Linear dynamic model of production-inventory with debt repayment: optimal management strategies

# Linear dynamic model of production-inventory with debt repayment: optimal management strategies

Ekaterina Tuchnolobova Victor Terletskiy Institute for Mathematics, Economics and Informatics, Irkutsk State University, Bulvar Gagarina 20, Irkutsk 664003, Russia Olga Vasilieva Department of Mathematics, Universidad del Valle, Calle 13 No. 100-00, Cali, Colombia
###### Abstract

In this paper, we present a simple microeconomic model with linear continuous-time dynamics that describes a production-inventory system with debt repayment. This model is formulated in terms of optimal control and its exact solutions are derived by prudent application of the maximum principle under different sets of initial conditions (scenarios). For a potentially profitable small firm, we also propose some alternative short-term control strategies resulting in a positive final profit and prove their optimality. Practical implementation of such strategies is also discussed.

###### keywords:
production-inventory system, debt repayment, linear dynamics, optimal control, profitability condition
###### Msc:
91B55, 49N90, 91B38
\newdefinition

remarkRemark \newdefinitionpropProposition

## 1 Introduction

In macro- and microeconomics, the continuous-time models involving ordinary differential equations naturally serve as a basis for understanding the behavior of economic systems where the dynamic aspects play an important role.

Mathematical models of microeconomic can help to explain macroeconomic phenomena and to improve the management of a particular production unit (plant, firm, family business, etc.). Among them, the major focus has been placed on production-inventory systems consisting of a manufacturing plant and a warehouse to store the finished goods which are produced but not immediately sold. Usually, the production rate is treated as a control variable while the purpose of control consists in meeting the existent level of product demand at the market or in maximizing the net profit of the production unit.

Traditional approach to solving the production-inventory problem in terms of optimal control is lucidly described in SethiThompson2000 () while a state-of-art review OrtegaLin2004 () identifies the major research efforts for applying control theoretic methods to productioninventory systems. During the last decade, various scholars have made essential contributions to this field. Among them it is worth to mention the control theory applications to the models with stock-dependent demand rate TengChang2005 () and with inventory-level-dependent demand rate KhmelnitskyGerchak2002 (); Urban2005 (). Other stream of research has been focused on optimization of inventory systems with product deterioration (see, e.g., Al-khedhairiTadj2006 (); BatenKamil2011 (); BakkerRiezebosTeuner2012 ()), systems with back-orders and lost sales ChangLo2009 (); BenjaafarElHafsiHuang2010 () or without them AdidaPerakis2007 ().

However, all previously mentioned works do not consider explicitly the dynamics of the firm’s debts acquired during the production period or prior to its commencement (overdue payables).

On the other hand, V. Tokarev Tokarev2001 (); Tokarev2002 () had proposed a microeconomic model of short-term crediting and debt repayment for a small firm where the rate of debt repayment has been treated as a control variable. This model does not consider inventory stock variation and mainly concentrates on the dynamic of production funds (firm’s basic assets) and current debts. The purpose of control consists in maximizing the final value of the firm’s production funds and the optimal control has a bang-bang structure naturally dependent on the loan’s interest rate. Namely, for low-interest banking rates it is more profitable to invest all available cash into production at the beginning of the period (in order to generate more profit) and then to repay the debts by the end of the period. For high-interest banking rates a reverse strategy is optimal: first comes the debt repayment and then the investment, since in this case the debts grow faster than the maximal possible profit obtained from production. The same model was further developed in GrigorievaKhailov2005 () under additional condition that all pending debts must be paid off by the end of the period.

In this paper, we propose a simple microeconomic model with linear continuous-time dynamics that explicitly includes the variation of current debt of a firm. Our model combines the traditional features of production-inventory systems with Tokarev’s approach and has three constrained state variables together with three bounded control variables. The model is premeditated for dealing with short-term planning of a profitable firm (such as small or family business) in economically stable market environment and may provide some guidelines for periodic short-term planning (weekly, monthly, etc.).

