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ol 7310 Ma sses der s th erti alg opo nly 1. Introduction tive is trol), stock rejec from ng, tha s exte a,b, 20 periods is fully backordered (Ha, 1997b; de Verikourt et al., 2002), reported in the literature deals with situations where the expected aggregate demand is assumed to be much less than the need and ive in ocate Recently the problems of joint production and admission control Contents lists available at ScienceDirect w.e Int. J. Productio Int. J. Production Economics 131 (2011) 663–673 that the optimal policy is characterized by monotone switchingE-mail address: efioan@dpem.tuc.gr or completely lost (Ha, 1997a, 2000). This is also the case in the production systems control literature in general. Most of the work have attracted some attention (see, e.g., Caldentey, 2001; Song, 2006; Ioannidis et al., 2008). For the problem of inventory rationing, Benjaafar et al. (2007) have incorporated order admis- sion decisions for two priority classes of customers and proved 0925-5273/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.ijpe.2011.02.011 turers producing new vehicles, and to repair shops as well. Most of the work in this area assumes that the demand during stockout the available capacity between competing requirements and to manage demand, so as to maximize profit and delivery reliability. Another example is manufacturers operating both as wholesalers and retailers. A similar situation is when a part supplier has to satisfy demand from both original equipment manufacturers and the so-called after-market. This is the case in the automobile industry, where a part supplier sells his products to manufac- to-variable cost ratio (Wheelwright and Hayes, 1985) and need for a highly skilled labor force. Thus, other alternatives to be considered. Ensuring coordination between marketing manufacturing functions appears to be an attractive alternat the above situations. The objective, then, is to efficiently all 2002; Lee and Hong, 2003; Melchiors, 2003; Gayon et al., 2009). Such problems arise in many occasions in the context of produc- tion systems. One example is delayed product differentiation. In that case a common stock of standard items is held and customer demands are satisfied through a rapid differentiation operation. that when demand is greater than capacity, the firm should expand capacity to solve the problem. Although this could be a viable long-term option, it is not a feasible option in the short term. Furthermore capacity expansion is often precluded as an economically viable option for some firms due to the high fixed- producing a single product type to two classes of customers. The objec or start producing (production con incoming order immediately from control), or backorder it, or even control), so as to maximize profit and backlog costs. The problem of inventory rationi customers from a common stock, ha the last decades (see, e.g., Ha, 1997 random demand from to decide when to stop whether to satisfy an (inventory rationing t it (order admission sales minus inventory t is, satisfying different nsively been studied in 00; de Verikourt et al., the range of 80–85% (or less) of the available capacity. Recent empirical data, however, suggests that many firms face total expected demand much higher than that (Sridharan, 1998). Fransoo et al. (1995) present the example of a glass- containers manufacturer in The Netherlands, where the actual aggregate demand level has been greater than 95% of the avail- able capacity. In other cases, such as high-fashion apparel industry, printing shops, and low volume component manufac- turing shops, the total demand is even greater than the available capacity as described in Balakrishnan et al. (1996). One may argue We consider a single-stage stochastic manufacturing system satisfy the available capacity. Typically, total demand is assumed to be in An inventory and order admission contr two customer