Optimal control of trade discount in a vertical distribution channel Igor Bykadorov 1, Andrea Ellero 2, and Elena Moretti 2 1 Sobolev Institute of Mathematics Siberian Branch Russian Academy of Science Acad. Koptyug prospect 4, Novosibirsk 630090, Russia bykad@math.nsc.ru 2 Dipartimento di Matematica Applicata Università Ca Foscari di Venezia Dorsoduro 3825/E, 30123 Venezia, Italy {ellero, emoretti}@unive.it Abstract. We consider a vertical control distribution channel in which a manufacturer sells a single kind of good to a retailer. We assume that wholesale price discount increases the retailer s sale motivation thus improving sales. We first analyze the manufacturer s profit maximization problem as an optimal control model in which the manufacturer s control is the discount on wholesale price (trade discount). We then embed the model in a Stackelberg game environment considering that manufacturer and retailer can both be leader of the channel, depending on the particular market structure. Keywords. Optimal control, sales motivation, vertical channel, trade discount. M.S.C. classification: 49N90. J.E.L. classification: C61, C73. 1 Introduction The aim of the paper is to investigate the relationships between the members of a distribution channel by means of optimal control models, the solution of which provide qualitative information on the channel members behaviors. We will consider a stylized vertical distribution channel: a manufacturer serves a single segment market through a single retailer. We investigate the incentive mechanism that can be used to drive channel members behaviors and consequently sales. Among the wide variety of marketing tools at disposal to manufacturers and retailers we will concentrate on simple contracts between suppliers and retailers based on trade discounts. A wholesale price discount is a promotional effort that has a double positive effect on sales: the first, on the market, is due to the fact This research has been supported by Università Ca Foscari di Venezia, CNR-NATO Senior Fellowships Programme 2003(no. 217.36 S), the Russian Federation for Basic Research (grant no.03-01-00877) and the Council for Grants (under RF President) and State Aid of Fundamental Science Schools (grant no. NSh-80.2003.6).
122 that a part of the wholesale price discount can be transferred to the final selling price (pass through). The second effect is on the retailer who becomes more motivated and then stimulated to work harder [6],[7]. We assume that an high price discount increases motivation and high motivation of retailer implies an higher effort in selling the product. The idea is that if the motivated retailer s performance is good in selling the product, the manufacturer s performance will be good as well. In Section 2 we propose an optimal control model in which the performance of the retailer is explicitly considered as a function of retailer s skill and motivation levels, taking also into account retailer s skill and other market parameters like consumer and retailer expectations. Optimal control models related to marketing problems have been widely studied in literature (see e.g. [4], [5]) also by the authors [1], [2]. A second way to analyze the relationship between manufacturer and retailer is the highlighting of their intrinsic conflict. In fact, their goals are different since each of the two firms aims at maximizing its own profit. In Section 3 we model the distribution channel as two different Stackelberg-games in which either the manufacturer or the retailer is the leader. In particular, in real distribution channels the retailer can become the leader when we consider large-scale retail stores. 