Linear structure of the model allows to apply the Pontryagin’s maximum principle as a necessary and sufficient condition of optimality. However, the application of the maximum principle for problems with mixed-type constraints can be rather challenging (see, e.g. Maurer1977 (); HartlSethiVickson1995 (); MaurerPesch2008 ()). Sometimes, there is no other way than to “guess” a possibly optimal control strategy and then to use the maximum principle for proving its optimality. For such “guesses”, we have just used a “common-sense” control strategies derived from the economic contexts of the problem. We have deliberately kept this model simple in order to obtain its analytical solutions under different scenarios and to propose some viable alternatives for better management in economically stable environment.

The paper is organized as follows. Section 2 provides the model description and formulates the maximum principle for optimal control problems with mixed-type constraints. Section 3 presents three basic scenarios of the firm management for different sets of initial conditions. In Section 4 we discuss some alternative strategies of the firm’s management and justify their optimality as well as practical implementation within the frameworks of the model. Section 5 contains the conclusion and briefly indicates the perspectives of future research.

## 2 Model description and preliminaries

In economically stable market environment, the accumulation of arrears by a small firm (family business, for example) clearly indicates the inefficiency of its short-term planning. Therefore, financial manager or self-employed entrepreneur should try to avoid the accumulation of unpaid debts while seeking to maximize the firms’ cumulative profit.

To examine this situation we propose to use a simple dynamic model of a firm producing a single good which operates under market stability111Under market stability, the inflation index for short-term periods is considered to be depreciable. Therefore, we omit the intertemporal discount factor in the model setting. and has a viable (deterministic) estimation for its product demand at the market. Such model can be posed in terms of optimal control as follows:

 J(u,v,w)=N(T)−D(T)→max (1)

subject to

 ˙N(t)=pw(t)−v(t)−Z(u),N(0)=N0, (2a) ˙D(t)=rD(t)+Au(t)−v(t),D(0)=D0, (2b) ˙S(t)=u(t)−w(t)−αS(t),S(0)=S0, (2c)

under state constraints

 N(t)≥0,D(t)≥0,0≤S(t)≤Smax,t∈[0,T] (3)

and control constraints

 0≤u(t)≤umax,0≤v(t)≤vmax,0≤w(t)≤wmax,t∈[0,T]. (4)

The quantities used in (1)-(4) are defines as

 u(t) – production rate (control variable) with maximum gross output given by umax>0 (constant); v(t) – rate of the debt repayment (control variable) with maximum repayment capacity given by vmax>0 (constant); w(t) – rate of sales (control variable) with maximum volume of demand given by wmax>0 (constant); N(t) – cumulative net profit of the firm by time t (state variable); D(t) – amount of overdue payables (debts) of the firm at time t (state variable); S(t) – volume of finished goods inventory at time t (state variable) with maximum storage capacity given by Smax>0 (constant); Z(u) – cost function, which includes other expenditures not related to the purchase of raw materials ( such as wages, social security costs, lease payments, etc.). For simplicity sake, we suppose that Z(u)=Ku+B where K,B>0 are given constants; p>0 – output (retail) price index (constant); r>0 – rate of overdue debt accumulation (constant); A>0 – average cost for the purchase of raw materials (constant); α>0 – outflow rate of finished goods inventory (constant) that includes sales and stock loss during storage.

Here the maximum rate of the debt repayment is supposed to be sufficiently large while the maximum volume of demand does not naturally exceed the maximum level of production capacity , that is, . It should be noted that is chosen as affine linear function in order to emphasize that even for , these indirect costs will not be zero.

Criterion (1) expresses the maximization of the cumulative net profit and minimization of overdue debts by final time . Here we do not impose the condition of total debt repayment by the final time (that is, as in GrigorievaKhailov2005 ()) and merely try to minimize the debt that may remain positive.

