OR Spectrum 28:223–240 (2006) DOI 10.1007/s00291-005-0004-5
REGULAR A RT ICLE
Jen-Ming Chen . Liang-Tu Chen . Jun-Der Leu
Developing optimization models for cross-functional decision-making: integrating marketing and production planning
Published online: 17 September 2005 © Springer-Verlag 2005
Abstract Although the lately evolved enterprise resource planning systems provide a unified platform for managing and integrating core business processes within a firm, the decision-making between marketing and production planning still remains rather independent. It is due in large part to the inherent weaknesses of ERP such as fixed and static parameter settings and uncapacitated assumption. To remedy these drawbacks, we propose decision model that solves optimally the production lot-size/scheduling problem taking into account the dynamic aspect of customer’s demand and the restriction of finite capacity in a plant. The joint decision model is developed based on the inventory followed by shortages (IFS) inventory policy. The model is practical and can possibly be used as an add-on optimizer like an advanced planning system in ERP framework that coordinates distinct functions with an aim at maximizing the total profit of a firm. In this study, two versions of the model, i.e., the decentralized and the coordinated decisionmaking policies, were derived. They were solved by using dynamic programming technique combined with iterative search. In addition, numerical study was carried out, comparative experiment was conducted, and sensitivity was analyzed with respect to major parameters. This research was partially supported by the National Science Council (Taiwan) under Grant NSC93- 2416-H-008-007. J.-M. Chen (*) . L.-T. Chen Institute of Industrial Management, National Central University, 320 Jhongli, Taiwan E-mail:
[email protected] L.-T. Chen Department of Industrial Engineering and Management, Chin-Min Institute of Technology, 351 Toufen, Taiwan J.-D. Leu Department of Business Administration, National Central University, 320 Jhongli, Taiwan
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Keywords Production . Marketing . Dynamic programming . Inter-functional coordination . Deterioration 1 Introduction The latest manufacturing technologies enhance cross-functional interaction between manufacturing and marketing such as flexible manufacturing system, justin-time, quick response, manufacturing resources planning, and enterprise resource planning. In spite of increasingly emphasizing on the aspect of customer demands, many production decision-making processes do not take marketing’s dynamic nature into account. For example, the classical economic order quantity model assumes a constant demand that is time-invariant, exogenous, and uncontrollable. The lately evolved ERP systems can optimize lot-size/scheduling problem with time-varying demand over a finite planning horizon, yet the dynamic aspect of pricing and other marketing related variables tend to be ignored in the highly computerized manufacturing planning systems. This paper is an attempt to bridge the gap between marketing and manufacturing decision making with the objective of developing mathematical model that can possibly serve as a solver or an optimizer in an add-on advanced planning system (APS) within ERP systems (Günther 2005). The basic architecture of an ERP system builds upon one database and a unified interface across the entire enterprise providing integrated business solutions for the core processes and the main administrative functions of an enterprise. In addition to increasing operational efficiencies, the highly integrated system can strengthen strategic advantages and generate financial and non-financial benefits that have been well documented in the literature (Gattiker and Goodhue 2004; Langenwalter 2000; Mandal and Gunasekaran 2002; Robinson and Dilts 1999; Stefanou 2001). However, ERP benefits cannot be fully realized without a finely tuned alignment and reconciliation between system configurations, organizational imperatives, and core business processes (Al-Mashari et al. 2003). Further, the fundamental basis of planning and scheduling in an ERP system is based on fixed and static parameter settings (e.g., lead time, lot size, safety stock, and costs) with infinite capacity (Hsiang 2001; Petty et al. 2000). As a result, the system generates suboptimal solutions to the production lot-size/scheduling problem. To remedy these drawbacks, abundant research has been devoted to methodology