Economic production quantity model with a shifting production rate

Transcription

1 Intl. Trans. in Op. Res. 15 (28) INTERNATIONAL TRANSACTIONS IN OPERATIONAL RESEARCH Economic production quantity model with a shifting production rate Mohamed Ben-aya a, Moncer Hariga b and Syed Naveed Khursheed a a epartment of Systems Engineering, King Fahd University of Petroleum & Minerals, hahran 31261, Saudi Arabia bendaya@ccse.kfupm.edu.sa, b Engineering Systems Management Program, College of Engineering, American University of Sharjah, PO Box 26666, Sharjah, United Arab Emirates Received 1 July 26; received in revised form 25 June 27; accepted 28 September 27 Abstract Typical models for determining the economic production quantity (EPQ) assume perfecroduct quality and perfecroduction processes. eteriorating processes may affecroduction systems in several ways. They may decrease the quality of the items produced, cause production stoppage and breakdowns and/or reduce the production rate due to production process inefficiency. The purpose of this paper is to present an EPQ model that incorporates the effect of shifts in production rate on lot sizing decisions due to speed losses. The cycle starts with a certain production rate and after a random time, the production rate shifts to a lower value. A mathematical model to determine the optimal production policy under these conditions is developed and analyzed. Numerical examples are presented for illustrative purposes. Keywords: production; inventory; deteriorating process; variable production rate 1. Introduction etermining the economic production quantity (EPQ) under various conditions has been of major interest to researchers. Recently, considerable efforts have been devoted to extending the classical EPQ model to address various practical considerations. Typical models for determining the EPQ assume perfecroduct quality and perfecroduction processes. eteriorating processes affect the system in many ways (Hall, 1983; Porteus, 1986; Rosenblatt and Lee, 1986). They can cause product defects that may affecroduct quality. They can cause process failure and stoppages that may affecrocess availability. They can also be the origin for speed losses (reduction in production output) and minor stoppages that may affecrocess efficiency. The overall equipment effectiveness is measured by the product of availability, efficiency, and quality rate. The effects on quality and process availability have been addressed in recent years by many r 27 The Authors. Journal compilation r 27 International Federation of Operational Research Societies Published by Blackwell Publishing, 96 Garsington Road, Oxford, OX4 2Q, UK and 35 Main St, Malden, MA 2148, USA.

2 88 M. Ben-aya et al. / Intl. Trans. in Op. Res. 15 (28) researchers (see Ben-aya and Rahim, 21). However, the process deterioration effect on process efficiency has not been adequately addressed in the literature. The main purpose of this paper is to investigate production deterioration affecting efficiency and its effect on lot sizing decisions in the context of the EPQ problem. In particular, we assume tharoduct deterioration leads to a shift in the production rate of the production process without affecting product quality. To the best of our knowledge, this problem has not been considered before. However, several papers analyzed the determination of the optimal production rate for the EPQ problem. This problem has been of interest to many researchers either to simply consider the production rate as a decision variable or to increase production rate to account for product deterioration due to a limited shelf-life, or in the context of certain machining problems in manufacturing. Khouja and Mehrez (1994) have extended the EPQ model to cases where the production rate is a decision variable. Uniroduction cost becomes a function of the production rate. Also, the quality of the production process deteriorates with increased production rate. They have shown that for cases where increases in production rate lead to a significant deterioration in quality, the optimal production rate may be smaller than the rate that minimizes uniroduction cost and also that for cases where quality is largely independent of the production rate, the optimal production rate may be larger than the one that minimizes uniroduction cost. Eiamkanchanalai and Benerjee (1999) have developed a model for simultaneous determination of optimal run length and production rate for a single item under the assumption tharoduction coser unit is a quadratic function of production rate. Larsen (1997) has investigated the use of production rate as a decision variable and its adjustment when a change in demand is observed. Al-Fawzan and Al- Sultan (1997) have extended the EPQ model when the demand must be delivered at a fixed interval of time by considering the production rate to be controllable. Eiamkanchanalai (1995) has extended the classical economic lot scheduling problem by treating the production rate of each item as a decision variable and incorporating a linear penalty function associated with unused capacity where each item s production cost function is assumed to be strictly convex in its production rate. In industry, some products spoil if they are stored beyond their shelf-life. If the shelf-life for the product is greater than the cycle time, it does not affect the inventory model, but if the shelf-life is less than the cycle time, it affects the whole inventory system. For an extensive review, the reader is referred to Goyal and Giri (21), Raafat (1991), and Nahmias (1982). Examples of work in this area include the following papers, among many others. Silver (1989) has considered a manufacturing equipment dedicated to a family of items, under a shelf-life constraint. He has shown that, under the assumption that the change of production rate does not incur any additional cost, reducing the production rate is more effective than reducing the cycle time. Silver (199) has extended his original work (1989) by allowing the individual item production rates to be controllable variables and has shown that, under certain circumstances, significanotential cost savings can be realized by just slowing down the production rate of just one key item in the family. Whether the production rate is determined optimally according to some criteria or dictated by the technological constraints of the production process, it may not remain constant during the production run if the process deteriorates over time. In this paper, we assume that the process starts with a production rate P 1 and after a random time, the production rate shifts to a value P 2, such that P 2 op 1. We consider the case where P 2 is known a priori. This type of deterioration is observed in machining systems (turning, milling, boring, etc.). For such systems, the production r 27 The Authors. Journal compilation r 27 International Federation of Operational Research Societies

