Int. J. Production Economics

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1 Int. J. Production Economics 133 (2011) Contents lists available at ScienceDirect Int. J. Production Economics journal homepage: Optimal inventory control of empty containers in inland transportation system Won Young Yun a,n, Yu Mi Lee a, Yong Seok Choi b a Department of Industrial Engineering, Pusan National University, Busan, Republic of Korea b Department of Logistics, Sunchon National University, Jeollanam-Do, Republic of Korea article info Available online 8 July 2010 Keywords: Empty containers Inventory level (s, S) inventory policy Arena OptQuest s abstract In this paper, we deal with an inventory control problem of empty containers in an inland transportation system. In inland container transportation, freights (containers) are transported between terminal and the customer s location by trucks, trains and barges. Empty containers are an important logistic resource and shipping companies try to operate and manage empty containers efficiently. Because of the trade imbalance between hub ports, empty containers should be periodically repositioned from surplus areas to shortage areas. However, it is not easy to exactly forecast the demand of empty containers, and we therefore need to build an efficient way to reposition the empty containers. In this paper, we consider a shortage area and propose an efficient inventory policy to control empty containers. We assume that demands per unit time are independent and identically distributed random variables. To satisfy the demand of empty containers, we reposition empty containers from other hubs based on the (s, S) inventory policy, and also consider the lease of empty containers with zero lead time. For the leased containers, we should return the number of empty containers leased to the leaser after the specified period. For a given policy, simulation is used to estimate the expected cost rate and we use the optimization tool, OptQuest s in Arena to obtain the near optimal (s, S) policy in numerical examples. & 2010 Elsevier B.V. All rights reserved. 1. Introduction The transportation demand of containers is rapidly increasing nowadays and the demand for empty containers is also increasing accordingly. Because of the trade imbalance, empty containers should be repositioned between shortage and surplus areas periodically and shipping companies need to have an inventory control policy to reposition the empty containers. Shipping companies reposition empty containers between hub areas, ports and depots. Because it usually takes a long time to reposition empty containers between hub areas and an efficient management of the empty containers is an important factor that can contribute to raising the productivity of shipping companies. Crainic et al. (1993) dealt with the allocation problem of empty containers according to the dynamic and uncertainty of demand. Cheung and Chen (1998) considered how the dynamic container allocation problem can be formulated as a two-stage stochastic network model. They also studied optimization problems for repositioning empty containers and determined how many leased containers are needed at ports. Shen and Khoong (1995) proposed a network optimization model between ports and solved the problem using A Mathematical Programming Language (AMPL). Lam et al. (2007) proposed dynamic and stochastic models for a simple two-port and two-voyage problem. Li et al. (2004), (2007) proposed a new (u, d) policy for the distribution problem of empty containers between ports. In this paper, we consider a port area that needs more empty containers, known as a shortage area (for example, Busan, Korea). Suppose that we should prepare a suitable number of empty containers to satisfy the customer s seasonally fluctuating demand. To satisfy the required number of empty containers, we can either reposition empty containers from the surplus area with a long lead time or lease empty containers. Thus, we consider the ordering (repositioning) and leasing policy under probabilistic demand and supply, with high and low demand seasons for the demand of empty containers. Holding, leasing and ordering are considered and we obtain optimum inventory policies to minimize the expected cost rate (long-run average cost per unit time) by an ARENA simulation. 2. Inventory control model n Corresponding author. Tel.: ; fax: addresses: wonyun@pusan.ac.kr (W. Young Yun), ymlee@pusan.ac.kr (Y. Mi Lee), drasto@sunchon.ac.kr (Y. Seok Choi). Fig. 1 shows an inland transportation network. Shipping companies store empty containers in depots, provide the empty /$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi: /j.ijpe