2 The optimal control model We consider a manufacturer that sells a single kind of good during a time interval [, t 2 ]. The manufacturer acts as a monopolist in a vertical channel selling to the only downstream firm, the retailer, and, to improve sales, can produce a promotional effort by means of a wholesale price discount; moreover, the retailer transfers a part of the discount to the consumer. The aim of the manufacturer is to maximize the total profit in the given time period. The problem is formulated as an optimal control problem in which the state variables are the total sales and the retailer s motivation. The manufacturer s control is the discount on wholesale price. Define x(t) = total sales during time period [, t], p w (t) = wholesale price at time t, c 0 = unit production cost, α(t) = trade discount at time t, α(t) [α 1, α 2 ] [0, 1]. This way p w (t) = p(1 α(t)), where p is the wholesale price when no trade discount is applied. Remark that ẋ(t) represents the sales at time t and coincides with the consumer s demand at time t. This means that we assume that the firm sells exactly the produced quantity. The total profit of the manufacturer is (p w (t) c 0 )ẋ(t)dt,
123 i.e., assuming x( ) = 0, (p c 0 )x(t 2 ) p ẋ(t)α(t)dt. We assume that retailer s sales motivation at time t is increasing with respect to both consumer s demand and trade discount. More precisely the retailer s motivation is summarized by the state variable M(t) whose dynamics is given by where Ṁ(t) = γẋ(t) + ɛ(α(t) α), γ = sales productivity in terms of retailer s sales motivation, γ > 0, ɛ = price discount productivity in terms of retailer motivation, ɛ > 0, α = trade discount expected by the retailer, α (α 1, α 2 ). The parameter α takes into account the fact that the retailer has some expectations on the trade discount applied by the manufacturer, his motivation decreases if his expectations are disappointed, i.e. if α(t) < α, while he is happy if α(t) α. The dynamics of the total amount of sales at time t is defined by where ẋ(t) = θx(t) + δm(t) + ηβα(t), δ = retailer motivation productivity in terms of sales, δ > 0, η = price discount productivity in terms of sales, η > 0, β = part of wholesale price discount transferred to the consumer and θ > 0 is a saturation aversion parameter: in fact this way the sales rate decreases as the sales increase, modeling the market saturation effect. We assume β (0, 1), which means that we do not allow the retailer to retain the total discount and that in any case he will keep a part of it. This way the following optimal control problem can be formulated P : maximize subject to (p c 0 )x(t 2 ) p ẋ(t)α(t) dt, ẋ(t) = θx(t) + δm(t) + ηα(t), Ṁ(t) = γẋ(t) + ɛ(α(t) α), x( ) = 0, M( ) = M, α(t) [α 1, α 2 ] [0, 1], where M is the initial motivation of the retailer (we assume M > 0) and η = ηβ.
124 2.1 Solution of the optimal control problem Let us observe that problem P is linear with respect to the state variables and is quadratic with respect to control. We will assume that the product sold in the market is such that if α(t) is constant t [, t 2 ], i.e. the wholesale price is constant during the sale period, then the total sales function, x(t), is concave. This leads to the following conditions (see [3]): where a > 0, aδm + bα 2 + δɛα 0, (1) a = θ γδ, b = aη δɛ. (2) Obviously, profit must be nonnegative α(t) [α 1, α 2 ]. In particular, we require that the maximum allowed discount is such that the corresponding profit is equal to zero. At this aim we will set (see [3]) α 2 = p c 0 p. (3) Another requirement is that demand ẋ(t) must always be nonnegative. As it is shown in [3] it is sufficient that { b 0 aδm + aηα1 e a(t 2 ) + (bα 2 + δɛα)(1 e a(t 2 ) ) 0, b < 0 aδm + aηα 1 + δɛ(α α 1 )(1 e a(t 2 ) ) 0. (4) Therefore we will assume that the parameters of the model satisfy (4). Pontryagin Maximum Principle [8] allows to obtain the optimal control α (t) of problem P [3]: Proposition 21 1) Let b 0, i.e. δ θη ηγ+ɛ. Then τ [, t 2 ] such that { α α1, t [t (t) = 1, τ), D 1 e λt + D 2 e λt + α+α 2 2, t [τ, t 2 ], where D 1 and D 2 are constants and λ = increases on [τ, t 2 ]. 2) Let b < 0, i.e. δ θη (5) aδɛ η. Moreover function α (t) strictly ηγ+ɛ. Then τ 1 and τ 2 such that τ 1 τ 2 t 2 and α 1, t [, τ 1 ), α (t) = D 1 e λt + D 2 e λt + α+α 2 2, t [τ 1, τ 2 ], (6) α 1, t (τ 2, t 2 ], where D 1 and D 2 are constants. Moreover function α (t) is strictly concave on [τ 1, τ 2 ] and α (τ 1 ) α (τ 2 ) (7)