development that adds “intelligence” into the current ERP systems. For examples, Petty et al. (2000) proposed a hybrid approach that offers finite capacity scheduling (FCS) scheme incorporated into MRP II packages. Such hybrid claims to give the benefits of FCS to established system users. Kim and Hosni (1998) formulated a multi-level capacitated optimization model and developed a relative efficient heuristic working under MRP II environment, which considers work center capacities and interrelationship between levels in lot-sizing computation. Vandaele and De Boeck (2003) proposed an advanced resource planning scheme explicitly capturing the stochastic nature of manufacturing systems. It is an ideal high-level tuning and planning tool and can be used in a variety of planning environments like JIT, MRP, ERP, and Theory of Constraints (TOC). Hu and Munson (2002) proposed an incremental quantity discount scheme rectifying the unrealistic assumption of fixed prices for MRP lot-sizing planning. In a com-
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parative study, Liberopoulos and Dallery (2003) proposed a unified modeling framework based on queuing network representation for describing, comparing, and contrasting classical multi-stage production-inventory control policies with lot sizing, including reorder point policy (RPP), JIT, MRP, and MRP II. For perishable goods management, Günther et al. (2005) incorporated shelf-life restrict into an advanced planning system. A recent review given by Jacobs and Bendoly (2003) suggests a major shift of ERP research from implementation issues toward the much richer area of ERP extendibility such as the model-based development for collaborative planning and decision support. In this paper, we develop a tactical-level decision model that solves the production lot-size/scheduling problem taking into account the dynamic nature of customer’s demand which is partially controllable through pricing scheme. As analogous to the sales and operations planning proposed by Van Landeghem and Vanmaele (2002) or the advanced planning and scheduling systems given by Gayialis and Tatsiopoulos (2004) and Günther et al. (2005), the proposed scheme can possibly be used for integrated planning between manufacturing and marketing like the APS system within a generic ERP framework (see Fig. 1), which integrates and coordinates distinct functions within a firm (Günther 2005; Langenwalter 2000). It is well documented (Eliashberg and Steinberg 1993; Federgruen and Heching 1999; Klassen and Rohleder 2001; Pullman and Moore 1999) that the coordinated approach is superior to the traditional decentralized approach in many dimensions such as minimizing cost or maximizing profit. Typical solution techniques in currently available APS systems include mixed integer programming formulation (Lütke Entrup et al. 2005; Simchi-Levi et al. 2000) and finite capacity heuristics such as simulated annealing and tabu search
Fig. 1 An integrated planning of the APS system
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(Cohen and Huchzermeier 1999). In this paper, we use calculus-based formulation coupled with dynamic programming and iterative search techniques to solve the cross functional decision problem. There is substantial literature on issues involving the coordination of pricing and production planning decisions. A comprehensive survey of this literature was given by Eliashberg and Steinberg (1993) and Yano and Gilbert (2004). To remain focus, we only review those works dealing with deteriorating items. Deterioration or decay defined by Raafat (1991) is the process that prevents an item from being used for its intended original use such as spoilage of foodstuffs, physical depletion due to pilferage, evaporation of liquids, etc. These phenomena of deterioration are prevalent and should not be disregarded in production lot-size/scheduling. This paper considers the effect of deterioration as a function of the on-hand stock level. Nahmias (1982) and Raafat (1991) provided comprehensive reviews of the related work dealing with perishable or deteriorating items. However, the aforementioned literature and most of the references therein do not incorporate pricing or any other marketing related decision. Some exceptions include the models proposed by Chen and Chen (2005), Cohen (1977), Kang and Kim (1983), Rajan et al. (1992) and Abad (1996). The models of Cohen and Rajan et al. jointly determine the optimal replenishment cycle and price for inventory