An Integrated Three-Tier Logistics Model

Transcription

1 An Integrated Three-Tier Logistics Model Sarawoot Chittratanawat and James S. Noble Department of Industrial and Manufacturing Systems University of Missouri Columbia, MO Abstract A three-tier logistics/production model is developed that addresses the integration of suppliers, manufacturers and distribution. The model can be applied at either micro (within facility) or macro (between facilities) levels. The multi-echelon problem is formulated to minimize overall inventory and setup cost. A simulated annealing based heuristic is developed, then experimentation is performed to determine operational guidelines and to compare performance versus a traditional EO/EP based approach. Keywords Logistics, Multi-echelon Inventory, Simulated Annealing 1. Introduction Many times logistics systems are treated as combinations of isolated functions. However, in today s competitive environment logistics has become a major component for success and an isolated approach will not give the desired results. In this paper an integrated 3-tier approach is presented that optimizes inventory cost throughout the logistics/production system. The "Three-Tier Integrated Logistics System" is formally defined as follows. 3-Tier Integrated Logistics System Three acyclically connected objects, where each object has its own unique function. A warehouse object orders material from outside the 3-Tier system and supplies material to the operation object. The operation object produces a product using material and processing operations. The product from the operation object is transported to the shipping object. The shipping object ships the product out of the 3-Tier system. The three-tier model is equally applicable at either the micro or macro levels of a production system. At the micro level it models warehouse, operation, and storage (shipping) functions. For multi-operation problems, groups of operation objects can be used to estimate the throughput for the operation group. At the macro level it models venders (suppliers), manufacturer and distributor/retailers. The dependent inventory problem is usually referred to as either the multi-level, multi-echelon or multi-stage inventory problem. In a dependent inventory environment, material flows from one stage to another causing the dependency of inventory between stages. The problem of multi-echelon inventory originated as stock-control at the warehouse (Clark and Scarf, 1960). Hence, the majority of multi-echelon research is based on inventory distribution using the Economic Order uantity (EO) model. Most research in "multi-stage production-inventory systems" has been for serial line production with a single item (Taha and Skeith 1970, Vickery and Markland 1986, Chen 1998). The multi-stage inventory problem reveals the importance of integration since "moves" connect one stage to another. Bozer and Kim (1997) studied the impact of transportation issues on inventory in order to determine the batch size in a trip-based system. Choi and Noble (2000a, 2000b) developed a comprehensive model to integrate the inventory problem with the material flow system. u et al. (1999) developed an integrated approach to solve the inventory and transportation problems. A dependent demand inventory system must integrate transportation since dependent demand involves moves from one-stage to another. As noted by Sarmineto and Nagi (2000), the production system must be explicitly and simultaneously considered since it accepts and processes raw material from a predecessor to create finished goods for a successor). However, past research has attempted to solve large-scale inventory network problems or the integration of inventory with other production domains via traditional inventory models which are by definition limited. In the following section, the 3TIPS formulation is developed to address the limitations.