classes Stratos Ioannidis Department of Production Engineering & Management, Technical University of Crete, a r t i c l e i n f o Article history: Received 1 April 2009 Accepted 9 February 2011 Available online 19 February 2011 Keywords: Inventory rationing Admission control Production control Markov chains a b s t r a c t In this paper we examine a two different customer cla of inventories and backor derived and used to asses parameters. Certain prop computationally efficient results show that the pr outperforms other commo journal homepage: ww policy for production systems with 0 Chania, Greece rkovian single-stage system producing a single item to satisfy demand of . A simple threshold type heuristic policy is proposed for the joint control s. Explicit forms of the steady-state probabilities under this policy are e average profit rate of the system and determine the optimal control es of the average profit rate are established and used to develop orithms for finding the optimal control parameter values. Numerical sed policy is a very good approximation of the optimal policy and used policies. & 2011 Elsevier B.V. All rights reserved. lsevier.com/locate/ijpe n Economics produce or stop producing, accept or reject an incoming order a th only one pur (1) adm (i) (ii) (iii) S. Ioannidis / Int. J. Production Economics 131 (2011) 663–673664 backorders of both customer classes. (2) A new modeling approach is presented for the assessment of the steady state probabilities and the average profit rate of the system. (3) Apart from the analysis approach, this paper presents a number of structural properties of the profit rate function, which give rise to efficient optimization algorithms to track down the optimal values of the control parameters so as to maximize the overall profit rate of the system in equilibrium. The rest of the paper is organized as follows. In Section 2, the problem is presented and formulated as a finite state Markov chain. In Section 3 steady-state probabilities of the Markov chain are derived and used to compute the average profit rate. Struc- tural properties of the average profit rate function are presented, which are used for the construction of a computationally efficient algorithm for the assessment of the optimal control parameters. In Section 4, the proposed policy is compared numerically with the optimal policy and other commonly used backordering policies. The results indicate that the proposed policy is a very good approximation of the optimal policy and superior to the other backordering policies. 2. Problem description Consider a production facility that produces a single product. Customers arrive at random times and each customer requests one ur work makes three contributions: In this paper, we propose a simple threshold type policy for a two-class system in which the production, service, and back- ordering decisions are integrated as in Benjaafar et al. (2007). The proposed policy has four control parameters. The first parameter determines when to switch off and on the produc- tion facility depending on the current inventory, the second one forces the system to maintain an inventory rationing level for high-priority customers, and the other two parameters determine when to reject incoming orders depending on the perf O dition under which such a policy can be useful for practical poses is when it achieves a good performance relative to the ormance of the optimal policy. gen con sholds and linear dependencies. For the example given above, reshold policy would be to accept an arriving class 2 order when the total backlog (linear function of the individual s) is below a prespecified threshold. Threshold policies are in eral easier to both optimize and implement. However, the key of high (class 1) or low (class 2) priority, and fill all orders from stock or reserve stock only for high priority (class 1) customers change dynamically with the state of the system. For example, when a