that is subject to continuous decay along product lifetime under standard EOQ cost assumptions. Abad extended the model of Rajan et al. by allowing shortages that can be partially backlogged at the end of the cycle. The model developed by Kang and Kim is a variant of Abad that deals with single period setting. Chen and Chen presented a dynamic version of the pricing-production decision model under multi-period setting, which is based on shortages followed by inventory (SFI) inventory policy. However, the SFI policy is impractical in many production/inventory systems. Instead, this paper adopts a more practical policy. We also take into account the time-varying deteriorating rate, selling price, production cost, and production rate. The remainder of this paper is organized as follows. Section 2 describes problem context. The mathematical model is developed in Section 3, in which two versions of the model, i.e., the decentralized and the coordinated decision policies, are derived and the solution procedures are developed. Section 4 conducts a comparative study between the solutions generated by the two policies, and performs sensitivity analysis with respect to major parameters. Concluding remarks and suggestions for future research are given in Section 5. 2 The problem context In this section, we describe the underlying problem and settings, including assumptions and necessary notations. We consider a manufacture firm who produces and sells a single product that is subject to continuous decay over lifetime, faces a price-dependent and time-varying demand function D(p, t) = f(p)g(t), and has the objective of determining price and production lot-size/scheduling so as to maximize the total profit stream over multi-period planning horizon. The demand function considered in this paper should satisfy the following realistic assumptions: (i) D(p, t) ≥ 0 and is continuous for p ∈ (0, pmax), t ≥ 0; (ii) D(p, t) is decreasing in p; and (iii) limp!0 Dðp; t Þ < þ1 and limp!1 Dðp; t Þ ¼ 0
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for t ≥ 0. The assumptions are assertions that demand is nonnegative, changes continuously with price and time, decreases as price increases, and is finite. There are many forms of demand functions that satisfy the assumptions (see Lau and Lau 2003). In this study, we assume the demand function is of the form: D(p, t) = f(p)g (t), where f(p) = a−bp, a > 0, b > 0, and p < pmax = a/b, and g ðt Þ ¼ et . Since et is nonnegative for all , the demand function Dðp; t Þ ¼ ða bpÞet satisfies the three realistic assumptions mentioned above. Besides, the sales trend over product lifecycle is characterized by , positive (or negative) value of which represents upward (or downward) demand rate over time, and = 0 represents time-invariant demand. Choosing this particular form is for its simplicity and for streamlining further discussions on the optimality of the problem. A typical behavior of the production schedule coupled with time-decreasing demand based on inventory followed by shortages (IFS) policy is illustrated in Fig. 2. Each cycle, say [zi−1, zi], of the production/inventory system starts with the production run lasting over period [zi−1, T1], followed by the consumption period over [T1, T2] due to demand requirements and deterioration loss, after which the system starts with the shortage period over [T2, T3], at which the production resumes over [T3, zi] to meet the backlogged and demand requirements. We assume the production rate is finite and shortages are allowed and completely backordered with an exception for the last cycle in which shortages are not allowed. To remain focus, we assume no inventory is held at the beginning and at the end of the time horizon. If the initial inventory level is positive in the system, no action will be taken until the depletion of inventory. In addition, we assume the deterioration of units occurs only when the item is effectively in stock, and there is no repair or replacement of deteriorated units during the planning period. To incorporate the multi-period setting in a typical ERP environment, we assume the inventory and selling price are reviewed periodically at time t, t = 0, 1, 2,..., H, where H is the planning horizon. At the beginning of each period, a joint decision is made regarding the lot-size and schedule of a new production
Fig. 2 Production schedule for a deteriorating item with time-decreasing demand based on inventory followed by shortages (IFS) policy
