Step 7. Step 8.
Assign any value to ~,v(0). Denote by I an empty set. Find the root of Function F t. If it exists, go to Step 10. Find all roots and corresponding moments tin for Functions F 2 and F 5. Denote by R the set of all moments t~n arranged in ascending order. If no root exists, go to Step 10. For every moment tin ~ R, define the interval [tin, tr] on which the conditions of the Corollary of Lemma 1 (for the roots of F2) or of Lemma 2 (for the roots of Fs) hold. Add the defined intervals to set L Find all roots tout of Functions F3, F 4 and F 6 that belong to the intervals of L and corresponding moments tin Functions F 3 and F 6. If no moment tin exists, then go to Step 6. Otherwise, add these moments to the set R and go to Step 4. Merge intervals [0, tint] and [ti.t, toutl], where t~,~ is a root of F2 or F5 and toutt E [tint, tr]. Assign j = 1. Merge intervals [to=j, tinj+~] and [t~j+l, to.tj+~], where t~nj+t is a root of F 3 o r F 6 and toutj+ I E [tinj+ I, if].
Step 9. If to,tj+t is a root of F4, then exit, otherwise j = j + 1 and go to Step 8. Step 10. Integrate the differential equations (1), (3), (6) and (7) on the time interval [0, T] to determine the prime and dual variables X(t), V(t), $x(t) and ~bv(t). For each moment of time, the control variables u(t), s(t), w(t) and y ( t ) are defined from the conditions derived in Section 3 (optimal production and capacity control regimes). Step 11. If the right-hand boundary condition $v(T) = 0 is satisfied with a given accuracy, then stop; the optimal solution has been found. Otherwise, correct the value of Sv(0) with respect to a root-finding method and go to Step 2. CAIE 2g/4~L
856
Konstantin Kogan and Eugene Khmelnitsky Table I. Second-order polynomial cost function parameters
a~ b, c~
C, (u)
C2(s)
C3(X)
0.00025 0.0 0.005
0.0007 0.003 0.0
0.000001/0.00002 0.0 0.0
5. N U M E R I C A L E X A M P L E
Numerical computations have been conducted to test the algorithm in a realistic production environment with complex demand profiles. An example from a soap manufacturing company was considered. The company utilizes a two-step process (three-level bill of materials) producing suds and several types of toilet soap which make up the aggregate product. At the beginning of the planning horizon 50 mixers produce the product at maximal rate V0 = 8 (product units per time unit). The liquid soaps obtained after mixing are pumped to an intermediate storage facility (aggregate buffer). The liquid is pumped from the buffer to a packing line of infinite capacity which releases the finished product. The cost dependencies were approximated by the quadratic polynomial functions of the form Ci(z) = ½ai22 + big + cl, i = 1, 2 and 3 (see Table 1, where t w o values for a 3 correspond to the costs incurred by surpluses and backlogs). For the linear costs, the parameters Ca, c5 and c6 are given as 0.05, 0.01 and 0.00001, respectively. The other parameters of the problem are set as follows: X0 = XT = 0 (product units), S ( t ) - 4 (product units per time unit), W = 0.1 (product units per square time units), p = 0.5% per time unit, ~ = 0.2% per time and 2 = 0.5% per time. In order to find values for the slack functions, the integration of the prime and dual systems was carried out by the Runge-Kutta method of the fourth-order with mesh points being equally distributed throughout the planning horizon [0, 50] and the step size being 0.25. The dichotomous search was adopted for solving root-finding problems, whereas, the accuracy of the search was chosen to be 0.001. The initial value for the dual variable assigned as Sv(0) = 0 was found on the optimal solution as Sv(0)= 0.0806. The results of the trajectory computations along with the given demand profile are presented in Fig. 3. The highest peak in the demand is compensated by both maximal capacity expansion (MCR) 17 13 9 5
i
,
1
v
I
i~ t
40 20 0
I 9
t
I
8 t
s(t) 4
0
P
w(t)
t
0.1
I 10
II
I
Ik
20
3o
I /40
It
-y(t)
t
Fig. 3. D e m a n d profile d(t), inventory levels X(t), capacity evolution V(t) and rates of production u(t), subcontracting s(t), capacity expansion w(t) and sell y(t).
