balancing facilities in aggregate production planning: make-to-order and make-to-stock environments

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Balancing facilities inaggregate productionplanning: Make-to-order and make-to-stockenvironmentsK. Kogan , E. Khmelnitsky & O. MaimonPublished online: 15 Nov 2010.

To cite this article: K. Kogan , E. Khmelnitsky & O. Maimon (1998) Balancingfacilities in aggregate production planning: Make-to-order and make-to-stockenvironments, International Journal of Production Research, 36:9, 2585-2596,DOI: 10.1080/002075498192715

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int. j. prod. res., 1998, vol. 36, no. 9, 2585± 2596

Balancing facilities in aggregate production planning: make-to-order and

make-to-stock environments

K. KOGAN² *, E. KHMELNITSKY ² and O. MAIMON²

The paper concerns an optimal control approach to continuous-time aggregateproduction planning in make-to-stock and make-to-order environments. Thedynamics of such a production system are modelled by purchased, in-processand ® nished inventory ¯ ows through multiple facilities of ® nite capacity. Theobjective of the make-to-order production is to track a given customer demandas closely as possible, while in the make-to-stock production it is to keep thepurchased inventory minimal when ® lling the stocks with volumes of productsscheduled to be completed by the end of the planning horizon. The optimalbehaviour of the system is studied with the aid of the maximum principlewhich allows new analytical results to be derived. Consequently, fast numericaland analytical algorithms for balancing facilities in the two corresponding envir-onments are suggested.

1. Introduction

This paper focuses on a general case of planning materials and capacity require-ments to balance multiple facilities. Balancing facilities in make-to-order and make-to-stock production environments under given demands for products is a typicalproblem arising in aggregate production planning and master scheduling. This isaccomplished by dividing an aggregate material ¯ ow into sub¯ ows and allocatingthem among manufacturing facilities. The goal is to track demands for productsalong a planning horizon in the make-to-order environment, or to ® ll the stocks withrequired volumes of products by the end of the planning horizon in the make-to-stock environment. The innovation of this paper is that with the aid of the maximumprinciple new analytical results for the corresponding environments are derived andfast solution methods which are based on these results are suggested.

Aggregate production planning and control exist in the literature, ranging fromdiscussions of the pioneering one-item continuous-time (Arrow and Karlin 1958)and discrete (Holt et al. 1960) models to more general analyses capturing di� erentlevels of aggregation in a hierarchical manner (Lasserre 1992, Sethi et al. 1992).Comprehensive reviews on models and achievements in this area can be found ina number of papers (Bitran and Hax 1977, Hax and Candea 1984, Graves 1993).Although it is impossible to list all models suggested, they can be commonly distin-guished by the form of the cost objective because it predetermines the solutionapproach.

In cases where the objective can be assumed to be linear, linear programmingtechniques are employed (Lipman et al. 1967, Hax 1978). More general cost analysisrevealing nonlinear dependencies typically leads to dynamic programming (Wagner

0020± 7543/98 $12.00 Ñ 1998 Taylor & Francis Ltd.

Revision received October 1997.² Department of Industrial Engineering, Tel-Aviv University, Tel-Aviv 69978, Israel.* To whom corespondence should be addressed.

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and Whitin 1958, Love and Turner 1993), whose complexity depends exponentiallyon the model dimensions. Growth of dimensions develops not only in multiplicationof facilities but also in detailing the planning horizon to follow rapidly changingcustomer demands.

On the other hand, the latest achievements in optimal control methods based onthe maximum principle theory, have allowed analytical investigation of aggregateproduction planning systems to be carried out in order to gain an insight into theiroptimal behaviour. As a result of such investigations, e� cient shooting numericalmethods have been suggested for single-item single-facility production planning pro-blems (Kogan and Khmelnitsky 1995, Khmelnitsky and Kogan 1996a). However,for larger problems the methods lose their e� ciency except for some special cases, asfor example, balancing multiple facilities under ® xed initial inventory conditions(Khmelnitsky and Kogan 1996b). Therefore, we extend the previous optimal con-trol-related works in three directions:

�Balancing multiple facilities in di� erent production environments;

�Deriving new analytical properties of the optimal solution;

�Developing a time-decomposition numerical method which is more e� cientthan shooting as well as dynamic programming.

In the next two sections of the paper, we consider a general single-item model forbalancing multiple facilities against a given demand rate along the planning horizon,and a time-decomposition method for solving it. Sections 4 and 5 deal with themake-to-stock environment where the demand is given by a cumulative volume atthe end of the planning horizon. The general balancing model is transformed thereinto dispatching a ® xed inventory ¯ ow through the facilities, and is solved analyti-cally in §5.