125 holds. Constants D 1, D 2 and the time of control s type changing (i.e τ if b 0 and τ 1, τ 2 if b < 0) can be found imposing continuity of the optimal control and the optimal trajectory (see [3]). Optimal sales level x (t) and optimal retailer motivation M (t) can then be calculated by substituting α (t) in the motion equations of Problem P. Observe that Proposition 2.1 states that the kind of optimal control depends on the value of δ. Parameter δ, retailer motivation productivity in terms of sales, describes retailer s selling skill, therefore the above proposition states that the kind of optimal control depends from the efficiency of the retailer. In particular with an inefficient retailer it is convenient to increase the discount during the selling period while with an efficient retailer it may be not convenient to increase the discount on wholesale price: in the latter case the manufacturer can improve profit reducing trade discount, the retailer is rather good and will sell a lot in any case. We provide now two examples which show the two possible optimal discount policies depending on the values of parameter δ as described in Proposition 21. In the first example we have increasing optimal trade discount during the sale period while in the second one the particular case in which the optimal discount is non-monotone is represented. Example 1. Let us consider problem P in the time period starting at = 0 and ending at t 2 = 2, with following values of the parameters: p = 2.9, c 0 = 0.8, β = 0.1215, γ = 0.7 (1 β) = 0.61495, ɛ = 0.6 (1 β) = 0.5271, α = 0.25, α 1 = 0.2, α 2 = p c0 p 0.7241, δ = 0.8, η = 0.8, θ = 10, M = 5. We have Case 1 of Proposition 21 and we obtain τ 1 0.50977. The optimal control of P is increasing in [, t 2 ] (see Figure 1). Example 2. Let us consider problem P with the same values of the parameters as in Example 1 except for: t 2 = 4, β = 0.15, γ = 0.7 (1 β) = 0.595, ɛ = 0.6 (1 β) = 0.51, θ = 1, M = 0.5. Now we have Case 2 of Proposition 21 and we can compute τ 1 0.7767, τ 2 3.7833. The dynamics of optimal control is reported in Figure 2. 3 Stackelberg-game approach A different point of view on the channel marketing activity can be obtained considering the manufacturer and the retailer as the two players of a Stackelberg game, assuming that each of them, in turn, takes the role of the leader. In this framework we will consider two controls: trade discount α(t), which is the manufacturer s control, and pass-through β(t), which is the retailer s control.
126 Fig. 1. Example 1: optimal discount policy. Fig. 2. Example 2: optimal discount policy.
127 The total profit of the manufacturer is, as in the optimal control problem P, (p c 0 )x(t 2 ) p ẋ(t)α(t)dt, while the total profit of the retailer can be written as follows p ẋ(t)α(t)(1 β(t))dt. We will consider two cases, in the first one the manufacturer is the leader of the channel and trade promotion has to be fixed at constant value α(t) = α, in the second one the retailer is the leader and his control has to be fixed at a constant value β(t) = β. 3.1 The manufacturer is leader Consider the manufacturer as the channel leader: we assume in this case that the manufacturer can only choose a constant trade discount during the whole sales period. This way we formulate the following open-loop Stackelberg game: ML : maximize (p c 0 )x(t 2 ) pα ẋ(t) dt = x(t 2 )((p c 0 ) pα) where, for each fixed α, functions x(t), M(t) and β(t) are optimal solution of maximize subject to pα ẋ(t)(1 β(t)) dt, ẋ(t) = θx(t) + δm(t) + ηαβ(t), Ṁ(t) = γẋ(t) + ɛ(α α), x( ) = 0, M( ) = M, β(t) [0, 1]. 3.2 The retailer is leader Consider the retailer as the channel leader: we assume in this case that the retailer can only choose a constant pass-through during the whole sales period. This way a new Stackelberg game can be formulated as follows: RL : maximize (1 β)p ẋ(t)α(t) dt where, for each fixed β, functions x(t), M(t) and α(t) are optimal solution of maximize subject to (p c 0 )x(t 2 ) p ẋ(t)α(t) dt, ẋ(t) = θx(t) + δm(t) + ηβα(t), Ṁ(t) = γẋ(t) + ɛ(α(t) α), x( ) = 0, M( ) = M, α(t) [α 1, α 2 ].