2 452 W. Young Yun et al. / Int. J. Production Economics 133 (2011) Lease Company Depot Terminal Customers Full container movement Empty container movement Fig. 1. Inland transportation network. Fig. 2. Ordering and leasing policy using the (s, S) ordering policy. containers for transportation of freights between terminal and customer locations, and sometimes lease empty containers if they need more empty containers immediately. In this paper, we consider a hub area, in which more empty containers are needed because the demand of empty containers is greater than the supply. To solve the imbalance problem, we should periodically reposition empty containers from surplus areas to a shortage area. In this paper, an inventory control problem of empty containers under probabilistic demand is studied under the following assumptions. Assumptions (1) 40 ft dry containers are considered. (2) There are two seasons: low and high demand seasons. (3) Demand and supply of empty containers per week are independent and identically distributed random variables. (4) (s, S) ordering policy is used. (5) There is a type of lease with zero lead time. (6) The lead time of repositioning is constant. Notation S 1 order-up-to level at low demand season s 1 order point at low demand season S 2 order-up-to level at high demand season s 2 order point at high demand season LT lead time of repositioning D i customer s demand in period i O i order amount in period i L i lease amount in period i H i stock level in period i N i net stock at the beginning in period i I i inventory level during the lead time in period i V i return amount from customers in period i C f fixed-ordering cost C o ordering cost on each unit C l leasing cost on each unit C n inventory holding cost on each unit TC i total cost in period i

3 W. Young Yun et al. / Int. J. Production Economics 133 (2011) We consider the (s, S) ordering policy (refer Silver et al. (1998)). Fig. 2 shows the ordering and leasing policy using the (s, S) ordering policy used in this paper. We also use different values of s, S in each season (high and low demand seasons). First, we estimate an inventory level at the specified time point (in this paper, four weeks late) every week. If the estimated inventory level is less than s, we order empty containers up to S. Otherwise, we do not order empty containers. If we need empty containers, but there are no empty containers, then we lease empty containers immediately and should return the number of the leased empty containers to the leasing company after the specific period. Various cost terms are related to this inventory problem, and we consider fixed-ordering, ordering on each unit, holding and leasing. We would like to find the optimal inventory policy minimizing the expected cost rate (long-run average cost per unit Fig. 3. Flow in simulation model.

4 454 W. Young Yun et al. / Int. J. Production Economics 133 (2011) time). But it is very difficult to obtain the expected cost rate analytically, and we use simulation to obtain the expected cost rate. The procedure to obtain the expected cost rate by simulation is as follows: Step 1: initialize all variables t¼0. Step 2: check whether low- or high-demand season. Step 3: calculate the inventory holding cost at time t. Step 4: calculate the current stock level of empty containers at time t. Step 5: calculate the estimated quantity of empty containers after lead time, 4. Step 6: check whether to order or not and calculate the ordering cost at time t. Step 7: generate demand and return amount at time t. Step 8: check whether to lease or not and calculate the leasing cost at time t. Step 9: calculate the net stock and total cost at time t. Step 10: update the total values of all to time t. Step 11: if the simulation stop condition is not satisfied, t¼t+1 and go to step 2. Otherwise, go to step 12. Step 12: calculate the estimated per unit time (total divided by simulation time). Fig. 3 shows the detail simulation flow to make decisions and calculate ( refer Kelton et al. (2004)). An ARENA was used to build the simulation model. 3. Numerical study 3.1. Numerical experiment In this section, we consider numerical examples using developed simulation model. Input data of model parameters are given in Tables 1 and 2. The simulation period is 5000 (weeks), the warm-up period is 500 (weeks) and the lead time of repositioning is 4 (weeks). For given input parameters at Tables 1 and 2, we consider different values of S 1, s 1,S 2,s 2, to obtain various per unit time (holding, leasing, ordering and total per unit time) by simulation and try to find the effect of model parameters to four cost factors. In Table 3, the value of S 1 changes from 100 to 500, S 2 from 130 to 530 with an increment by 50, and for given S 1 and S 2, s 1 changes from 0 to 90 and s 2 from 0 to 100. From Table 3 and Fig. 4, we can find the following trends: Table 1 Distributions of demand and supply. Low demand season High demand season Demand Normal (200, 20 2 ) Normal (300, 30 2 ) Supply Normal (150, 20 2 ) Normal (220, 30 2 ) If s 1 and s 2 increase for given S 1 and S 2, the ordering and holding increase, but the leasing cost decreases. If S 1 and S 2 increase for given s 1 and s 2, the holding cost increases, but the leasing and ordering decrease. The total cost is the lowest when S 1 ¼100, s 1 ¼70, S 2 ¼130, s 2 ¼80. Table 3 Costs terms for different values of S 1, s 1,S 2,s 2. S 1 s 1 S 2 s 2 Holding Leasing Ordering Total , , , , , , , , , , , , ,915 93, , , ,380 76, , , ,060 68, , , , , , , , , , , , , , , ,625 78, , , ,306 63, , , ,468 48, , , , , , , , , , , ,339 89, , , ,581 69, , , ,388 52, , , ,788 40, , , , , , , , , , , ,820 80, , , ,303 63, , , ,101 46, , , ,853 36, , , , , , , ,333 97, , , ,524 74, , , ,338 57, , , ,220 39, , , ,207 31, , , , , , , ,523 90, , , ,483 69, , , ,013 51, , , ,736 37, , , ,633 29, , , , , , , ,449 84, , , ,464 62, , , ,334 46, , , ,266 34, , , ,356 26, , , ,344 90, , , ,064 78, , , ,080 58, , , ,018 43, , , ,477 33, , , ,702 23, , , ,769 81, , , ,658 72, , , ,616 53, , , ,720 39, , , ,604 29, , , ,730 20, , ,549 Table 2 Costs factors. Costs factors Inventory holding Leasing Fixed-ordering Ordering Costs 2000 (unit/week) 12,000 (unit/month) 50,000 (number of times/week) 4000 (unit/week)