class 2 order arrives during a stockout period, the fewer the number of pending orders, the easier it is to make the decision to accept the new order (monotonicity). However, the threshold that separates the state space into a region of acceptance decisions and a region of rejection decisions depends monotoni- cally on the specific backlogs of class 1 and class 2 orders encountered upon the arrival of the new order. Unfortunately, such dependencies are nonlinear and do not have analytical expressions. An approach to this problem involves the use of threshold type policies, which are switching policies with fixed thre curves. Another interesting result of their work is that order admission control significantly improves the performance of the system. In a monotone switching policy, decisions as to whether unit of product. Two customer classes denoted 1 and 2 arrive excess of s. The above policy will be referred to as the dependent double base backlog policy (DDBB). DDBB is a fixed-parameter, make-to- stock policy that includes some commonly used policies as special cases like complete backordering (CB) policies that have c1¼c2¼N. The net profit rate of the system is a function of the four control parameters s, s, c1, and c2. We define the following financial parameters: ai unit profit from selling a product to customers of class i (selling price less cost of purchasing raw material and proces- sing per item), bi cost, if any, per lost sale (due to, e.g., customer loss of g (those originally pending and those that have arrived in the meantime) are filled and the system is left only with class 2 orders pending. Thereafter, the system starts to produce to stock and keeps all accepted class 2 orders pending until the inventory reaches a certain stock rationing level s, sZ0. This stock is set aside to satisfy future class 1 orders without delay. Finally, class 2 pending orders are satisfied one at a time only when the production facility produces items in tory level. An arriving class 2 order is accepted when the backlog position is less than a base backlog threshold c2 and rejected otherwise. Accepted orders of class 1 have priority and are served immediately if stock is available. Accepted class 2 orders are satisfied only after all class 1 orders are filled and a sufficient amount of inventory is built up in the system. Specifically, if pending orders of both customer classes are present and the system produces one item, then this item will fill a class 1 order. This continues until all class 1 orders ission control, and inventory rationing: Production control uses a safety stock to protect the system against stockouts but also against excessive inventories. The production facility produces to stock as long as the number of finished items is less than a certain threshold s, called the base stock, and stops when the inventory level reaches s. It is assumed that no cost or delay is incurred when the facility is switched on and off. The system avoids excessive backlogs by employing a class- dependent admission control policy. A class 1 order arriving during a stockout period is accepted if the number of class 1 pending orders is less than a base backlog c1 and rejected otherwise; if there is stock available, the order is accepted and satisfied immediately. The admission of class 2 customers depends on the backlog position, which is defined as the total number of pending orders of both classes minus the inven- according to Poisson processes with rates l1 and l2, respectively. When orders of both customer classes are pending, the system gives priority to class 1 customers. Processing times are indepen- dent, with exponential random variables with mean 1/m. Finished items are stored in an output buffer. The system incurs holding costs when products are made to stock in anticipation of future demand, backordering costs when orders are filled with delay, and, possibly, costs of lost sales for rejected customer orders. The