2. Make-to-order environment: problem formulation

Consider a production system consisting of I distinct facilities F, i = 1, . . . ,I,which produce an aggregate product. Figure 1 shows the product ¯ ow through thefacilities to satisfy customer orders. The time parameters of the system are modelled

2586 K. Kogan et al.

Figure 1. Product ¯ ow in the make-to-order environment.

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by corresponding rates, i.e. the number of items per time unit. In this model, thesuppliers lead time of the purchased inventory R(t) is modelled by the purchasingrate z(t) , while the processing time of the in-process inventory Xi (t) on the ithfacility, i = 1, . . . ,I is de® ned by the production rate ui (t) of the facility. Finally,the customer orders speci® ed by their ® nished product volumes and due dates aretransformed into a given demand pro® le d (t) .

The dynamics of such production is described by mass conservation equationsfor purchasing, processing and transporting processes.

2.1. Production process: purchased inventory ¯ owThe purchased inventory is replenished by the purchasing rate and is withdrawn

by transporting the inventory to facility i with controllable rate wi (t) :

ÇR(t) = z(t) - åi

wi (t), R(0) = 0. (1)

2.2. Production process: in-process inventory ¯ owThe work-in-process ¯ ows through a facility and is described by the di� erence

between the rate of transporting this work to the facility and its production rate ui (t) :

ÇXi (t) = wi (t) - ui (t), Xi (0) = X0i . (2)

2.3. Production process: ® nished inventory ¯ owThe ® nished inventory balances the total production against the demand:

ÇY (t) = åi

ui (t) - d (t), Y (0) = 0. (3)

2.4. Production constraintsThe purchasing rate is bounded from above by the time-dependent purchasing

capacity:0 £ z(t) £ Z(t) . (4)

The transporting rate to a facility is limited by the delivery capacity:

0 £ wi (t) £ W i. (5)

The production rate of a facility is restricted by the production capacity:

0 £ ui (t) £ Ui . (6)

The objective of aggregate production planning in the make-to-order environ-ment is to track the customer demand as closely as possible while minimizing thetotal cost of carrying inventories C1 (R(t) ) , C2,i (Xi (t) ) and C3 (Y (t) ) in the systemalong the planning horizon [0, T]:

òT

0[C1 (R(t) ) + å

iC2,i (Xi (t) ) + C3 (Y (t) )]dt ® min . (7)

The convex cost functions C1 (R(t) ) , C2,i (Xi (t) ) and C3 (Y (t) ) re¯ ect inventorycarrying costs when their arguments are positive and costs of shortages otherwise.

Balancing facilities in aggregate production planning 2587

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3. Make-to-order environment: numerical solution approach

The problem of balancing facilities is stated here in the form of an optimalcontrol with rates of purchasing z(t) , transporting wi (t) and production ui (t) con-trollable along the planning horizon. Since the right-hand sides of the dynamicequations for the state variables (R(t),Xi (t), Y (t)) are di� erentiable with respectto the control variables (z(t),wi (t),ui (t) ) , the maximum principle (see literaturesurvey by Hartl et al. 1995) can be applied to develop a numerical solution method.

3.1. The maximum principleAccording to the maximum principle, the optimal values of control must max-

imize at each point of time the Hamiltonian:

H = - C1 (R(t) ) - åi

C2,i (Xi (t) ) - C3 (Y (t) ) + w R(t) z(t) - åi

wi (t)( )+ w X

i (t) (wi (t) - ui (t) ) + w Y (t) åi

ui (t) - d (t)( ) , (8)

where dual variables w R(t) , w Xi (t) and w Y (t) satisfy the dual di� erential equations:

Çw R(t) = CÂ1 (R(t) ), w R(T ) = 0; (9)

Çw Xi (t) = CÂ2,i (Xi (t) ), w X

i (T ) = 0; (10)

Çw Y (t) = CÂ3 (Y (t) ), w Y (T ) = 0. (11)

The solution approach which is based on the Hamiltonian maximization can beconstructed either numerically for the general statement (1) ± (7), or analytically forsome special cases. An example of such a case is found when dealing with dispatch-ing in the make-to-stock environment, and is considered in the next section.

To optimize the Hamiltonian numerically, we employ a time-decompositionmethod which improves gradually the objective by projecting the Hamiltoniangradient on the set of admissible controls given by constraints (4) ± (6).

The sequence of the following steps describes the application of the time-decomposition method to the problem.

Step 1. Guess a nominal (feasible) solution (e.g. all control variables are identicallyequal to zero), and integrate the primal system (1) ± (3) from left to right.