Equation (2a) describes the net profit accumulation: total revenue minus the debt repayment and other costs . Equation (2b) provides the dynamics of accumulated arrears: the debt increases due to the interest rate and purchase of raw materials and decreases with repayments . Equation (2c) describes changes in finished goods inventory: the stock increases with production and decreases with sales and losses during storage.

By introducing formal notation and (vector of adjoint variables associated with state variables and , respectively), the Hamiltonian of the optimal control problem (1)–(4) is linear in and separable with respect to :

 H(Ψ,X,U)=ψ1[pw−v−Ku−B]+ψ2[rD+Au−v]+ψ3[u−w−αS]
 =[−Kψ1+Aψ2+ψ3]u−[ψ1+ψ2]v+[pψ1−ψ3]w−Bψ1+rDψ2−αSψ3,

where the factors to the control components are the switching functions:

 θu=−Kψ1+Aψ2+ψ3,θv=−ψ1−ψ2,θw=pψ1−ψ3. (5)

Note that the switching vector only depends on the adjoint vector (whose components are also called “shadow prices” in economics). By assigning a vector of Lagrange multipliers to four state constraints (3), we can define the Lagrangian for our problem as

 L(Ψ,X,U,Λ)=H(Ψ,X,U)+λ1N+λ2D+λ3S+λ4[S−Smax]

For our linear control problem (1)–(4), the maximum principle serves as necessary and sufficient condition for optimality and can be rigorously justified using direct adjoining approach described, e.g., in Maurer1977 (); HartlSethiVickson1995 (); MaurerPesch2008 (). Therefore, a piecewise continuous (or bang-bang) control defined by switching functions (5) that maximizes the Hamiltonian almost in all points of is optimal in (1)–(4) if and only if there exist an absolutely continuous costate trajectory of the adjoint system

 ˙ψ1(t)=−∂L∂N=−λ1(t), ψ1(T)=μ1+1, ˙ψ2(t)=−∂L∂D=−rψ2(t)−λ2(t), ψ2(T)=μ2−1, (6) ˙ψ3(t)=−∂L∂S=αψ3(t)−λ3(t)+λ4(t), ψ3(T)=μ3−μ4,

as well as a piecewise continuous vector function of multipliers , and a nonzero vector such that the following conditions of complementary slackness are satisfied:

 λ1(t)N(t)=0μ1N(T)=0λ2(t)D(t)=0μ2D(T)=0λ3(t)S(t)=0μ3S(T)=0λ4(t)[S(t)−Smax]=0μ4[S(T)−Smax]=0λi(t)≥0μi≥0,i=1,2,3,4. (7)

This classical result will be very essential for further design of optimal control strategies under which the objective functional (1) attains its maximum value.

## 3 Optimal functioning of a profitable firm: case studies.

Optimal functioning of a firm significantly depends on whether an external demand for its products ensures a positive profit. In mathematical formalization (see, e.g. Kessides1990 ()), a firm is profitable if the following inequality holds:

 pw(t)>Z(u)+Au(t), (8)

that is, if its sales profit fully covers all underlying expenses (such as purchase of raw materials, equipment, and other indirect costs). Otherwise, the firm is unprofitable.

Generally speaking, optimal management strategies will also depend on the initial values of state variables, such as presence or absence of initial profits, debts, and finished goods inventory. Therefore, it is interesting from the economic point of view to revise three basic scenarios and obtain optimal strategies satisfying the maximum principle. In the subsequent case-studies the condition (8) will be in force.

### 3.1 Scenario 1: absence of initial debt, presence of stock at t=0

Suppose that at initial time the firm has no arrears and possesses some stock of finished goods and cash resources, that is,

 N(0)=N0>0,D(0)=D0=0,S(0)=S0>0.