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run (if any) and its associated selling price p. The problem is equivalent to determining the optimal sequence of times zi−1, i = 1, 2,..., n, at which a new inventory cycle starts, the selling price is reset, and the production schedule (i.e., T1, T2, and T3) and lot-size Q are specified simultaneously so that the total profit stream over [0, H] is maximized. It is worthy noting that n ≤ H, z0 = 0, zn = H, zi−1 is integer and zi−1 ∈ [0, H), and n is the total number of productions to be scheduled over the horizon. To facilitate further discussions, the following notations are defined and will be used throughout the paper. θ(τ(t)) = The deteriorating coefficient of the item over product lifetime τ(t). I(p, t) = The inventory level at time t when the selling price is p. θ(τ(t))I(p, t) = The deterioration rate at time t when product lifetime is τ(t) and its selling price is p. ρ(t) = The production rate at time t with ρ(t) > D(p, t) for 0 ≤t≤ H. ω(t) = The variable production cost per unit of item at time t. K = The fixed production setup cost per run. h = The holding cost per unit of time and unit of item. s = The shortage cost per unit of time and unit of item. 3 The model In the underlying scenario, we first develop model to solve the pricing and scheduling problem over inventory cycle [zi−1, zi] in a static fashion. The dynamic decision for the sequence zi−1, i = 1, 2,..., n, is then solved using dynamic programming formulation. In the proposed system, an inventory cycle can be divided into four periods (see Fig. 2): the demand, deterioration, and production simultaneously incurred over period [zi−1, T1], the demand and deterioration incurred over [T1, T2], the demand only over shortage period [T2, T3], and the demand and production over period [T3, zi]. The variation of the inventory level I(p, t) with respect to time t can be described by the following system of differential equations: @ I ðp; t Þ þ I ðp; t Þð ðt ÞÞ ¼ ðt Þ ða bpÞet ; @t @ I ðp; t Þ þ I ðp; t Þð ðt ÞÞ ¼ ða bpÞet ; @t @ I ðp; t Þ ¼ ða bpÞet ; @t
for
@ I ðp; t Þ ¼ ðt Þ ða bpÞet ; @t
for for
zi1 t T1 ; T1 t T2 ;
T2 t T 3 ; for
T 3 t zi :
(1) (2) (3) (4)
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Solving the system of Eqs. (1–4) with the initial and boundary conditions I(p, zi−1) = 0, I(p, T2) = 0, and I(p, zi) = 0, yields Zt I ðp; t Þ ¼
R u ððvÞÞd ðvÞ ðuÞ ða bpÞeu e t du; for
zi1 t T1 ;
(5)
zi1
ZT2 Ru uþ ð ðvÞÞd ðvÞ t du; ¼ ða bpÞe
for
T1 t T2 ;
(6)
t
ZT2 ¼ ða bpÞeu du;
for
T2 t T3 ;
(7)
t
Zt ðuÞ ða bpÞeu du; ¼
for
T 3 t zi :
(8)
zi
Equating Eqs. (5) and (6) at time t = T1 generates ZT1
ðt Þ ða bpÞet e
Rt T1
ð ðvÞÞd ðvÞ
Rt ZT2 tþ ð ðvÞÞd ðvÞ T1 dt¼ ða bpÞe dt:
zi1
(9)
T1
The optimal stopping time of production T1 can be obtained by solving Eq. (9) through classical numerical search techniques. It is worth mentioning that the stopping production time is a function of p and T2. Likewise, equating expressions (7) and (8) at time t = T3, we have ZT2 ZT3 t ða bpÞe dt ¼ ðt Þ ða bpÞet dt: T3
(10)
zi
The optimal starting time of production T3 can be obtained in a similar manner through numerical search, which is also a function of p and T2. Accordingly, the production lot-size can be obtained by integrating the production rates over [T3′, T1], in which T3′ is the starting production time just before time epoch zi−1: ZT1 Qðp; T2 Þ ¼
ðt Þdt;
(11)
0
T3
which transforms time-variables T1(p, T2) and T3(p, T2) into quantity-variable Q(p, T2), representing the economic production quantity over the cycle with two variables p and T2 therein. In what follows, the two-variable maximization problem
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over cycle [zi−1, zi] is formulated and solved optimally. Determination of the optimal sequence of inventory starting time, zi−1, i = 1, 2,..., n, for the multi-period problem is then solved by using dynamic programming. For an arbitrary cycle [zi−1, zi], the total revenue generated from the sales is: Zzi zi1 ð pÞ ¼
pða bpÞet dt:
(12)
zi1
The total cost due to variable production cost, holding cost, shortage cost, and setup cost over [zi−1, zi] is: ! RT1 Rzi RT1 RT2 !ðt Þðt Þdt þ !ðt Þðt Þdt þ h I ðp; t Þdt þ I ðp; t Þdt zi1 ðp; T2 Þ ¼ zi1 zi1 T3 T1 ! Rzi RT3 ðI ðp; t ÞÞdt þ ðI ðp; t ÞÞdt þ K: þs ð13aÞ T2
T3
Substituting Eqs. (5–8) into Eq. (13a), interchanging labels and combining terms yield ZT2 Zzi t C1 ðT1 ; t Þða bpÞe dt þ C2 ðT3 ; t Þða bpÞet dt (13b) zi1 ðp; T2 Þ þ zi1
T2
ZT1 þ zi1
Zzi C3 ðT1 ; t Þ dt þ C4 ðT3 ; t Þdt þ K; T3