Step 2. Integrate the dual system (9) ± (11) with respect to the found primal variablesfrom right to left.

Step 3. Vary the control values in the direction of the Hamiltonian increase:

d z(t) = e w R (t), d wi (t) = e ( w Xi (t) - w R(t) ), d ui (t) = e ( w Y (t) - w X

i (t) ),where e is a positive number.

Step 4. If the varied control is out of the feasible interval, then ® nd the nearestfeasible point, e.g. if

z(t) + d z(t) > Z(t), then set z(t) + d z(t) = Z(t) ;if

z(t) + d z(t) < 0, then set z(t) + d z(t) = 0.

Step 5. Integrate the primal system with the varied control.


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Step 6. Calculate the objective function. If the objective has not decreased thendecrease e and go to step 3. Otherwise, consider the varied solution as anew nominal one and go to the next step.

Step 7. Check the standard stop condition:

òT

0[d z(t)2 + å

id wi (t)

2 + åi

d ui (t)2]dt < TOL ,

where TOL is a given tolerance. If it holds then exit, otherwise go to step 2.

4. Make-to-stock environment: problem formulation

The previous section deals with aggregate demand rate de® ned at every point ofthe planning horizon. If the customer orders are accumulated in volumes XT

i to beful® lled on each facility i by the end of the planning horizon T , then the problemreduces to dispatching the purchased inventory only, so minimalizing it along theplanning horizon. Figure 2 shows the product ¯ ow for this case.

The problem is formulated as follows:

Production process: purchased inventory ¯ ow:

ÇR(t) = z(t) - åi

ui (t), R(0) = 0, R(t) ³ 0. (12)

Production process: in-process inventory ¯ ow:

ÇXi (t) = ui (t), Xi (0) = X0i , Xi (T ) = XT

i . (13)

Production constraints:

0 £ ui (t) £ Ui . (14)

Objective:

òT

0R(t)dt ® min . (15)


Figure 2. Product ¯ ow in the make-to-stock environment.

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Note that in contrast to the problem formulation in a make-to-order environment, alinear form of the cost function in the objective along with a given purchasing policyz(t) in production process equation (12) are adopted here. This is due to the fact thatthe model of make-to-stock production with a nonlinear cost objective and control-lable purchasing rate can be solved by the time-decomposition numerical methodpresented in the previous section for the make-to-order production system. On theother hand, the linear cost function case studied with the aid of the maximumprinciple enables us to gain an insight into the problem and to prove an importantmanagerial rule of the critical ratio type, based on which the analytical optimalsolution to the problem is developed.

5. Make-to-stock environment: analytical solution algorithm

Prior to analysing the extremal behaviour of the production system, we derive anequivalent statement of the problem to simplify the analysis. First, we rewrite theobjective by consecutively replacing R(t) with its equation (12) and expressing ui (t)from equation (13):

òT

0R(t)dt = ò

T

0 òt

0z(¿) - å

iui (¿)[ ]d¿dt = ò

T

iZ(t) - å

i

(Xi (t) - X0i )[ ]dt,

where Z(t) = ò t0 z(¿)d¿ is a cumulative amount of the purchased inventory by time t.

Secondly, we change the variable Xi (t) to XTi - ^Xi (t)Ui , and then eliminate the

uncontrollable parameters X0i , XT

i and Z(t) from the objective, because they do notin¯ uence the optimal solution. Thus, the objective takes the following form:

òT

0 åi

^Xi (t)Uidt ® min . (16)

The new variable ^Xi (t) re¯ ects the minimal time required at moment t to replenishthe in-process inventory Xi (t) up to the ordered volume XT

i ; that is commonlyreferred to as a critical ratio. By also changing the absolute production rate ui (t)the relative one ui (t) /Ui , we obtain the equivalent statement of the problem:

òT

0 åi

^Xi (t)Uidt ® min, (17)

subject to

Ç ^Xi (t) = - ui (t), ^Xi (0) =X0

i

Ui, ^Xi (T ) ³ 0, (18)

0 £ ui (t) £ 1, åi

^Xi (t)Ui ³ åi

XTi - X0

i - Z(t) . (19)

Note that the necessary and su� cient condition for the solution of the derivedproblem to exist is that the total purchased inventory volume Z(T ) does not exceedthe cumulative orders for the system, i.e:

åi

XTi - X0

i ³ Z(T ) .