In this case, it will be profitable for the firm to start the production at the moment by which the whole existing stock of finished goods is sold, thus generating no new debt before . The production is then started at the moment with maximum volume of demand in order to avoid overproduction. Then the production costs of the firm at any given time will be (since is rather large). Therefore, we anticipate that optimal control will be of the form:

 u∗={0,t∈[0,tS)wmax,t∈[tS,T],v∗={0,t∈[0,tS)Awmax,t∈[tS,T],w∗=wmax. (9)

Under this strategy, no arrears will arise (that is, ), because the firm is able to pay its debts on time, while interest payments on the debt are economically unprofitable. In order to prove the optimality of (9) and find a point , we must determine Lagrange multipliers and that satisfy the complementary slackness conditions (7):

 N(t)>0impliesλ1(t)=0,μ1=0;
 D(t)=0impliesλ2(t)≥0,μ2≥0;
 S(t)

The underlying adjoint system (6) can be written as

 ψ1(t)≡1,˙ψ2(t)=−rψ2(t)−λ2(t),ψ2(T)=μ2−1,˙ψ3(t)={αψ3(t),t∈[0,tS)αψ3(t)−λ3(t),t∈[tS,T],ψ3(T)=μ3, (10)

and the switching functions (5) must satisfy the conditions:

 θu(t)=Aψ2(t)+ψ3(t)−K{<0,t∈[0,tS),=0,t∈[tS,T], (11) θv(t)=−ψ2−1{<0,t∈[0,tS),=0,t∈[tS,T], θw(t)=p−ψ3(t)>0.

According to (11), for it is fulfilled that

 Aψ2(t)+ψ3(t)−K=0,ψ2(t)=−1

and in view of (10) we have On the other hand, using in (11) it is obtained that . By substituting this expression in the last equation of (10) it becomes clear that for and . Thus, we have found a set of multipliers that satisfy the complementary slackness conditions (7):

The latter proves the optimality of control (9). In order to find the switching point let us consider the ODE system (2) under optimal control (9) within the interval :

 ˙N(t)=pwmax−B,N(0)=N0>0,˙D(t)=rD(t),D(0)=0,˙S(t)=−αS(t)−wmax,S(0)=S0>0.

whose solution is given by

 N(t)=N0+(pwmax−B)t,D(t)=0,S(t)=αS0+wmaxαexp[−αt]−wmaxα.

Apparently, the switching point must be a unique root of equation where is given above, since this real function is strictly decreasing. Therefore, exactly by the moment

 tS=1αlnαS0+wmaxwmax>0 (12)

the firm’s stock of finished goods becomes empty. {remark} In effect, if it occurs that in (12), then optimal control (9) simply becomes

 u∗=0,v∗=0,w∗=wmax

and implies no production, only sales of existent stock of finished goods until final time . This situation may arise when the rate of outflow of finished goods inventory is very slow while initial stock is replete; in other words, when To evaluate the objective functional (1) (whose value is solely defined by in this case) in optimal control (9), we should find the solution of the corresponding ODE system (2) within the interval

 ˙N(t)=(p−A−K)wmax−B,N(tS)=N0+(pwmax−B)tS,˙D(t)=rD(t),D(tS)=0,˙S(t)=−αS(t),S(tS)=0.

Its solution is

 N(t)=N0+(pwmax−B)t+(A+K)wmax(tS−t),D(t)=0,S(t)=0

and yields

 J(u∗,v∗,w∗)=N0+(pwmax−B)T+(A+K)wmax(tS−T). (13)

### 3.2 Scenario 2: presence of initial debt and stock at t=0

Suppose that at initial time the firm has non-zero arrears and possesses some stock of finished goods and cash resources, that is,

 N(0)=N0>0,D(0)=D0>0,S(0)=S0>0.

In this case, it will be profitable for the firm to start immediately the debt repayment and avoid further accumulation of arrears. The latter can be done by changing the second component of control strategy (9) resulting in

 u∗={0,t∈[0,tS)wmax,t∈[tS,T],v∗={vmax,t∈[0,tD)Awmax,t∈[tD,T],w∗=wmax. (14)

Here has the same meaning as before (that is, ) and indicates the moment of full repayment of arrears (that is, ).