3 M. Ben-aya et al. / Intl. Trans. in Op. Res. 15 (28) rate is a function of the speed rate, which in turn depends on the tool life. Consequently, any tool failure due to wearout will disrupt the machine operations, resulting in a gradual decrease in the production efficiency. This is often the case (Iakovou et al., 1996) when the tool s operating life is terminated by multiple injuries instead of a single one. In other words, the tool is deteriorating by a gradual and cumulative wear process. Using tool wear sensoring devices, the exact working condition of the cutting tool can be determined. Based on the detected tool s wear, the decision maker can lower the cutting speed (and, therefore, lower the production rate) to prolong the tool s life. We will develop and analyze a mathematical model to determine optimal production policy (i.e. the production time). Numerical examples are presented for illustrative purposes. The remainder of this paper is organized as follows: in the next section, we define the problem and state the specific assumptions and notation that will be used in the development of the mathematical models. In Section 3, we develop the mathematical model for the general probability distribution of the time of the shift. We also analyze the model and give numerical examples for both exponential and Weibull distributions. A summary of the paper and some concluding remarks are presented in Section Problem definition, notation, and assumptions We consider the EPQ model where the production rate may shift at a random point in time to a lower production rate because of speed losses due to some sort of process deterioration. The new production rate, if it happens, is assumed to be known a priori from past experience. Lost production during the lower production rate period is assumed to incur some penalty cost due, for example, to delivery delays, reduced manpower and machine productivity, etc. Product quality is not affected by this shift in production rate. The process deterioration can be detected earlier by conducting minor inspections during the inventory uptime period and restoring the system to its perfect state. The same idea was used in several papers dealing with shifts in the process quality (Khouja, 25). However, in this paper, it is assumed that the production process is restored to the original production rate only at the beginning of the next cycle. The model with multiple inspection and restoration can be considered as an intersting avenue for future research. As is evident from Fig. 1, the cycle starts with production rate P 1 and after a random time t, with a known probability distribution, the production rate shifts to a value P 2, such that P 2 op 1 ; during the whole cycle the demand is assumed to be fixed and known. Shortages are not allowed. The problem is to determine the optimal production run in order to minimize the total expected cost comprised of setup cost, holding cost, and production costs Assumptions and notation The notation used in developing the model is as follows: P 1 5 production rate at the beginning of the cycle t 5 random time at which the process shifts from higher to lower production rate f(t) 5 the probability density function of the time to shift 5 production rate after the shift P 2 r 27 The Authors. Journal compilation r 27 International Federation of Operational Research Societies