where Zt R t ð ðvÞÞd ðvÞ C1 ðT1 ; t Þ ¼ h e u du; T1
C2 ðT3 ; t Þ ¼ sðT3 t Þ; C3 ðT1 ; t Þ ¼ ð!ðt Þ C1 ðT1 ; t ÞÞðt Þ; C4 ðT3 ; t Þ ¼ ð!ðt Þ C2 ðT3 ; t ÞÞðt Þ:
(13b)
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The total profit over [zi−1, zi] is the net value of the revenue and the cost as expressed below: zi1 ðp; T2 Þ ¼ zi1 ð pÞ ¼
RT2
zi1 ðp; T2 Þ
ðp C1 ðT1 ; t ÞÞða bpÞet dt þ
zi1
Rzi T2
RT1
ðp C2 ðT3 ; t ÞÞða bpÞet dt (14)
Rzi C3 ðT1 ; t Þdt C4 ðT3 ; t Þdt K:
zi1
T3
(14)
Model (14) is a function of selling price p and time-variable T2, which can be determined either by the decentralized policy or by the centralized, coordinated policy. In the decentralized decision process, the marketing department sets the price by maximizing its gross profit function disregarding the production cost, the market responds with a specific demand, and the production department makes the lot-sizing and scheduling decision with the objective of minimizing the total production cost while satisfying the demand. As contrast to the sequential process, the coordinated policy makes the pricing and production decisions at a time. 3.1 The decentralized policy The gross profit function over [zi−1, zi] can be expressed below: Zzi zi1 ð pÞ ¼
ðp !ðt ÞÞða bpÞet dt:
(15)
zi1
The optimal price over the cycle can be obtained by differentiating Eq. (15) with respect to p and setting the results equal to zero: d z ð pÞ ¼ dp i1
Zzi
ða 2bp þ b!ðt ÞÞet dt ¼ 0:
(16)
zi1
Let p be the solution of Eq. (16) which represents the optimal price over the cycle. To show the optimality of the solution we shall demonstrate that the gross profit function in Eq. (15) is concave in p, i.e., its second order sufficient condition is strictly less than zero. *
Proposition 1 The gross profit function zi1 ð pÞ is concave in p. Proof The second partial derivative of the gross profit function with respect to p is zi d zz ð pÞ ¼ 2b R et dt zi1 dp zi1
¼ 2bðezi ezi1 Þ :
(17)
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Since b > 0 and ðezi ezi1 Þ > 0 for zi > zi−1, regardless of the value of , Eq. (17) is strictly negative. We have shown that the solution of the first order necessary condition p* is optimal provided the demand function is specified. If the firm uses a sequential process in its decision-making process, decisions of pricing and production schedule are made separately. The objective of the production department in the decentralized policy is to minimize the total cost. By differentiating Eq. (13b) with respect to T2 and setting the result equal to zero, the optimal production schedule, denoted by T2*, is the solution of the first order differential equation. Since we are unable to obtain a closed form solution to this problem, an iterative search procedure is developed to determine variable T2. The procedure starts with solving Eq. (16) for p* and an initial guess of T2 ¼ T20 where T20 is a reasonable value for T2, based on which we compute the corresponding T10 and T30 as the initial maximum inventory and shortage time epochs. Letting T1 ¼ T10 and T3 ¼ T30 ; and differentiating Eq. (13b) with respect to T2 and setting the result equal to zero, we can solve the time-variable T2 ¼ T21 : The process continues iteratively until the solutions converge, which is detailed below. Iterative Search { Step 0: Let l = 1, ɛ = 10−4. Step 1: Solve Eq. (16) for p*. Step 2: Let the first order necessary condition of Eq. (13b) with respect to T2 be (13c). Step 3: Give an initial guess of T2 ¼ T2l1 where T2l1 is a reasonable value for T2, substitute p*, T2l1 into Eqs. (9) and (10), and solve the equations to obtain T1l1 and T3l1 : Step 4: Substitute p*, T1l1 and T3l1 into Eq. (13c), and solve the equation to obtain T2l . Step 5: Substitute p*, T2l into Eqs. (9) and (10), and solve the equations to obtain T1l and T3l : Step 6: If Abs T1l T1l1 T1l1 > "; Abs T2l T2l1 T2l1 > "; or Abs ½T3l T3l1 = T3l1 > "; then l = l+1 and go to Step 3; otherwise, output T1*, T2*, T3*, and p*; stop. } The search procedure generates optimal schedule and price: T1*, T2*, T3*, and p* over inventory cycle [zi−1, zi], only if Eqs. (9) and (10) and the first order necessary condition of (13b) with respect to T2 are uni-modular. Since the optimality property depends on the forms and parameters of demand and deteriorating functions, we cannot prove it analytically for the general functions. Instead, we take a numerical approach to verify the optimality in our experiment study, where all functions and parameters are specified explicitly.