Lagrange multipliers and must satisfy the complementary slackness conditions (7):

 N(t)>0impliesλ1(t)=0,μ1=0;
 S(t)

The underlying adjoint system (6) can be written as

 ψ1(t)≡1,˙ψ2(t)={−rψ2(t),t∈[0,tD)−rψ2(t)−λ2(t),t∈[tD,T],ψ2(T)=μ2−1,˙ψ3(t)={αψ3(t),t∈[0,tS)αψ3(t)−λ3(t),t∈[tS,T],ψ3(T)=μ3,

and the switching functions (5) must satisfy the conditions:

 θu(t)=Aψ2(t)+ψ3(t)−K{<0,t∈[0,tS),=0,t∈[tS,T], θv(t)=−ψ2−1{<0,t∈[0,tD),=0,t∈[tD,T], θw(t)=p−ψ3(t)>0.

By employing a technique similar to the one used in the analysis of Scenario 3.1 and considering two cases ( and ), we can find a set of multipliers and that satisfy the complementary slackness condition (7):

 λ1(t)=λ4(t)=0,λ2(t)={0,t∈[0,tD)r,t∈[tD,T],λ3(t)={0,t∈[0,tS)α(A+K),t∈[tS,T],μ1=μ4=0,μ2=0,μ3=A+K.

The latter proves the optimality of control (14). However, it is not quite clear which switching time ( or ) will occur first. Common sense suggests that smaller initial debt can be repaid faster. Therefore, if is relatively small, then ; otherwise, for relatively large . Eventually, it may also happen that and there will only one switching point. The following proposition summarizes this idea and provides exact formulae for calculation of and in terms of problem entries, as well as their position with respect to each other.

###### Proposition 1

The stock of finished goods becomes empty at the moment given by (12) independently of initial debt amount . The total debt repayment occurs at the moment that depends on in the following way:
(a) if then and there is only one switching point;
(b) if then and

 tD=1rln(vmaxvmax−rD0); (15)

(c) if then and

 tD=1rln(vmax−Awmaxvmax−rD0−Awmaxexp[−rtS]). (16)

Formal proof of this proposition can be consulted in the Appendix. {remark} In effect, if it occurs that , then optimal control (14) simply becomes

 u∗={0,t∈[0,tS),wmax,t∈[tS,T]v∗=vmax,w∗=wmax.

and implies constant debt repayment at maximum rate for all In this case, initial debt must be very large: and, therefore, current debts will not be repayed by final time . Additionally, Remark 3.1 remains valid under this scenario as well. Otherwise, if , then regardless of position of with respect to and the value of criterion (1) is solely defined by the cumulative net profit at final time . The following proposition provide an explicit formula for evaluation of the objective functional.

###### Proposition 2

For and regardless of its position with respect to , we have

 J(u∗,v∗,w∗)=N0+(Awmax−vmax)tD+KwmaxtS+wmax(p−A−K)T−BT, (17)

where is given by (12) and is defined either by (15) or by (16) according to Proposition 1.

The proof of this proposition is rather straightforward and its key features a given in the Appendix.

### 3.3 Scenario 3: presence of initial debt and absence of initial stock at t=0

Suppose that at initial time the firm possess some cash resources and has no stock of finished good along with non-zero arrears, that is,

 N(0)=N0>0,D(0)=D0>0,S(0)=S0=0.

In this case, it will be profitable to start the production immediately and to pay off the existing debts straightaway with maximum rate of repayment. Therefore, we anticipate that optimal control will be of the form:

 u∗=wmax,v∗={vmax,t∈[0,tD)Awmax,t∈[tD,T],w∗=wmax. (18)

Here indicates the moment of full repayment of all debts (that is, ).