The maximum principle applied to the problem (17) ± (19) states that if a trajec-tory ( ^Xi (t) and ui (t) ) is the optimal one then piecewise continuous functions w i (t)


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(dual variables) and a non-negative measure-function d¹ (t) (Lagrange multiplier)exist so that the following is true:

The dual systemd w i (t) = Uidt - Uid¹ (t) . (20)

The transversality condition and complementary slackness:

If ^Xi (T ) = 0 then w i (T + 0) ³ 0,if ^Xi (T ) > 0 then w i (T + 0) = 0. (21)

The maximum principle: The Hamiltonian

H = - åi

^Xi (t)Ui + åi

w i (t)ui (t)

is maximized for each t by the control variables ui (t) subject to the capacity con-straint from (19).Optimal production regimes: With respect to the maximum principle, three optimalproduction regimes can be readily derived by maximizing the Hamiltonian as afunction of ui (t) :

ui (t) =1, if w i (t) > 0 (maximal dispatching for facility i) ;0, if w i (t) < 0 (no-dispatching for facility i) ;a , a Î [0,1], if w i (t) = 0 (partial dispatching for facility i) .

ìïíïî

(22)

Note that while dispatching for a facility is unambiguously determined forw i (t) /= 0 (either maximal or no-dispatching regime), the partial dispatching (singularregime) requires additional consideration. The ambiguity is resolved in Lemmas 1and 2 presented in the appendix. Lemma 1 shows that the ordering relation betweeninitial values of ^Xi (0) remains unchanged for the optimal solution along the entireplanning horizon that is used in Lemma 2 to determine the controls on the partialdispatching regime.

Properties of the optimal solution proven in the Lemmas sustain, in fact, a wide-spread critical ratio rule, which relates the amount of work needed to be completedand the maximal rate with which this work can be carried out (see, for example,Tersine 1985). Indeed, it is also common sense to save capacity and dispatch ® rst tothose facilities which require greater time for replenishing them. This insightfulproperty also becomes optimal in our case according to Lemma 2 and is realizedin the following analytical algorithm.

The algorithm consists of the following steps:

Step 1. Find the interval of time ¿ = [t1,t2]where the purchasing rate exceeds thetotal production capacity: z(t) > å i Ui.

Step 2. If interval ¿ does not exist then go to step 5. Otherwise, there always exists afacility which produces according to the partial dispatching regime.Therefore, the optimal controls are assigned with respect to Lemmas 1and 2.

Step 3. Order all the facilities as follows: ^XI (0) ³ ^XI- 1 (0) ³ ´´´ ³ ^X1 (0) .Step 4. Integrate the equation (18) with the given left-hand boundary condition.

During the integration, assign the optimal control values at every point of


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time on the overall planning horizon t Î [0, T]according to the followingconditions:

ui (t) = 1, if z(t) - åI

j=iUj ³ 0;

ui (t) =

z(t) - åi|ui=1

Ui

M , if ^Xi (t) ³ ^Xj (t), j Î I \ {m|m > i},where M is the number of the facilities for which this condition holds;uij (t) = 0, otherwise. Stop, the optimal trajectory is found.

Step 5. The partial dispatching regimes does not exist on the optimal trajectory.Integrate the equation (18) with the given left-hand boundary condition.During the integration, assign the optimal control values at every point oftime on the overall planning horizon t Î [0, T]according to the followingconditions:

ui (t) = 1 if ^X(t) > 0, and ui (t) = 0 otherwise.

Stop, the optimal trajectory is found.

6. Numerical example

To illustrate the general numerical method for balancing multiple facilities in themost complex make-to-order production environment, we consider a ® ve-facilityproblem with parameters shown in table 1.

The surplus and backlog cost coe� cients in table 1 are due to a quadratic form ofthe cost functions C2,i (X) = 1

2 c+i X2, if X ³ 0, and C2,i (X) = 1

2 c-i X2, if X < 0adopted in the example. The same form is also used for C1 (R) and C3 (Y ) withcost coe� cients c+

1 = c+3 = 0.5 and c-1 = c-

3 = 2.5The maximal Z(t) and optimal z(t) purchasing rates are depicted in ® gure 3

against demand rate d (t) given along the planning horizon T = 25. The solutionobtained by the suggested numerical algorithm after two seconds of computing onIBM PC-486 is depicted in ® gures 3± 7.

Finally, to test the computational e� ciency of the developed time-decompositionmethod, we compared it with GAMS, a powerful software considered to be one ofthe best for solving mathematical programming problems. The accuracy of optimi-zation was set at 10% for both methods and the integration was approximated bythe Euler scheme with mesh points equally distributed over the planning horizon.