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* Determination of the optimal inventory schedule zi−1 , i=1, 2,..., n, and the integer subscript n over the planning horizon can be accomplished by solving the dynamic programming model: (18a) zi ¼ Max zi1 þ zi1 p* ; T2* : 0 zi1 < zi < H
and H ¼ Max zi1 þ zi1 p* ; H : 0 zi1 < H ;
(18b)
with boundary condition ∏0 = 0. In Eq. (18a,b), zi represents the maximal cumulated value of the profit generated over period [0, zi], noting that zi ¼ Pi j ¼ 1 zj1 ðp; T2 Þ: The recursive procedure works in a forward fashion to determine the maximal total profit over [0, zi]. On the last stage of the procedure, zn ¼ H is found that is the maximal accumulated profit over planning horizon [0, H]. It is worth noting that if shortages are allowed in the last cycle, then Eq. (18b) should be omitted in tracking forward recursive procedure. The optimal * schedule zi−1 and the associated lot-size Q(p*, T2*) in Eq. (11) can be determined by tracking backward from time H to time 0. The solution procedure is detailed in Fig. 3. 3.2 The coordinated policy The multi-variate maximization problem under the coordinated policy can be formulated as follows: Maximize
zi1 ðp; T2 Þ
(19)
Under coordinated policy, selling price and production lot-size are determined simultaneously by solving the first order differential equation of model (19) with respect to p and T2. The optimal solution, denoted respectively by p** and T2**, can be obtained by using iterative search. The concavity property of the profit model can be proved by verifying the determinant of the Hessian matrix and one of the following sufficient conditions: @2 z ðp; T2 Þ < 0; @p2 i1
@2 z ðp; T2 Þ < 0: @T22 i1
(20)
Due to the high-power expression of the exponential function and the sufficient condition is strongly dependent on the values of parameters and the form of demand and deteriorating functions, the signs of Hessian matrix and Eq. (20)
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Fig. 3 Solution procedure based on dynamic programming combined with iterative search
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cannot be verified analytically. Instead, we will take a numerical approach to ensure the concavity of the problem to be studied in our experiments. The search procedure generates converged solutions into optimal, say T1**, T2**, ** T3 , and p**, for an arbitrary cycle [zi−1, zi], where T1** and T3** can be transformed into production lot-size Q(p**, T2**) using Eq. (11). The production schedule and associated price trajectory and lot-sizes over the planning horizon can be determined by solving the dynamic programming model: (21a) zi ¼ Max zi1 þ zi1 p** ; T2** Þ : 0 zi1 < zi < H and H ¼ Max zi1 þ zi1 p** ; H : 0 zi1 < H ;
(21b)
with boundary condition ∏0 = 0. The dynamic procedure follows the analogous fashion as stated previously which has been detailed in Fig. 3. 4 Numerical study The dynamic solution procedure for both policies was implemented on a personal computer with a Pentium CPU at 2.0 GHz under Windows XP operating system using Mathematica version 5.0. In our experiments, we first verified numerically the concavity of the models, and then determined the optimal price p* or p** and * ** or zi−1 , i = 1, associated lot-size Q(p*, T2*) or Q(p**, Tz** ), and the schedule zi−1 2,..., n. For the iterative procedure, the process continued until the absolute value of relative error between consecutive iterates was less than or equal to 10−4. The process took about 3–5 iterations to converge in all experiments being studied. The computer time required for the dynamic programming was, on the average, less than 5 s. Several experiments were conducted to attend qualitative insights into the structures of the proposed policies and their sensitivity with respect to major parameters. We focused in particular on investigating the solution property as well as the benefit of the coordinated policy compared to the decentralized policy in settings with time-decreasing demand. Further, we explored the impacts of pricesensitivity coefficient of demand, coefficient of production rate, and production unit cost on the profits generated by the two policies. 