Lagrange multipliers and must satisfy the complementary slackness conditions (7):

 N(t)>0impliesλ1(t)=0,μ1=0;
 S(t)=0impliesλ3(t)≥0,μ3≥0;
 S(t)

The underlying adjoint system (6) can be written as

 ψ1(t)≡1,˙ψ2(t)={−rψ2(t),t∈[0,tD)−rψ2(t)−λ2(t),t∈[tD,T],ψ2(T)=μ2−1,˙ψ3(t)=αψ3(t)−λ3(t),ψ3(T)=μ3,

and the switching functions (5) must satisfy the conditions:

 θu(t)=Aψ2(t)+ψ3(t)−K=0,θw(t)=p−ψ3(t)>0,θv(t)=−ψ2−1{<0,t∈[0,tD),=0,t∈[tD,T].

By mean of the same technique employed in previous case-studies, we have found a set of multipliers and that satisfy the complementary slackness condition (7):

 λ1(t)=0,λ2(t)={0,t∈[0,tD)r,t∈[tD,T],λ3(t)=α(A+K),λ4(t)=0μ1=0,μ2=0,μ3=A+K,μ4=0.

The latter proves the optimality of control (18). In order to find the switching point we integrate (2b) under optimal control (18) and obtain

 D(t)=D0exp[rt]+1r(vmax−Awmax)(1−exp[rt]),t∈[0,tD)

where indicate the exact moment when hits zero, that is, Thus,

 tD=1rlnvmax−Awmaxvmax−rD0−Awmax. (19)

Naturally, for smaller initial debt this point will be closer to zero, and for larger one it will be farther from zero. {remark} Eventually, it may occur that . In this case, the initial debt must be substantially large: Therefore, it will not be totally paid off by final time even under constant debt repayment at maximum rate:

 u∗=wmax,v∗=vmax,w∗=wmax.

Clearly, if , then and the value of criterion (1) is solely defined by the cumulative net profit at final time . Direct integration of (2a) under optimal control (18) over results in

 J(u∗,v∗,w∗)=N0+(Awmax−vmax)tD+wmax(p−A−K)T−BT. (20)
{remark}

Actually, Scenario 3.3 can be treated as a special case of Scenario 3.2 when and hence for all In this case, (16) coincides with (19) and (17) becomes (20). On the other hand, Scenario 3.1 cannot be treated as a special case of Scenario 3.2 by merely setting . The latter becomes obvious by setting in (17) and comparing this result with (13). In all three scenarios considered above, the optimal control strategies result in total absence of debts in the end of period, that is (except the situation when initial arrears are extremely high, see Remarks 3.2 and 3.3). On the other hand, positive cumulative profit and, consequently, positive value of the objective functional can only be guaranteed in Scenario 3.1 under the “profitability condition” (8). Effectively, second summand in (17) (as well as in (20)) will be negative if is very large. The latter may result in negative overall profit even under the condition (8).

Therefore, if there is an initial debt and available cash while the firm’s capacity of debt repayment is almost unlimited (that is, ), one should think about alternative control strategies in order to guarantee the positivity of overall profit.

## 4 Alternative control strategies and their optimality

Let us consider again the optimal control problem (1)-(4) under Scenario 3.2 when the firm has almost unlimited capacity of debt repayment. Mathematically, it means that and also implies that . In other words, this passage to the limit yields discontinuities in the state variables and at the initial point :

 limtD→0D(tD)=0(by definition of tD)
 limtD→0N(tD) = N0+limvmax→∞tD→0tD∫0(pwmax−vmax−B)dt = N0+limtD→0(pwmax−B)tD−limvmax→∞tD→0vmaxtD = N0−limvmax→∞vmax⋅1rlnvmaxvmax−rD0=N0−1rlimvmax→∞lnvmaxvmax−rD01vmax = N0−1rlimvmax→∞rD0v2maxvmax(vmax−rD0)=N0−D0.