4 9 M. Ben-aya et al. / Intl. Trans. in Op. Res. 15 (28) Fig. 1. Inventory profile with a shift in production rate at t: (a) ta[, ] and (b) ta[, 1]. 5 production time per cycle 5 demand rate h 5 carrying coser item per unit time S 5 setup cost a 5 the increase in unit machining cost due to increase in the production rate b 5 the per unit cost of running the machine independent of the production rate including labor and energy costs g 5 ratio of the overtime uniroduction cost to the original uniroduction cost before the shift C l 5 losroduction coser uniroduced TC 5 total coser cycle CL 5 total cycle time ETC ( ) 5 expected total coser unit time In developing the model, the following assumptions are made: 1. The process begins with a production rate P After time t, which is a random variable following a general distribution, the process shifts to a lower production rate P emand is deterministic and constant with P 1 XP 2 X. 4. The uniroduction cost is assumed to be convex in the production rate (the unit production cost first decreases with increasing production rate until a minimum unit production cost is reached, after which the uniroduction cost starts to increase with the production rate). We develop an EPQ model involving two production rates where the production capacity is reduced after a shift in the production rate and a penalty cost is incurred as a result. This model is derived in the next section. r 27 The Authors. Journal compilation r 27 International Federation of Operational Research Societies

5 3. Mathematical formulation and analysis M. Ben-aya et al. / Intl. Trans. in Op. Res. 15 (28) There are two possible scenarios for the occurrence of the time of the shift. Scenario 1. The shift occurs during the production run, i.e. ta[, ]. Scenario 2. The shift occurs after the production run, i.e. ta[, 1]. We derive costs per cycle for each of these two scenarios Scenario Production cost The uniroduction cost is assumed to be a convex function of the production rate P, as different production rates result in differenroduction costs. As the production rate is increased, some costs such as the labor and energy costs are spread over more units while per unit tool/die costs increase. The net result is that uniroduction costs decrease until an ideal design production rate of the machine is reached, beyond which uniroduction cost increases. The function considered here is of the form ap þ b P Production cost ¼ ap 1 þ b P 1 P 1 t þ ap 2 þ b P 2 ð tþ: P 2 ð1þ Holding cost The average holding coser cycle is obtained using the area under the inventory curve of Fig. 1 and is given by " ( ) HC ¼ h t 2 ðp 1 P 2 Þ 2 # ðp 1 P 2 Þ P 2 ðp 1 P 2 Þ þ t þ t 2 P 2 ðp 2 Þ p : ð2þ Losroduction cost In many practical cases, a penalty is incurred if the producer fails to deliver the agreed quantity in time. Also, the item may not be the only item using the production facility and is having some reserved time. The losroduction cost captures the penalty involved due to the above reasons and it represents the opportunity cost for noroducing the planned quantity, P 1. Mathematically, it can be written as P 1 ½P 1 t þ P 2 ð tþš; or Losroduction cost ¼ C l ðp 1 P 2 Þð tþ: ð3þ r 27 The Authors. Journal compilation r 27 International Federation of Operational Research Societies

6 92 M. Ben-aya et al. / Intl. Trans. in Op. Res. 15 (28) Adding a fixed setup cost, S per cycle, the total coser cycle is given by TC 1 ¼S þ ap 1 þ b P 1 t þ ap 2 þ b P 2 ð tþ P 1 P 2 " ( ) þ h t 2 ðp 1 P 2 Þ 2 ðp 1 P 2 Þ P 2 ðp 1 P 2 Þ þ t 2 # þ t 2 P 2 ðp 2 Þ p þ C l ðp 1 P 2 Þð tþ: 2 ð4þ From Fig. 1, the cycle length is given by CL 1 ð ; tþ ¼ P 2 þðp 1 P 2 Þt : ð5þ 3.2. Scenario 2 A similar approach can be used to derive the total coser cycle in case tx. In this case, there is no losroduction penalty and the total coser cycle is given by ( ) TC 2 ð Þ¼Sþ ap 2 1 þ b P 1 ðp 1 Þt 2 p þ h : ð6þ 2 and the cycle length in this case is simply CL 2 ð Þ¼ P 1 : ð7þ 3.3. Expected total coser unit of time The expected total coser cycle in general is obtained using the time of the shifrobability distribution as follows: E½TCð ÞŠ ¼ TC 1 ð ; tþf ðtþ dt þ TC 2 ð Þf ðtþ dt; where TC 1 ð ; tþ and TC 2 ð Þ are given by Equations (4) and (6), respectively. Similarly, expected cycle length is given by E½CLð ÞŠ ¼ CL 1 ð ; tþf ðtþ dt þ CL 2 ð Þf ðtþ dt; where CL 1 ð ; tþ and CL 2 ð Þ are given by Equations (5) and (7), respectively. r 27 The Authors. Journal compilation r 27 International Federation of Operational Research Societies