4.1 An illustrative example The example is based on the settings: number of periods H=12, cost parameters K = 60, h = 0.1, s = 10, and time-decreasing production cost !ðt Þ ¼ cet ¼ 5e0;05t ; i.e., c = 5 and β = −0.05, fixed production rate ðt Þ ¼ fe t ¼ 300 (i.e., γ = 0), constant deteriorating rate θ(τ(t)) = α = 0.1, and demand function Dðp; t Þ ¼ ða bpÞet ¼ ð100 10pÞe0:05t : The scenario simulates the diminishing effect of demand function toward the end of product lifecycle. The solutions generated by the two policies are summarized in Tables 1a and b where the selling price p and
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Table 1a Numerical results of optimal price and production scheduling Period
1
2
Decentralized policy p 7.33 7.33 T1 0.26 T2 T3 Coordinated policy p 7.83 7.83 T1 0.21 T2 T3
3
4
5
6
7
8
9
10
11
12
7.33
7.00 3.25
7.00
7.00
6.72 6.24
6.72
6.72
6.48 9.24
6.48
6.48
6.86 9.21
6.86
6.86
2.75 2.98 7.83
5.78 5.98 7.45 3.22
7.45
2.75 2.98
7.45
8.80 8.99 7.12 6.21
7.12
5.78 5.99
7.12 8.80 8.99
production scheduling (T1, T2, and T3) are reported in Table 1a and demand D and lot-size Q over the horizon are in Table 1b. Several key performance measures are given in Table 2, including the second order sufficient conditions and the determinant of Hessian matrix, maximal inventory level Imax, service level (i.e., product availability) SL, and the accumulated profit ∏. In the example, both policies generated identical production cycles (zi−1 = 0, 3, 6, 9) and the same number of productions n = 4. Comparing the solutions generated by the two policies, the coordinated policy generates higher price and more net profit (373.42 vs 356.37). Intuitively, the higher price yields less quantity demanded in the market as well as smaller production lot-size and less amount of deteriorating units. In addition, the coordinated policy produces smaller value of Imax (59.39 vs 70.01) which implies less storage capacity required and, as a result, less investment needed in warehousing and material handling. 4.2 Comparative study Using the base settings above, we compare the net profits generated by the two policies with respect to major parameters, such as the demand parameters a and b, production unit cost c, holding cost h, setup cost K, and deteriorating coefficient α. As shown in Fig. 4, the coordinated policy outperforms the decentralized
Table 1b Numerical results of demand and optimal lot-size Period 1
2
3
4
5
6
7
8
9
10
11
12
Decentralized policy D 26.08 24.81 23.60 25.17 23.94 22.78 23.68 22.52 21.43 21.87 20.81 19.79 Q 77.82 81.74 77.12 75.85 Coordinated policy D 21.17 20.13 19.15 21.43 20.38 19.39 20.80 19.79 18.82 19.51 18.56 17.66 Q 63.26 69.48 67.71 67.71
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Table 2 Key performance measures [zi−1, zi]
[0,3]
[3,6]
[6,9]
[9,12]
Decentralized policy @2 @p2 @ @T22
Imax SL ∏ Coordinated policy @2 @p2 @2 @T22
Hessian Matrix Imax SL ∏
−55.72
−47.96
−41.28
−35.53
23.58
221.20
207.51
188.42
70.01 0.93
68.58 0.94
65.58 0.95
65.60 1.00
356.37 −53.23
−46.32
−40.19
−34.75
−190.30
−190.72
−184.07
−169.41
10128.40 58.09 0.93
8832.76 59.39 0.94
7397.96 58.37 0.95 373.42
5810.88 59.18 1.00
significantly in the total net profit as b and c increase and as a decreases. Other parameters (h, K, and α) seem insignificant to the net profit improvement between the two policies.