Optimal control method for aggregate production planning
857
and partial subcontracting (PSR), while on the overall planning horizon, the regular production capacity is fully utilized. As a result, two curves u ( t ) and V ( t ) coincide in Fig. 3. Close to the end of the planning horizon, an extra capacity is sold (MSR). Formally, the optimal sequence of the production and capacity control policies computed for the given demand is as follows: PSR
-- MCR
=~ P S R
-- NES
=~ F P R
-
NES
=~ F P R
-
MSR
=~ P S R
-- MSR,
where the moments of changing over are equal to 13.5, 28.5, 38.5 and 40.5 time units. More than 100 runs of the algorithm have been carried out for different demands, cost functions and boundary conditions. The time required for an IBM PC-486 computer to compute the optimal solutions ranged from 0.6 to 9.0 min. 6. CONCLUSIONS
A continuous-time optimal control model has been stated to manage a large-scale manufacturing system where product capacity deteriorates with a rate proportional to the capacity itself. The system is controlled by production smoothing, subcontracting and capacity expansion and sell at one hierarchical level of production planning. Concurrent consideration of these means with respect to their linear and nonlinear costs allows correct production policies to be formulated in response to the arbitrary change in demand. The analysis of the maximum principle for the problem results in derivation of the optimal production and capacity control regimes and conditions for their changing over. A fast numerical algorithm which is based on finding the roots of specially constructed monotonic slack functions is suggested. The algorithm takes advantage of the analytical properties of the optimal policies for locating and sequencing them. REFERENCES 1. K. J. Arrow and S. Karlin. Smooth production plans. Studies in the Mathematical Theory oflnventory and Production, pp. 70-85. Stanford Universtiy Press, CA (1958). 2. C. C. Holt, F. Modigliani, J. F. Muth and H. A. Simon. Planning Production, Inventories and Work Force. Prentice-Hall, Englewood Cliffs, NJ (1960). 3. J. B. Lasserre. An integrated model for job-shop planning and scheduling. Mgt Sci. 38, 1201-1211 (1992). 4. S. P. Sethi, M. I. Taksar and Q. Zhang. Capacity and production decisions in stochastic manufacturing systems: an asymptotic optimal hierarchical approach. Product. Oper Mgt 1, 367-392 (1992). 5. S. C. Graves et al. (Eds). Handbooks in OR and MS. Elsevier, Amsterdam (1993). 6. A. C. Hax and D. Candea. Production and Inventory Management. Prentice-Hall, Englewood Cliffs, NJ (1984). 7. J. Bitran and A. Hax. On the design of hierarchical production planning systems. Decis. Sci. 8, 28-55 (1977). 8. J. O. McClain, L. J. Thomas and J. B. Mazzola. Operations Management: Production of Goods and Services. Prentice-Hall, Englewood Cliffs, NJ (1992). 9. A. C. Hax. Aggregate production planning. Handbook of Operations Research. Van Nostrand Reinhold, NY (1978). 10. M. I. Kamien and L. Li. Subcontracting, coordination, flexibility and production smoothing in aggregate planning. Mgt Sci. 36, 1352-1363 (1990). 11. J. C. Bean and R. L. Smith. Optimal capacity expansion over an infinite horizon. Mgt Sci. 31, 1523-1532 (1985). 12. S. A. Lipman, A. F. Rolfe, H. M. Wagner and J. S. C. Yuan. Algorithm for optimal production scheduling and employment smoothing. Opers Res. 15, 1011-1029 (1967). 13. W. H. Taubert. A search decision rule for the aggregate scheduling pattern. Mgt Sci. 14, B343-B359 (1968). 14. A. E. Bryson and Y.-C. Ho. Applied Optimal Control. Ginn & Co, Waltham (1969). 15. C. L. Hwang, L. T. Fan and L. I. Erickson. Optimum production planning by the maximum principles. Mgmt Sci. 13, 751-755 (1967). 16. A. Y. Dubovitsky and A. A. Milyutin. Theory of the maximum principle Methods of Theory of Extremum Problems in Economy, Nauka, Moscow (1981). 17. E. Khmelnitsky and K. Kogan. Necessary optimality conditions for a generalized problem of production scheduling. Optimal Control Applic. Meth. 15, 215-222 (1994). 18. R. L. Burden and J. D. Faires. Numerical Analysis. Boston, PWS-KENT (1989). 19. H. Luss. Operations research and capacity expansion problems: a survey. Opers Res. 30, 908-928 (1982). 20. H. M. Wagner. Principles of Operations Research with Applications to Managerial Decisions. Prentice-Hall, Englewood Cliffs, NJ (1969). APPENDIX
Lemma 1 Let the partial sell regime (PSR) be realized on a time interval [q, t2] then y ( t ) = - f l V ( t ) - [ C ~ [ X ( t ) ] + p{c 6+ c5([~ + p) + C~[V(t)}]/C~[V(t)].
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Konstantin Kogan and Eugene Khmelnitsky
/'roof 1. Differentiating the condition of the PSR, ~,v(t)= c5 e -p', and taking into the account the dual equation (7), the following equation results: [c6 + c5(fl + p)]e -pt = a~(t) + 2ar(t ).
(A1)
Because the PSR is under consideration, ay(t) equals 0 [see equation (9)]. Next, equation (A1) cannot be satified on the no production and underproduction regimes because a,(t) is equal to zero on these regimes [see equation (8)]. Thus, the FPR is the only regime that can occur along with the PSR [i.e. u(t) = V(t)] and a~(t) = Ox(t) - C~ [V(t)]e -p'. Consequently, equation (A1) takes the following form:
~kx(t ) = {c6 + c5(fl + p) + C~ [V(t)]}e -p'.