2 2. Three-Tier Integrated Logistics/Production System Formulation The 3-tier production system consists of 3 facilities with the key components defined as follows. Material flow - The warehouse facility, W, supplies "raw material" to an operation facility, O. The operation facility accepts the raw material, processes it, and sends it to a shipping facility, S. Inventory cost parameters - There are two inventory cost parameters, fixed cost and holding cost. The inventory fixed costs at the warehouse, operation and shipping departments are s W, s O and s S, respectively; and they are defined as per order, setup, or shipping, respectively. The inventory holding cost parameters at warehouse, operation and shipping are h W, h O, and h S, respectively; and they are defined as per item per hour. Production cycle - The production rate of the operation facility is in "items/hour", denoted by P. The production cycle is defined by a lot size of items/lot. Annual demand - The 3-tier system must fill a demand of D items over the planning time T. Unit load size movement - The movement of materials/products between facilities is based on a unit load size, u, which is assumed to be at material handling equipment capacity and can be different between different facilities. Specification of material handling equipment - Material moves are made using one or more units of material handling equipment having a velocity of v feet per hour. Distance between facilities - Move distances are calculated between the centroid coordinates of facilities. Fixed transfer time interval - The schedule between moves is based on a fixed transfer time interval". However, the time interval from the warehouse to operation (t WO ) and from operation to shipping (t OS ) facilities can be different. Order cycle and warehouse storage - The warehouse facility receives material in any quantity less than the warehouse storage capacity, W W ; and continuously supplies material to operation facilities to avoid starving machines. The order cycle of the warehouse facility is a function of the production cycle, waiting time between cycles, waiting time for demand allocation, and waiting time during echelon utilization. Storage for operation/shipping facilities - The operation facility accepts material up to its storage capacity (W O ) from the warehouse. The shipping facility accepts product (up to its capacity, W S ) from the operation facility. The shipping schedule is a function of the production cycle, N S. Figure 1 illustrates the interaction between the different inventory levels at the warehouse, operation and shipping facilities. The shaded areas represent the echelon inventory for the operation facility. Warehouse inventory Operation inventory t Shipping inventory t Figure 1 Interaction of inventory levels in a 3-tier production/distribution system t Objective function - The objective function minimizes the total cost associated with inventory in the system (both fixed cost and variable holding cost). Inventory cost at an operation facility is based on both echelon ("beforeprocess" material) and production ("after-process" material) inventory. Inventory cost at the warehouse facility is based on the number of cycles per order, N W. Inventory cost at the shipping facility consists of both shipping and holding cost. In order to allocate product throughout the planning horizon, the production plan must be balanced. Therefore, the production facility is assumed to produce one cycle and wait until the product is needed.

3 Problem constraints - Time intervals (t WO and t OS ) are constrained so that the time interval must be greater than or equal to the handling time of a move. In addition to the handling time restrictions, the time interval must satisfy the production rate for one shipment. From warehouse to operation, one shipment must be greater than or equal to the units produced in one time interval. When considering the transfer from an operation to the shipping department (O S), the shipment (un OS ) cannot be made unless the operation department produces a sufficient amount of product (Pt OS ). The time interval from warehouse to operation provides a feasible range, while from the operation to shipping department it provides a lower bound on the time interval. However, for a multidepartment problem, the production rate of a successor operation will be the upper bound for the transfer time interval and a loose upper bound can be established for the feasibility of the one department problem. Available storage space at warehouse, shipping and operation facilities restricts the production lot size and the number of cycles per order/shipment with respect to the unit volume. The storage space constraint for both the warehouse and shipping is a function of production lot size and number of cycles per order. The storage space required at an operation facility is not dependent on a single unit, since the operation facility has both echelon and production inventory. The maximum echelon inventory is the production lot size less "items used during the arrival period. The maximum production inventory is the production lot size less "units shipped during the production period". It is difficult to model the combination of both echelon and production inventory simultaneously since they occur at the same time, but with different time intervals. This model constrains the operation inventory by using the maximum echelon and production inventory levels. Transfer constraints are imposed to limit the number of items moved so that it is less than the production lot size. Finally, there are constraints that account for the fact that the number of production cycles per order/shipment limits the number of order/shipping cycles. 3. approach The formulation contains several discrete variables (N W, N S, n WO, and n OS ) which make it non-differentiable. Therefore, in this research an iterative solution approach based on a neighborhood search (improvement algorithm) is used. In order to have an efficient neighborhood search, a set of discrete variables is selected and then optimized based on a continuous, unconstrained formulation. The unconstrained, continuous cost function was analyzed for its convexity and then a lot size function was developed. Then the impact on the cost function of each continuous variable was examined and finally a strategy to optimize the discrete variables was determined. 3.1 Simulated Annealing algorithm A Simulated Annealing (SA) (Dowsland 1993) approach allows for the generation of quality solutions for the problem even though the problem contains a range of complicating issues. Simulated Annealing was adopted it has minimal memory requirements and is a theoretically proven convergent algorithm. Even though the solution quality is not proved in this research, the results of pilot-runs show convergence in reasonable time. Initial condition and solution The initial solution (s 0 ) is randomly selected from the neighborhood. Only feasible solutions are selected. The initial temperature (t 0 ) is set to be very high (= 1). Neighborhood sampling: N = {N W, N S, n WO, n OS } A combination of discrete variables is considered as a neighborhood for the algorithm. Each discrete variable is sampled from a variable set based on the problem constraints. Evaluation functions The time interval functions, lot size function, test of feasibility test function and cost function are used as evaluation functions. The time interval functions are derived from the problem constraints. Cooling schedule (nrep, α) The cooling schedule consists of a single nrep and α(t) = t/(1+βt); β is a suitable small value (0.01). Stopping condition Deterministic stopping (based on a given number of iterations, i.e. 200,000) is used. The SA-based neighborhood search algorithm has initial solutions evaluated by the cost function and a time function that is used to solve the time interval for each iteration. The cooling routine solves the lot size and evaluates