Fig. 4 Impact of major parameters on profit improvements: coordinated vs decentralized
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Fig. 5 Sensitivity of major parameters in the net profit generated by the coordinated policy
4.3 Sensitivity analysis Fig. 5 reports the impacts of the major parameters (a, b, c, h, K, and α) on the total profit generated by the coordinated policy. It suggests that the total profit is more sensitive to the demand parameters and production unit cost, but is less sensitive to holding cost, setup cost, and deteriorating coefficient. 5 Conclusions In this paper, two versions of the joint decision-making model were developed, which are based on a more practical inventory followed by shortages policy. The model determines simultaneously or sequentially the optimal price and production lot-size/scheduling for a perishable item over multi-period planning horizon. We formulated the profit-maximization problem as dynamic programming models, and provided efficient solution procedures. An illustrative numerical example and the sensitivity with respect to major parameters have been conducted. Numerical results have shown that the solution generated by the coordinated policy outperforms that by the decentralized in maximizing net profit. Further, we have shown that the percentage of profit difference between the two policies increases significantly in the demand parameters and production unit cost. The proposed model is based on a periodic review policy which makes it applicable in many manufacturing planning and control practices. It can be used as an add-on optimizer like the advanced planning system to an ERP system (Günther
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2005). The main restriction for practical implementation of the model is subject to its dynamic pricing mechanism which may be unacceptable by the contract-based customers. Further, the deterministic demand function may make the research impractical and restrict its potential applicability, due to unavailable data to specify the parameters. However, as with most management science applications, the proposed scheme is purely theoretical. In our study, the demand was deterministic, the pricing policy was periodic, and the proposed scheme can only apply to the designated scenario. These assumptions and restrictive conditions may limit its applicability, yet does not degrade its merit. Our exploratory research has demonstrated technical feasibility of the proposed scheme. A natural extension of this research is to consider more complicated and practical demand and deterioration functions in the model, such as the uncertain demand, stochastic demand, and fuzzy-modeling deterioration. In addition, the value drop effect considered by Rajan et al. (1992) can possibly be incorporated into the model. Another direction of this research is to develop a prototype of an advanced planning system with an ERP system (Günther 2005) that integrates the management science techniques into commercial software for collaborative and robust planning. References Abad PL (1996) Optimal pricing and lot-sizing under conditions of perishability and partial backordering. Manage Sci 42(8):1093–1104 Al-Mashari M, Al-Mudimigh A, Zairi M (2003) Enterprise resource planning: a taxonomy of critical factors. Eur J Oper Res 146:352–364 Chen JM, Chen LT (2005) Pricing and production lot-size/scheduling with finite capacity for a deteriorating item over a finite horizon. Comp Oper Res 32:2801–2819 Cohen MA (1977) Joint pricing and ordering policy for exponentially decaying inventory with known demand. Nav Res Logist Q 24:257–268 Cohen M, Huchzermeier A (1999) Global supply chain management: a survey of research and applications. In: Tayur S, Ganeshan R, Magazine M (eds) Quantitative Models for Supply Chain Management. Kluwer Academic Publishers, Dordrecht, pp 671–702 Eliashberg J, Steinberg R (1993) Marketing-production joint decision-making. In: Eliashberg J, Lillien GL (eds) Handbooks in operations research and management science. Elsevier Science Publishers, B V, Marketing, Amsterdam, pp 5 Federgruen A, Heching, A (1999) Combined pricing and inventory control under uncertainty. Oper Res 47(3):454–475 Gattiker TF, Goodhue DL (2004) Understanding the local-level costs and benefits of ERP through organizational information processing theory. Inf Manage 41:431–443 Gayialis SP, Tatsiopoulos IP (2004) Design of an IT-driven decision support system for vehicle routing and scheduling. Eur J Oper Res 152:382–398 Günther HO (2005) Supply chain management and advanced planning systems: a tutorial. In Günther HO, Mattfeld DC, Suhl L (eds) Supply Chain Management und Logistik. Physica, Heidelberg, pp 3–40 Günther HO, van Beek P, Grunow M, Entrup M (2005) Coverage of shelf life in APS systems. Günther HO, Mattfeld DC, Suhl L (eds) Supply Chain Management und Logistik. Physica, Heidelberg, pp 135–156 Hsiang T (2001) The illusion of power. OR/MS Today February Hu J, Munson CL (2002) Dynamic demand lot-sizing rules for incremental quantity discounts. J Oper Res Soc 53:855–863 Jacobs FR, Bendoly E (2003) Enterprise resource planning: developments and directions for operations management research. Eur J Oper Res 146:233–240 Kang S, Kim IT (1983) A study on the price and production level of the deteriorating inventory system. Int J Prod Res 21:899–908
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