(A2)
2. Differentiating equation (A2) results in the following equation:
C][X(t)] = - p {c6 + cs(fl + p) + C~ [V(t)]} + CT[V(t)][-y(t) -/~V(t)].
(A3)
3. From equation (A3), it immediately follows that:
y(t) = - f l V ( t ) -
C~[X(t)] + p {c6 + c5(fl + p) + C'l [V(t)]}
c','[v(t)]
Corollary If, on an interval of time, the partial capacity sell regime is realized, then the following three conditions hold: (1) ~bv(t ) = c 5 e-pt;
(2) ~kx(t ) = {c6 + c5(fl + p) + C~[V(t)l}e-Pt;
C; [X(t)] + p {c6 + c s (fl + p) + C~[V(t)]} <~2V(t). C'([V(t)]
(3) 0 ~< --flV(t)
Proof The first condition holds by definition of the PSR. The second one was proved in Item I of Lemma 1. The third condition emanates from equation (2). Lemma 2 Let the PER be realized C~[V(t)]}]/C'~[V(t)].
on
a
time
interval
[tl,/2],
then
w(t)=flV(t)-[-[C~[X(t)]~-p{c6q-c4(fl+p)+
Proof 1. Differentiating the condition of the SPR, ~,v(t) = c4 e -p', and taking into account the dual equation (7), the following equation results:
[c6 + c4(fl + p)]e -pl = a~(t) + Aay(t).
(A4)
Because the PER is under consideration, %(t) equals 0 [see equation (9)]. Next, equation (A4) cannot be satisfied on the no production and underproduction regimes because a~(t) is equal to zero on these regimes [see equation (8)]. Thus, the F P R is the only regime that can occur along with the PER [i.e. u(t) = V(t)] and au(t ) = ~x(t) - C~ [V(t)]e -p'. Consequently, equation (A4) takes the following form:
~x(t) = {c6 + c,(fl + p) + C~[V(t)]}e -pt.
(A5)
2. Differentiating equation (A5) results in the following equation:
C'3[X(t)I = - 0 { c 6 + e4(/~ + p) + C~[V(t)]} + CT[V(t)][w(t) -/~V(t)l.
(A6)
3. From equation (A6), it immediately follows that:
w(t) = flV(t) +
C;[X(t)] + p{c6 + c4(/~ + p) + C~ [V(t)]}
c','[v(t)]
Corollary If, on an interval of time, the partial capacity expansion regime is realized, then the following three conditions hold: (1) ~kr(t) = c4 e-P'; (3) 0 ~
(2) ~bx(t ) --- {c6 + c4(fl + p) + C~[V(t)]}e-P';
C~[X(t)] + p {c, + c4([3 + p) + C~ [V(t)]} <<_w. C~[V(t)]
Proof The first condition holds by definition of the PER. The second one was proved in Item 1 of Lemma 2. The third condition emanates from equation (2). Lemma 3 Slack Functions F t , F 2, F3, F4, F5 and F 6 are nondecreasing functions of their arguments.
Optimal control method for aggregate production planning
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Proof Let the system of ordinary differential equations (i), (3), (6) and (7) be given, and [Xi (t), VI (t), ~bxl (t), ~bvl (t)] and IX2 (t),
V2(t), tpx2(t), ~ ( t ) ] be two solution of the system with the same initial conditions except ~xl (0)> ~x2(0): 1. Consider a small interval of time [0,t~] on which ~x~(t)> ~x2(t).
2. From maximizing the Hamiltonian (see Section 3, optimal production control regimes) it follows that u(t)+ s(t) is a nondecreasing function of ~x(t), i.e. ul (t) + sL(t) >/u 2(t) + s2(t), t ~ [0, tj ]. 3. From equation (3), it immediately follows that ~l(t)/> )(2(t) on the same interval of time. 4. Because Xl(0) = :(2(0) and )(l(t) >t )C2(t), then Xl(t) >tX2(t). 5. Because C3(X) is a convex differentiable function, then C~(X) is a nondecreasing function. From equation (6) the following results: ~xl (t) 1> ~x2(t). 6. Because ~xl (0) > ~:a (0) and ~xl (t)/> ~x2(t), then ~xl (tl) - Ip~ (tl) >~~xl (0) - ~x~(0). 7. It has been proven that when ~bxl(0) > ~ ( 0 ) , then Xl(tl)>I Xz(tl) and ~xl(tl)- ~bx2(tl)>~~x~(0)- ¢'x2(0)> 0. Therefore, the proof can be continued in the same way on subsequent time intervals each of which is no shorter than [0, tl] until the planning horizon [0, T] is covered. Thus, Xt(T) >1X2(T) and Ft is a nondecreasing function.
The proof for the remaining slack functions is analogous and, therefore, is omitted here.