5 W. Young Yun et al. / Int. J. Production Economics 133 (2011) ,000 (cost) 800, , , ,000 (S 1, S 2 ) 100, , , , , , , , , , , , , (s 1, s 2 ) 0, 01 0, 20 30, 40 50, 60 70, 80 90, 100 Fig. 4. Trends of total by S 1, s 1,S 2,s 2. Table 4 Cost terms for different values of simulation period. Simulation period (weeks) Holding cost Leasing cost Ordering cost Total cost ,693 53, , , ,980 57, , , ,948 59, , , ,502 68, , , ,976 70, , , ,996 70, , , ,982 71, , , ,123 71, , , ,027 72, , , ,907 74, , , ,980 75, , , ,352 76, , , ,711 76, , , ,533 74, , , ,206 76, , , ,446 75, , , ,270 74, , , ,504 73, , , ,291 75, , , ,983 76, , , ,538 76, , , ,380 76, , , ,227 76, , , ,797 76, , , ,467 77, , , ,912 76, , , ,448 76, , ,237 In simulation, the simulation period is 500 and the warm-up period is 50. Table 4 and Fig. 5 show the estimated cost terms for different simulation periods in case that S 1 ¼100, s 1 ¼70, S 2 ¼130, s 2 ¼80 in Table 3. We can find that all are stable at 5000 (around 100 years because the time unit is week) Near optimal inventory policies To obtain the optimal inventory policies, we should find the values of S 1,S 2,s 1 and s 2 to minimize the expected rate (the expected total cost per unit time). In this problem, because it is difficult to find the closed form of the expected rate, simulation is used to estimate the expected rate. Based on simulation results, we can find the near optimal inventory policy by using an optimization tool of Arena, OptQuest s. OptQuest s executes the simulation model, obtains the estimated value of the objective function and changes the values of decision variables iteratively until the stopping condition is satisfied. As input variables for OptQuest s, we give the upper bounds (300, 150) of S 1,s 1 and S 2,s 2 and the lower bounds (70, 30), respectively. The automatic stop rule of OptQuest s is used and the simulation period is 5000 (weeks) with 500 (weeks) warm-up period. In order to find the effect of cost parameters to the near optimal values of S 1,S 2,s 1 and s 2, we consider three cases and find near optimal solutions. Firstly, we change the leasing cost from 6000 to 18,000 and the fixed-ordering cost from 50,000 to 150,000 for given holding and ordering. Table 5 shows the near optimal values of S 1,S 2,s 1 and s 2 and we can find that if the fixed-ordering cost increases, the values of S 1 and S 2 increase, and the values of s 1 and s 2 decrease. Moreover if the leasing cost increases, all values of S 1, S 2, s 1, s 2 increase. Secondly, we change the holding cost from 1000 to 3000 and the fixed-ordering cost from 50,000 to 150,000 for given leasing and ordering. Table 6 shows that if the holding cost