7 M. Ben-aya et al. / Intl. Trans. in Op. Res. 15 (28) ETC Fig. 2. Plot of expected total coser unit time. Using the renewal reward theorem, the expected coser unit time ETC( ) will be ETCð Þ¼ E½TCðÞŠ E½CLð ÞŠ : ð8þ Given the complex mathematical form of ETC( ), it is difficult to establish its convexity analytically. However, we show below that the first-order condition for ETC( ) has at least one solution. Moreover, even for simple time of the shift distributions, such as the exponential distribution, the problem is very intractable analytically. For distributions such as Weibull, it is more appropriate to use simulation-optimization techniques to determine the optimal production run time minimizing the expected coser unit time. The following proposition asserts the existence of a solution for this problem and will be helpful in justifying an appropriate solution procedure. Proposition 1. The first-order condition for ETC( ) has at least one solution. The proof is provided in the Appendix. Based on the results of several numerical examples, we observed that the expected coser unit time function for the exponential distribution behaves as shown in Fig. 2 and has a unique solution Exponential time of the shift We will take an example of exponential time of the shift, for which the probability density function is given by f ðtþ ¼le lt. In this special case, the total expected coser unit time is given by r 27 The Authors. Journal compilation r 27 International Federation of Operational Research Societies

8 94 M. Ben-aya et al. / Intl. Trans. in Op. Res. 15 (28) " ( ) ETCð Þ¼ S þ C 2 P 2 þ C 1 ðp 1 P 2 Þ þ h P 2ðP 2 Þt 2 p 2 ð1 e l Þ þ C 1 P 1 C 2 P 2 C 1 ðp 1 P 2 Þþ hp 2ðP 1 P 2 Þ 1 e ltp l l e l ( ) þ C 1 P 1 þ hp 1ðP 1 Þt 2 p 2 e l n o8 h ðp 1 P 2 Þ 2 ðp 1 P 2 Þ < e ltp 2 2 þ 2l þ l þ 2 : 2 l l 2 2 e l 4 P 1 þ 1 l e ltp l 2 t 2 p e lt p ðp 1 P 2 Þ þ 1 3 e l tp P 2 5: 93 = 5 ; ð9þ where C 1 ¼ ap 1 þ b P 1 and C 2 ¼ ap 2 þ b P Numerical results A numerical example along with sensitivity analysis will be presented to illustrate the developed model. The golden section search method is used to solve the problem. The data used for this numerical example are P , P , 5 2, S 5 37, h 5 2, a 5.2, b 5 15, C l 5 2 and l 5 1. The solution obtained for this parameter setting is 5.278, with a total expected cost per unit time equal to To observe the effect of important model parameters on the solution, we changed the parameter of the shift distribution l from.1 to 2. The results are summarized in Fig. 3. As we increase l, the mean time of the shift decreases and less time is available for producing at the rate P 1. Because the per uniroduction cost is lower for P 1 than P 2, so logically, the time of production should decrease to decrease the overall cost. Also, as l is increased further, the portion (a) ETC ( ) 6 5 (b) λ λ ETC ( ) Fig. 3. Effect of l on (a) ETC and (b) r 27 The Authors. Journal compilation r 27 International Federation of Operational Research Societies