overall system performance is defined as the long-run rate of profit from sales less the costs outlined above. Next, we describe a control policy that strikes a balance among the components of the overall performance measure while prioritizing class 1 customers. This policy is characterized by three types of decision: production control, oodwill, contractual penalties, etc.) h unit holding cost rate, which is the cost per unit time per finished item held in the buffer, and bi unit backlog cost rate of customer class i, which is the cost per unit time of delay of a pending order of class i. In steady state, the net profit rate of the system is given by G¼ a1TH1�b1ðl1�TH1Þþa2TH2�b2ðl2�TH2Þ�hH�b1B1�b2B2 where THi is the average rate of sales (throughput) to customers of class i, li�THi is the rate of lost sales, H is the average inventory, and Bi is the average backlog of class i customers. All these quantities depend on the four control parameters of DDBB. However, because the components bili of G do not depend on the control policy, instead of G we examine the following perfor- mance measure: J¼ r1TH1þr2TH2�hH�b1B1�b2B2 ð1Þ where ri¼ai+bi. For simplicity, we shall refer to J as the average finished items if n40, or the number of class 1 pending orders if probabilities can be computed from the normalization condition and the following Chapman–Kolmogorov (C–K) equations: Pðsþc1,0Þðl1þl2Þ ¼ Pðsþc1�1,0Þm ð2Þ Pðm,0Þðl1þl2þmÞ ¼ Pðmþ1,0Þðl1þl2ÞþPðm�1,0Þm, m¼ sþc1þ1,. . .,sþc1�1 ð3Þ Pðsþc1,0Þðl1þl2þmÞ ¼ Pðsþc1þ1,0Þðl1þl2Þ þPðsþc1�1,0ÞmþPðsþc1,1Þm ð4Þ Pðm,0Þðl1þl2þmÞ ¼ Pðmþ1,0Þl1þPðm�1,0Þm,m ¼maxf1,c1�c2þ1g,. . .,sþc1�1 ð5Þ Pðsþc1,kÞðl1þl2þmÞ ¼ Pðsþc1�1,kÞmþPðsþc1,k�1Þl2 þPðsþc1,kþ1Þm,k¼ 1,. . .,sþc2�1 ð6Þ Pðsþc1,sþc2Þðl1þmÞ ¼ Pðsþc1�1,sþc2ÞmþPðsþc1,sþc2�1Þl2 λ1 μ λ1 μ λ1 μ S. Ioannidis / Int. J. Production Economics 131 (2011) 663–673 665 no0. The second parameter k represents the number of class 2 pending orders. In our case the form of the Markov chain and the corresponding Chapman–Kolmogorov equations depend on the relative values of the two base backlogs. The flow diagram of this Markov chain for the case c14c2 is presented in Fig. 1. For convenience, we introduce an auxiliary variable m¼n+c1 instead of the state variable n, and compute the steady-state probabilities P(m, k). Since the system is Markovian, these n, 0 λ1 μ λ2 s, 0 σ, 0 λ1+λ2 μ μ λ2 λ1 μ λ1+λ2 μ n, k λ1 μ λ2 σ, k μ λ2 λ1 μ μ λ2 λ2 n,σ+c2 λ1 μ σ,σ+c2 λ1 μ μ λ2 profit rate. The assumption that class 1 customers have priority over class 2 customers could be based on the fact that the cost of delaying a class 1 order by one time unit is greater than the corresponding cost for a class 2 order, i.e., b1Zb2. A similar relation may also hold for the unit profit margins, that is, r1Zr2. However, custo- mer priorities could also be based on strategic or marketing decisions, regardless of the validity of these inequalities. 3. Steady state probabilities and average profit rate function assessment Our aim is to determine the four control parameters of DDBB, which maximize the average profit rate. To do that we first derive expressions for the steady-state probabilities and the average profit rate of the system operating under DDBB. The system can be described by a Markov chain with states (n, k). The parameter n is an integer that represents the inventory of Fig. 1. Markov chain of the system operat ð7Þ Pðc1�c2þk,kÞðl1þmÞ ¼ Pðc1�c2þkþ1,kÞl1þPðc1�c2þk�1,kÞm þPðc1�c2þk,k�1Þl2, k¼maxf1,c2�c1þ1g,. . .,sþc2�1 ð8Þ Pðc1�c2,0Þðl1þmÞ ¼ Pðc1�c2þ1,0Þl1þPðc1�c2�1,0Þm, if c2oc1 ð9Þ Pð0,c2�c1Þm¼ Pð1,c2�c1Þl1þPð0,c2�c1�1Þl2, if c24c1 ð10Þ Pð0,kÞðl2þmÞ ¼ Pð1,kÞl1þPð0,k�1Þl2,k¼ 1,. . .,c2�c1�1,c24c1 ð11Þ Pð0,0Þðl2þmÞ ¼ Pð1,0Þl1, if c24c1 ð12Þ Pðm,kÞðl1þmÞ ¼ Pðmþ1,kÞl1þPðm�1,kÞm, k¼maxf0,c2�c1þ2g,. . .,sþc2, m¼ 1,. . .,c1�c2þk�1 ð13Þ Pð0,kÞm¼ Pð1,kÞl1,k¼maxf0,c2�c1þ1g,. . .,sþc2 ð14Þ Pðm,kÞðl1þl2þmÞ ¼ Pðmþ1,kÞl1þPðm�1,kÞmþPðm,k�1Þl2, k¼ 1,. . .,sþc2�2, m¼maxf1,kþc1�c2þ1g,. . .,sþc1�1 