In the above expressions we have used the form of given by (15) (since ) together with L’Hôspital’s rule. By permitting such finite jumps in the initial states of and , we can now adjust the optimal control strategy (14) in a way that its implementation will result in a positive value of the objective functional (1).

From the above formula, it is clear that optimal solution must depend on the relationship between and ; namely, there are two options to be revised: and .

### 4.1 Total debt repayment at initial time: N0≥D0

Suppose that the firm is capable to settle all its debts at once, and still to have a non-negative profit () at the beginning of the period. Then it will be profitable first to sell the existing stock of finished goods, without starting the production and, thus, not generating new arrears. Exactly at the moment (when the stock is cleared out, that is, ) the production is started at the rate equal to the maximum volume of demand . In other words, we arrive to Scenario 3.1 with one difference only: the initial cash resources are now given as . It is easy to prove that control strategy (9) will be optimal and to do so one should merely repeat all the deductions made in Subsection 3.1 with instead of .

Finally, the objective functional will have positive value under optimal control (9):

 J(u∗,v∗,w∗)=N0−D0+(pwmax−B)T+(A+K)wmax(tS−T)>0

due to the profitability condition (8).

### 4.2 Partial debt repayment at initial time: N0<d0

This case looks more challenging than the previous one. The firm does not have enough cash to settle all its debts right away. Therefore, it spends all available cash to repay a part of the initial debt . Thus, the firms profit at becomes equal to zero, while its initial debt is reduced to If then there should be no production up to the moment that marks a full clearance of the finished goods stock. From the production at a rate (maximum volume of demand) is started. Meanwhile, all the profit obtained from sales is spent on repayment of previous and new debts right up to the moment at which all the debts are paid off, that is, . Note that new debts are generated from the commencement of production, that is, for . Thus, for all the firm’s disbursements related to the production process will become equal to . In consequence, we propose the following optimal control:

 (21)

that have more sophisticated structure since depends also on .

Lagrange multipliers and must satisfy the complementary slackness conditions (7):

 S(t)

The underlying adjoint system (6) can be written as

 ˙ψ1(t)={−λ1(t),t∈[0,tD)0,t∈[tD,T],ψ1(T)=1,˙ψ2(t)={−rψ2(t),t∈[0,tD)−rψ2(t)−λ2(t),t∈[tD,T],ψ2(T)=μ2−1,˙ψ3(t)={αψ3(t),t∈[0,tS)αψ3(t)−λ3(t),t∈[tS,T],ψ3(T)=μ3,

and the switching functions (5) must satisfy the conditions:

 θu(t)=−Kψ1(t)+Aψ2(t)+ψ3(t){<0,t∈[0,tS)=0,t∈[tS,T],θv(t)=−ψ1(t)−ψ2=0,θw(t)=pψ1(t)−ψ3(t)>0.

Here (as well as in Scenario 3.2) we have two switching points and ; therefore, the optimal solution will essentially depend on their position with respect to each other. Consequently, one should revise two cases ( and ) and find only one underlying set of multipliers that satisfy the complementary slackness conditions (7) in both cases. The latter results in the following set:

 λ3(t)={0,t∈[0,tS)α(A+K),t∈[tS,T],λ4(t)=0,μ1=μ2=μ4=0,μ3=A+K.

Existence of the above multipliers clearly proves the optimality of control (21). This case bears strong resemblance to the Scenario 3.2 since there are two switching points ( and ) with the same meanings and , respectively. Their position with respect to each other will naturally depend on the amount of initial debt . Eventually, it may also happen that and there will only one switching point. The following proposition extends the Proposition 1 for this case and provides exact formulae for calculation of and in terms of problem entries.

###### Proposition 3

The stock of finished goods becomes empty at the moment given by (12) independently of initial debt amount . The total debt repayment occurs at the moment that depends on in the following way:
(a) if then and there is only one switching point;
(b) if then and

 tD=1rln(pwmax−Bpwmax−B−r(D0−N0)); (22)

(c) if