9 Table 1 Effect of changing P 1 /P 2 on the solution M. Ben-aya et al. / Intl. Trans. in Op. Res. 15 (28) P 1 /P 2 t p ETC Setup cost Production cost Holding cost Losroduction cost Table 2 Effect of changing P 2 / on the solution P 2 / t p ETC Setup cost Production cost Holding cost Losroduction cost of time during which P 1 is available becomes even smaller, in order to meet the demand the production time starts to increase. The increase of l will lead to an increase in the expected total coser unit time because of higher uniroduction cost of P 2. The effects of changing P 2 and on the solution are summarized in Tables 1 and 2, respectively. Increasing P 2 decreases the uniroduction cost because the uniroduction cost is convex in P and is minimum at P This leads to a reduction in the expected total coser unit time and higher production times, as expected. As we increase the demand keeping other parameters fixed, the cycle length would increase to satisfy the demand. As the cycle length increases, the time the inventory has to be held increases, thereby increasing the cost Weibull time of the shift Simulation has been used extensively to evaluate the performance of complex systems such as manufacturing and supply chain systems. However, besides the answers to the what if questions that simulations can provide, responses to the what is questions related to the best operating policy are more practical and needed for these complex systems. The answers to these questions determine the best values for the decision variables of the system that maximize or minimize a single or multiple performance measures. Recently, simulation practitioners tried to find answers to these kinds of questions through the use of simulation optimization, which is defined in the literature as the process of searching for the best decision variable values from among all possibilities withouerforming a complete evaluation search. Swisher et al. (2) presented a survey of the search techniques used in simulation optimization. Carson and Maria (1997) reviewed the methods used and the application developed in this growing area of simulation. r 27 The Authors. Journal compilation r 27 International Federation of Operational Research Societies

10 96 M. Ben-aya et al. / Intl. Trans. in Op. Res. 15 (28) Evolver Algorithm: Generate t s until a stopping criterion is satisfied. Monte Carlo Simulation: Replicate t and average the coser unit of time over multiple replications Mean of the coser unit of time Fig. 4. Relationship between Evolver algorithm and Monte Carlo simulation. Fig. 5. Excel output for the exponential case. uring the last decade, several optimization procedures have been incorporated into commercial simulation packages. Examples of such optimization software include ProModel, AutoModel, Micto Saint, LayOPT, FactoryOPT, and Risk Optimizer. A brief description of these softwares optimization modules can be found in Carson and Maria (1997). In this paper, we will use Risk Optimizer from Palisade to minimize the expected total coser unit of time given by (8). RISK Optimizer is a simulation optimization add-in for Microsoft Excel that combines Monte Carlo simulation and a genetic optimization algorithm (Evolver). The relationship between these two modules used in Risk Optimizer is shown in Fig. 4. Before using Risk Optimizer to solve the problem with a Weibull distribution, we first validate the model for the case when the time of the shift is exponentially distributed for which an analytical solution is available. Figure 5 shows an excel spreadsheet showing the input data as well as the output for the above illustrative example. Observe also that Risk Optimizer output is very close to the analytical solution with regard to both average coser unit of time and run time. Table 3 shows the results of the comparison between the analytical Risk Optimizer solutions for different values of l. Table 3 shows that the simulation optimization model provides accurate measures for the coserformance of the inventory system. In fact, the simulation results exhibited little deviation from the analytical r 27 The Authors. Journal compilation r 27 International Federation of Operational Research Societies