ð15Þ We distinguish three distinct cases for the relative values of the backlog thresholds c1 and c2: (i) when c14c2 Eqs. (10)–(12) are not valid; (ii) when c24c1 Eq. (9) is not valid; and (iii) when c1¼c2 none of Eqs. (9)–(12) is valid. − c2, 0 λ1 μ λ1 μ λ2 k− c2,k λ1 μ −c1,σ+c2 λ1 μ −c1, k λ1 μ λ1 μ −c1, 0 λ1 μ ing under DDBB policy when c14c2. S. Ioannidis / Int. J. Production Economics 131 (2011) 663–673666 The C–K Eqs. (2)–(15) are partial difference equations in m and k. To solve them, we start from the equations for k¼0 and then we work our way up to k¼s+c2. Eq. (3) is a second-order homogeneous difference equation in m. Its general solution has the form (see, e.g., Jagerman, 2000): Pðm,0Þ ¼ K1xm1 þK2xm2 ,m¼ sþc1,. . .,sþc1 ð16Þ where x1¼r¼m/(l1+l2) and x2¼1 are the roots of the character- istic equation (l1+l2)x2�(l1+l2+m)x+m¼0. Substituting Eq. (16) into the boundary Eq. (2) yields K2¼0; thus, Eq. (16) becomes Pðm,0Þ ¼ Krm,m¼ sþc1,. . .,sþc1 ð17Þ where we have set K¼K1 for simplicity in the notation. Eq. (13) is also second-order homogeneous in m, with general solution P(m, k)¼Dk,1z1m+Dk,2z2m, where z1¼m/l1¼r1 and z2¼1 are the roots of the characteristic equation l1z2�(l1+m)z+m¼0. With the use of boundary Eq. (14) we have that Dk,2¼0. Therefore the general solution of Eq. (13) is given by Pðm,kÞ ¼Dk,1rm1 ,k¼maxf0,c2�c1þ2g,. . .,sþc2,m¼ 0,. . .,c1�c2þk ð18Þ Similarly, Eq. (5) has a general solution: Pðm,0Þ ¼ A0,1,0ym1 þA0,2,0,ym2 m¼maxf0,c1�c2g,. . .,sþc1 ð19Þ with y1 and y2 the solutions of the characteristic equation l1y2�(l1+l2+m)y+m¼0: yi ¼ ðl1þl2þmÞ7 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðl1þl2þmÞ2�4l1m q 2l1 Eqs. (17) and (19) have to agree for m¼s+c1. Also, when c14c2, Eqs. (18) and (19) for m¼c1�c2 and k¼0 have to agree with the boundary Eq. (9). Thus, we can express the constants A0,1,0, A0,2,0 and D0,1 as linear functions of K. When c1rc2, Eq. (13) is not valid for k¼0. Alternatively, (13) is valid with all its constituent terms equal to zero. Hence the probabilities P(m,k) in Eq. (18) are zero from which we deduce that D0,1¼0, and therefore we need expressions only for A0,1,0 and A0,2,0. Substituting the expression of Eq. (19) into Eq. (15) for k¼1 we obtain an inhomogeneous second-order difference equation for the probabilities P(m,1), in which P(m,0) are the inhomogeneous terms. The solution of this equation is Pðm,1Þ ¼ A1,1,0ym1 A1,2,0ym2 þA1,1,1mym1 þA1,2,1mym2 The first two terms in the expression given above are the solution of the homogeneous equation and the last two terms are the particular solution of the inhomogeneous equation. It is convenient to rewrite the expression for P(m,1) in the following form: Pðm,1Þ ¼ A1,1,0ym1 þA1,2,0ym2 þA1,1,1ðmþ1Þym1 þA1,2,1ðmþ1Þym2 ¼ X2 i ¼ 1 X1 j ¼ 0 A1,i,j mþ1 j ! ymi , m¼maxf0,c1�c2þ1g,. . .,sþc1 Next we substitute this expression into Eq. (15) for k¼2 in order to get the general expression of P(m, 2). Repeating this procedure we can work our way up for k taking values from 1 to s+c2�2. This leads to the general solution of Eq. (15), which is Pðm,kÞ ¼ X2 i ¼ 1 Xk j ¼ 0 Ak,i,j mþk j ! ymi ,k¼ 0,. . .,sþc2�2, m¼maxf0,kþc1�c2g,. . .,sþc1 ð21Þ where for k¼1,y,s+c2�2 and the constants Ak,i,j satisfy Ak,i,k ¼ Ak�1,i,k�1l2 l þl þm�2y l ,i¼ 1,2,. . . 1 2 i 1 Ak,i,j ¼ Ak,i,jþ1yil1þAk�1,i,j�1l2 l1þl2þm�2yil1 ,i¼ 1,2,j¼ 1,. . .,k�1 The remaining constants Ak,1,0, Ak,2,0, and Dk,1 can be expressed as linear functions of K with the use of the boundary equations. When k¼1 we use Eqs. (4) and (8) and the fact that P(c1�c2+1, 1) should satisfy both Eqs. (18) and (21). For k¼2,y,s+c2�2 we use Eq. (6) instead of (4), with the second variable equal to k�1. When c1oc2, Dk,1¼0 and we have to use Eq. (11) instead of (8) for koc2�c1 or Eq. (10) for k¼c2�c1 to get expressions for constants Ak,1,0 and Ak,2,0. For the special case in which k¼s+c2�1, we may use Eq. (5) for k¼s+c2�2 and Eq. (8) in order to express Ds+c2�1,1 and P(s+c1, s+c