Production Delivery Surplus cost Backlog costFacility i capacity, Ui capacity, Wi coe� ., c+

i coe� ., c-i

1 1.5 1.7 0.5 2.42 1.6 1.4 0.7 2.33 1.4 1.1 0.3 1.74 1.9 1.9 0.4 2.75 2.4 2.2 0.6 2.5

Table 1. Production system parameters.

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Figure 3. The maximal Z(t) and optimal z(t) purchasing rates against demand rate d (t) .

Figure 4. The optimal transporting rates wi (t) .

Figure 5. The optimal production rates ui (t) .

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Table 2 presents the average computational times for both methods when thenumber of facilities changes from 5± 25, and the number of mesh points is 50 and 100.

7. Conclusions

A continuous-time production planning problem has been stated to balancefacilities in two typical make-to-order and make-to-stock production systems. In


Figure 6. The optimal in-process inventories Xi (t) .

Figure 7. The optimal purchased R(t) and ® nished Y (t) inventories.

Number of facilities, INumber of

Method mesh points 5 10 15 20 25

Time-decom- 50 1 3 4 6 11position 100 2 5 7 11 18

GAMS 50 1 3 8 24 99100 4 32 148 497 1370

Table 2. Computational time (s).

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the make-to-order environment, the objective is to control purchasing, transportingand production rates so as to track dynamic customer demand as closely as possible.In the make-to-stock environment, the objective is to keep the purchased inventoryminimal by dispatching it to the facilities so as to ® ll the stocks with the requiredvolumes of products by the end of the planning horizon.

Presentation of the problem in the form of production ¯ ow control allows one tostudy analytically and numerically the optimal behaviour of the two systems. Withthe aid of the maximum principle the original dynamic problem is reduced to max-imization of the Hamiltonian at every point of time. Having a set of such static dualproblems, the make-to-order balancing is then tackled by an e� cient, projected-gradient-based numerical procedure. Furthermore, when dealing with the make-to-stock environment, the introduction of specially chosen variables, which re¯ ectthe relation between the production rate and the time needed to replenish thein-process inventories, gives insight into the analytical properties of the optimalsolution. Based on those properties, a simple analytical algorithm is developed foroptimal inventory dispatching.

Appendix

Lemma 1: Let ^Xi (0) ³ ^Xj (0) , then on the optimal solution ^Xi (t) ³ ^Xj (t) ," t Î [0, T].Proof: Let there exist a moment t0 such that ^Xi (t0) = ^Xj (t0) . Then choosew i (0)Uj = w j (0)Ui. From the chosen initial condition on dual variables and dualequation (20) it immediately follows that:

w i (t)Ui

ºw j (t)Uj

. (23)

By choosing further ui (t) = uj (t) when w i (t) = w j (t) = 0, we ensure that all the con-ditions of the maximum principle hold. We then obtain from expressions (18), (22)and (23) that ui (t) = uj (t) and ^Xi (t) = ^Xj (t) , t Î [t0, T]. Thus, if ^Xi (0) > ^Xj (0) , thenon the optimal solution either this relation remains along the entire planninghorizon, or ^Xi (t) and ^Xj (t) become equal at moment t0 and remain equal until theend of the planning horizon. h

Lemma 2: Let I1, I2 and I3 denote sets of facilities on the maximal, no- and partialdispatching regimes respectively. If N facilities i1, i2, . . . , iN are on the partial dis-patching regime on an interval of time and are ordered by ^Xi (t) :^XiN (t) ³ ^XiN- 1 ³ ´´´ ³ ^Xi1 , then the optimal control values are de® ned according

to the following conditions:

�uij (t) = 1, if z(t) - åi Î I1

Ui - Uij ³ 0 and ^Xij (t) ³ ^Xik , k Î I3\ {im|m > j};

�uij (t) =

z(t) - åi Î I1

Ui

Mif ^Xij (t) ³ ^Xik , k Î I3\ {im |m > j}, where M is the number

of facilities for which this condition holds;

�uij (t) = 0, otherwise.

Proof: Let us di� erentiate the condition w ij (t) = 0 of the partial dispatching regimeand take into account the dual equation (20):


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d¹ (t) = dt or Ç¹ (t) = 1.

From the last fact and the complementary slackness, it follows that the state con-straint in equation (8) holds as an equality along the partial dispatching regime:

åi

^Xi (t)Ui = åi

XTi - X0

i - Z(t) . (24)

This means that partial dispatching is realized if and only if the surplus of thepurchased inventory vanishes. Di� erentiating now the equality (24), we eventually® nd the condition for the control values on the partial dispatching regime:

åi

ui (t)Ui = z(t) . (25)

One can easily observe that the control values de® ned by the lemma satisfy condition(25) and keep the order proved in Lemma 1. h

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