11 Table 3 Validation of Risk Optimizer s solution M. Ben-aya et al. / Intl. Trans. in Op. Res. 15 (28) l Analytical solution Risk Optimizer solution ETC ETC Fig. 6. Excel output for the Weibull case. results for the different values of l. When the time of the shift for the production process follows a Weibull distribution, an analytical solution to the problem minimizing the expected coser unit of time given by (8) is extremely difficult to obtain. Therefore, for such a situation, a simulation optimization procedure can be used to generate a least-cosroduction run time. Using the same data as in the illustrative example with the parameters of the Weibull given as c (shape parameter) and b (scale parameter) 5.1, we built the optimization problem in a Microsoft Excel spreadsheet and used Risk Optimizer to search for the optimal production run time. The results for this example are shown in Fig. 6. It can be observed from Fig. 6 that running the production process for a period of.2856 units of time will result in a least expected total coser unit of time of We also solved the same model for different values of the shape and scale parameters. The result of this experimental work is reported in Table 4. From Table 4, it can be noticed that for a given shape parameter, the expected total increases as the scale parameter increases. Moreover, for a given scale parameter, the shape parameter does not have a significant effect on the expected total coser unit of time. r 27 The Authors. Journal compilation r 27 International Federation of Operational Research Societies

12 98 M. Ben-aya et al. / Intl. Trans. in Op. Res. 15 (28) Table 4 Effect of the shape and scale parameters on the problem s output c b Mean m ETC Conclusions eteriorating processes affecroduction systems in several ways. They can cause product defects affecting product quality or cause process failure and stoppages affecting system availability. They can also cause speed losses and minor stoppages affecting process efficiency. Overall equipment effectiveness is defined as the product of availability, efficiency, and quality rate. The effects on quality and process availability have been addressed in recent years by many researchers. However, the process deterioration effect on process efficiency has not been adequately addressed in the literature. In this paper, we presented a mathematical model dealing with process deterioration affecting efficiency and its effect on lot sizing decisions in the context of the EPQ problem. In particular, we assume tharocess deterioration leads to a shift in the production rate of the production process without affecting product quality. It is assumed that the production cycle starts with a certain production rate and after a random time, the production rates shift to a lower value. A mathematical model designed to determine optimal production policy under these conditions was developed and analyzed. Numerical examples were presented for illustrative purposes. Models involving shifts to a random production rate are under investigation. We are also investigating processes where process deterioration affects production rate and product quality simultaneously. Acknowledgment The authors are grateful to an anonymous referee for his constructive comments and would like to acknowledge the support of King Fahd University of Petroleum and Minerals. References Al-Fawzan, M.A., Al-Sultan, K.S., Economic production quantity for a manufacturing system with a controllable production rate. Production Planning and Control 8, r 27 The Authors. Journal compilation r 27 International Federation of Operational Research Societies

13 M. Ben-aya et al. / Intl. Trans. in Op. Res. 15 (28) Ben-aya, M., Rahim, M.A., 21. In Rahim, M.A., Ben-aya, M. (eds) Integrated Modeling in Production Planning, Inventory, Quality and Maintenance. Kluwer Academic Publishers, Boston, pp Carson, Y., Maria, A., Simulation optimization: methods and applications. Proceeding of the 1997 Winter Simulation Conference, pp Eiamkanchanalai, S., The Economic Lot Scheduling Problem with variable production rates. Ph.. thesis, rexel University. Eiamkanchanalai, S., Benerjee, A., Production lot sizing with variable production rate and explicit idle capacity cost. International Journal of Production Economics 59, Goyal, S.K., Giri, B.C., 21. Recent trends in modeling of deteriorating inventory. European Journal of Operational Research 134, Hall, R., Zero Inventories. ow Jones, Homewood, IL. Iakovou, E., Ip, C.M., Koulomas, C., Optimal solutions for the machining economics problem with stochastically distributed tool lives. European Journal of Operational Research 92, Khouja, M., 25. The use of minor setups within production cycles to improve product quality. International Transactions in Operational Research 12, Khouja, M., Mehrez, A., Economic production lot size model with variable production rate and imperfect quality. Journal of the Operational Research Society 45, Larsen, C., Using a variable production rate as a response mechanism in the economic production lot size model. Journal of the Operational Research Society 48, Nahmias, S., Perishable inventory theory: A Review. Operations Research 3, Porteus, E.L., Optimal lot sizing, process quality improvement, and setup cost reduction. Operations Research 34, Raafat, F., Survey of literature on continuous deteriorating inventory models. Journal of the Operational Research Society 42, Rosenblatt, M.J., Lee, H.L., Economic production cycles with imperfecroduction processes. IIE Transactions 17, Silver, E.A., Shelf life considerations in a family production context. International Journal of Production Research 27, Silver, E.A., 199. eliberately slowing down output in a family production context. International Journal of Production Research 28, Swisher, J., Hyden, P.., Jacobson, S.H., Schruben, L.W., 2. A survey of simulation optimization techniques and procedures. Proceeding of the 2 Winter Simulation Conference, pp r 27 The Authors. Journal compilation r 27 International Federation of Operational Research Societies