128 Remark that the follower problem coincides with problem P of Section 2. 3.3 The particular case in which both controls are constant We consider now the particular case in which both controls must be constant during the whole period [, t 2 ]. In this case the solution of problems ML and RL becomes straightforward and allows to obtain some preliminary considerations on the models outcomes. In particular for what concerns problem M L, in which the manufacturer is the leader, the optimal trade discount and pass-through can be easily computed by integration of the motion equations and by substituting into the two functionals the expression of the optimal total sales trajectory x (t). If the optimal trade discount is an interior point of the interval [α 1, α 2 ] then the profit of the manufacturer is given by x (t 2 )((p c 0 ) pα ) where α = p c 0 2p K 2(H + L) (8) with H = η [ 1 e a(t2 t1)] > 0 a L = δɛ [ 1 a a 1 ] a e a(t2 t1) (t 2 ) > 0 K = δ [ M(e a(t 2 ) 1) + ɛα( 1 a a + 1 ] a e a(t 2 ) + (t 2 )). The optimal value of the total sales, x (t 2 ), depends explicitly from α. More precisely where the optimal pass-through β is x (t 2 ) = Hα β + Lα + K β = Hα Lα K 2Hα. (9) We can observe that in the above formulas the parameter η, which describes the reaction of the market to price discounts, affects only the value of H. The parameters α and M, instead, which describe threshold levels for motivation and trade discount expectations and are therefore characteristic attributes of the retailer, affect only K. This enables sensitivity analysis with respect to different properties of the market actors. For example it is easy to obtain from (8) that a trade discount higher than α 2 /2 = (p c 0 )/(2p) can only be obtained if K is
129 negative. This means for example that α, i.e. the expected trade discount, must be sufficiently high. Moreover the sign of K is crucial in determining the relation between α and β (see (9)). The optimal reaction of the retailer to increasing values of the optimal trade discount can increase or decrease depending on the sign of K. Similar computations allow to provide analogous results can be obtained also if we consider constant controls in problem RL where the retailer is leader. 4 Concluding remarks In this paper we explore the bilevel programming approach to the optimal control of trade discounts. In section 3 we have formalized the particular situation in which one of the controls is constant and in section 3.3 we present some results related to a preliminary study of the case in which both controls are constant. A deeper insight into the problem could be obtained by means of sensitivity analysis of the profit of retailer and manufacturer. Another subject of future research could be to allow piecewise constant controls both for the leader and/or for the follower. References 1. Bykadorov, I., Ellero, A. and Moretti, E.: Minimization of communication expenditure for seasonal products. RAIRO Operations Research 36 (2002) 109 127. 2. Bykadorov, I., Ellero, A. and Moretti, E.: A model for the marketing of a seasonal product with different goodwills for consumer and retailer. Journal of Statistics & Management Systems 6 (2003) 115 133. 3. Bykadorov, I., Ellero, A. and Moretti, E.: Optimal control of retailer s motivation by trade discounts. Report n.119/2004, Dipartimento di Matematica Applicata, Università di Venezia (2004). 4. Feichtinger, G., Hartl, R. F. and Sethi, S. P.: Dynamic optimal control models in advertising: recent developments. Management Science 40 (1994) 195 226. 5. Jørgensen, S., Zaccour, G.: Differential games in marketing. Kluwer Ac. Pub., London (2004). 6. Lawler E. E.: Motivation in work organizations. Brooks/Cole, Monterey - Cal (1973). 7. Mitchell T. R.: Motivation: new directions for theory, research and practice. Academy of management review 7 (1982) 80 88. 8. Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V. and Mishchenko, E.F.: The mathematical theory of optimal processes. Pergamon Press, London (1965).
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