6 456 W. Young Yun et al. / Int. J. Production Economics 133 (2011) , ,000 (Cost) 400, ,000 Ordering cost Leasing cost Holding cost 200, , (Simulation period) Fig. 5. Trends of cost factors by simulation period. Table 5 Optimal policies for various values of leasing and fixed-ordering. Table 8 Optimal policies for various values of demand and supply. Holding Ordering Leasing Fixedordering S 1 s 1 S 2 s 2 Total Low demand season High demand season S 1 s 1 S 2 s 2 Total , , , , , ,766 12,000 50, , , , , ,987 18,000 50, , , , , ,008 Gap between demand and supply , , , , ,170 Table 6 Optimal policies for various values of holding and fixed-ordering Leasing Ordering Holding Fixedordering S 1 s 1 S 2 s 2 Total 12, , , , , , , , , , , , , , ,933 10, , , ,591 (Cost) Total Cost 50,80 55,88 60,96 65,104 70,112 (Low, High) Fig. 6. Total for various values of demand and supply. Table 7 Optimal policies for various values of holding and leasing. Fixedordering Ordering Holding Leasing S 1 s 1 S 2 s 2 Total 50, ,048 12, ,728 20, , ,522 12, ,641 20, , ,593 12, ,933 20, ,908 increases, all values of S 1, S 2, s 1, s 2, decrease, and if the fixedordering cost increases, the values of S 1 and S 2 increase, and the values of s 1 and s 2 decrease. Thirdly, we consider different values of holding and leasing for given unit and fixed-ordering in Table 7. Table 7 shows that if the holding cost increases, the values of S 1 and S 2 decrease, but the values of s 1 and s 2 are almost unchanging, and if the leasing cost increases, all values of S 1, S 2, s 1, s 2 increase. Finally, we also consider five cases with different gap between mean values of demand and supply of two seasons. The first case with 50 and 80 means the standard case with same values of model parameters in Tables 1 and 2, because the gap between

7 W. Young Yun et al. / Int. J. Production Economics 133 (2011) mean demand and supply is 50 and 80 for two seasons. For other four cases, we make the gap larger by increasing the mean demands of two seasons. Table 8 and Fig. 6 show that if the gap between demand and supply increases in each season, then all values of S 1,s 1 and, S 2,s 2 increase and the total cost also increase. Acknowledgements This work was supported by the Grant of the Korean Ministry of Education, Science and Technology (The Regional Core Research Program/Institute of Logistics Information Technology) (The Regional Research Universities Program/Research Center for Logistics Information Technology). 4. Conclusions In this paper, we considered an inventory control problem of empty containers. For probabilistic demand of empty containers with low and high seasons in a hub area, we used the (s, S) inventory policy to order empty containers from other hub areas. While there is a lead time for repositioning, we can lease empty containers immediately. Holding, leasing and ordering are considered and the expected cost rate (long-run average cost per unit time) is an optimization criterion. Simulation is used to obtain the expected cost rate and some numerical examples are studied. For given values of model parameters, OptQuest s is used to find the near optimal inventory policy based on simulation results. For further studies, we will consider coordinate inventory problems of empty containers in multi-depot cases and in this paper, an independent and identical distribution is assumed for probabilistic demand, but more complicate stochastic processes can also be used and similar inventory problems can be studied. References Cheung, R.K., Chen, C.Y., A two-stage stochastic network model and solution methods for the dynamic empty container allocation problem. Transportation Science 32, Crainic, T.G., Gedreau, M., Dejax, P., Dynamic and stochastic models for the allocation of empty containers. Operations Research Society of America 41, Kelton, W.D., Sadowski, R.P., Sturrock, D.T., Simulation with Arena, third ed. McGraw-Hill. Li, J.A., Liu, K., Leung, Stephen C.H., Lai, K.K., Empty container management in a port with long-run average criterion. Mathematical and Computer Modeling 40, Li, J.A., Leung, Stephen C.H., Wu, Y., Liu, K., Allocation of empty containers between multi-ports. European Journal of Operational Research 182, Lam, S.W., Lee, L.H., Tang, L.C., An approximate dynamic programming approach for the empty container allocation problem. Transportation Research Part C 15, Silver, E.A., Pyke, D.F., Peterson, R., Inventory Management and Production Planning and Scheduling, third ed. John Wiley & Sons. Shen, W.S., Khoong, C.M., A DSS for empty container distribution planning. Decision Support Systems 15,