14 1 Appendix M. Ben-aya et al. / Intl. Trans. in Op. Res. 15 (28) Let E½TCð ; E½CLð : Note ETCð Þ¼ E½CLðÞŠE½TCð ÞŠ E½TCð ÞŠE½CLð E½CLð ÞŠ 2 ; E½CLð ÞŠ ¼½CL 1 ð ; Þ CL 2 ð ÞŠ f ð Þþ CL 1 ð ; tþf ðtþ dt þ E½TCð ÞŠ ¼½TC 1 ð ; ÞŠ TC 2 ð ÞŠ f ð Þþ TC 1 ð ; tþf ðtþ dt þ Note that TC 1 ð ; Þ¼SþðaP 2 1 þ bþ þ h P 1ðP 2 Þ t 2 p 2 ¼ TC 2ð Þ; and CL 1 ð Þ¼ P 1 ¼ CL 2 ð Þ: Let cð Þ¼E½CLð ÞŠE½TCð ÞŠ E½TCð ÞŠE½CLð ÞŠ ; then CL 2 ð Þf ðtþ dt; TC 2 ð Þf ðtþ dt: cð Þ¼ " # CL 1 ð ; tþf ðtþ dt þ CL 2 ð Þf ðtþ dt " tp Z # 1 TC 1 ð ; Þ TC 2 ð Þ f ðtp ÞþZ TC 1 ð ; tþf ðtþ dt þ TC 2 ð Þf ðtþ dt " # TC 1 ð ; tþf ðtþ dt þ TC 2 ð Þf ðtþ dt " tp Z # 1 TC 1 ð ; Þ TC 2 ð Þ f ðtp ÞþZ TC 1 ð ; tþf ðtþ dt þ TC 2 ð Þf ðtþ dt : r 27 The Authors. Journal compilation r 27 International Federation of Operational Research Societies

15 Now let us look at the limit of c( )as!. E½CLðÞŠ ¼ E½TCðÞŠ ¼ CL 2 ðþf ðtþ dt ¼ ; TC 2 ðþf ðtþ dt ¼ E½CLðÞŠ ¼½CL 1 ð; ÞCL 2 ðþš þ Hence, ¼ P 1 P 1 f ðþþ lim cðþ¼ SP 1! : For the limit of c( )as!1, we have lim cðþ¼ lim!1!1 M. Ben-aya et al. / Intl. Trans. in Op. Res. 15 (28) Sf ðtþ dt ¼ S; P 1 f ðtþ dt; P 1 f ðtþ dt ¼ P 1 : Z P tp 2 f ðtþ dt þ P Z 1 P tp 2 tf ðtþ dt ap 2 1 þ b TC 1 ð ; tþf ðtþ dt þ C1 ðp 1 P 2 Þþh P 2ðP 1 P 2 Þ P 2 f ðtþ dtg: Retaining only higher order terms, i.e. those involving t 2 p, we have lim cðt P 2 pþ lim!1!1 h P 2ðP 2 Þ h P 2ðP 2 Þ t 2 P 2 p 2 ; ¼ lim h P2 2 ðp 2 Þ!1 2 2 t 2 p ¼þ1: There is at least one for which the derivative is equal to zero. t þ P 2ðP 2 Þ f ðtþ dt r 27 The Authors. Journal compilation r 27 International Federation of Operational Research Societies