4 feasibility. The algorithm stops once it reaches the maximum number of iterations. The SA algorithm is implemented in C++. The code is compiled and run on Pentium II 400 Mhz with Microsoft Windows The running time is approximately 10 seconds for each problem. 4. Example problem and experimentation The example problem consists of three-tiers for single product: warehouse, operation, and shipping (figure 2). Warehouse (W) storage space for warehouse WW two nw wwo Operation (O) storage space for operation WO tos nos wos Shipping (S) storage space for shipping WS Figure 2 Example problem illustration Variation of fixed cost (F0X) - Nine different combinations of fixed cost were explored, where the fixed cost at the warehouse is an order cost; the fixed cost at shipping is the shipping cost, and the fixed cost at the operation department is the setup cost. Variation of the fixed cost resulted in a slight impact on total cost and consistently resulted in higher fixed cost giving higher total cost in all cases, except one that had the lowest fixed cost at warehouse and shipping, but the highest total cost due to high fixed and variable inventory cost at operations. Variation of unit load size (U0X) - To test the sensitivity and impact of partial capacity shipment, the unit load size (ULS) was varied from 10 to 90 items/trip. The ULS was positively correlated with total cost, i.e., the higher ULS, the more expensive. This result supports the desirability of using a small ULS. However, the 3TIPS model does not consider the capital cost of Material Handling Equipment (MHE), which increases since the number of MHE increases as the ULS decreases to support the same move quantity. Therefore, MHE integration is necessary to optimize the overall cost. Variation of equipment velocity (V0X) - The equipment velocity was varied from 20% (720 ft/hr) to 500% (18,000 ft/hr) of the base problem (3,600 ft/hr). The results from varying the velocity were logical, i.e., faster speeds reduce total inventory cost. In practice, faster equipment implies better equipment, and thus, higher cost. Variation of storage space (W0X)- The storage space experiment was conducted based on a combination of department type. Nine combinations were explored. The lowest cost solution was for the case (W07) where the storage at the warehouse greater than operation, but less than the shipping facility. Large warehouse and shipping space resulted in less cost than having large operation space (W01, W02, W03). Using the same space for all facilities resulted in average cost (W05). Variation of operation facility location (Z0X) - The experiment considered up to 13 possible locations using 5 unique zones. The results show that the closer operation facilities are to the shipping facility the better the solution (less inventory cost). Looking at the cost components, a large cost savings is due to production inventory holding cost both during- and non- production. Since the value of material is higher after production (increasing inventory holding cost), it is desirable to move the operation facility closer to shipping. Summary of results - An analysis of the results reveals that a significant component of inventory cost is not only due to inventory cost parameters, but also structural factors, such as unit load size, equipment speed, space, and location. The following are a few general guidelines for integrated system design that were found:

5 1. The impact of fixed inventory cost is less than that of the structural factors. 2. Using small unit load size with highly frequent moves (or high number of equipment) gives better solutions than large using large unit load size and less frequent moves (or less number of equipment). 3. Faster equipment reduces total inventory cost and overall total cost. 4. Large warehouse and shipping facilities provide better solutions than having a large operation facility. 5. It is desirable to have operation facilities closer to successor facilities rather than having the operation facility closes to predecessor facilities. 4.1 Traditional inventory system experiment In order to explore the quality of 3TIPS solutions, an experiment was designed to compare 3TIPS to a traditional inventory approach. The traditional inventory system approach was defined as that which uses either the Economic Order uantity (EO) and/or Economic Production uantity (EP). EO and EP address single item, single facility problem. So for multi-item or multi-facility problems, additional assumptions or techniques are required to avoid overlapping cycle time solutions (Silver et al., 1998). To overcome the overlapping cycle time problem, the Common Cycle assumption (CC) is used and the formulation and lot size function were revised accordingly. The solution to all cases based on this formulation are given in table 1. Due to the traditional EO/EP formulation, the solutions are identical if the relevant problem parameters (cases F0x) are unchanged. Overall, the production lot size was found to quite low, 426 items/lot, resulting in a high number of production cycles and thus higher total cost. Since the production lot size is low, the solution is storage space feasible. Problem Traditional Total Table 1 Traditional solution vs. 3TIPS solution 3TIPS Total Traditional % Imp Problem Total 3TIPS Total % Imp F ,726 1,500 12, % V ,803 1,222 22, % F ,699 1,413 13, % V ,803 1,222 18, % F ,008 1,323 12, % V ,803 1,209 11, % F ,008 1,320 13, % V ,803 1,225 13, % F ,803 1,224 12, % V ,803 1,225 13, % F ,651 1,118 12, % V ,803 1,216 10, % F ,008 1,320 11, % V ,803 1,225 12, % F ,651 1,103 9, % W , , % F , , % W , , % U ,803 1,225 10, % W , , % U ,803 1,222 10, % W , , % U ,803 1,225 10, % W , , % U ,803 1,225 13, % W , , % U ,803 1,221 13, % W ,803 1,216 10, % U ,803 1,225 12, % W ,803 1,189 17, % U ,803 1,225 10, % W , , % U ,803 1,225 15, % Z ,803 1,225 12, % U ,803 1,216 18, % Z ,803 1,225 12, % V ,803 1,196 18, % Z ,803 1,224 12, % V ,803 1,223 10, % Z ,803 1,224 10, % Z ,803 1,224 10, % Overall Average 43.7% Overall Max 60.5% Overall Min 2.7% Table 1 also shows a cost comparison between the traditional inventory system (EO/EP with CC assumption) and the integrated 3-tier approach. In general, the traditional inventory system with a CC assumption results in an

6 inferior solution compared to the 3-tier integrated approach (CC gives higher total inventory cost). Even though the CC assumption is an upper bound for any inventory system without overlapping, the solution for the 3-tier integrated approach is a non-overlapping solution as well. The 3-Tier solution is on average 43% better than the traditional approach, with a best case of 60% (cases F01, V08, U01, U02, U07, F07, F08 and W07). The 3-Tier approach results in better solutions since it does not continuously hold WIP at the operation facility (as assumed by traditional approach). Moreover, the 3-Tier approach separately optimizes each facility, instead of using a common cycle for all facilities. Therefore, the 3-Tier solution is very close to an Independent (IS) approach, which is the lower bound to the multi-facilities inventory problem. In thirty-eight of forty-one cases (92.68% of the cases), the 3-Tier approach eliminated the inventory holding cost due to waiting by using a one-to-one ordering (or shipping) strategy (one order per production lot). The three cases with more than one production cycle per order (or shipping) showed significantly higher total cost due to waiting time (cases W02, W03 and W04). 5. Summary In this paper a 3-Tier production system has been modeled and analyzed. s for the 3-Tier problem were obtained using a simulated annealing based approach. The solutions revealed that there is a significant inventory cost impact not only due to inventory cost parameters, but also structural factors, i.e., unit load size, equipment speed, space, and location. The results of the 3-Tier approach were then compared to a traditional inventory system using the CC assumption and the 3-Tier solutions were better than those for the traditional approach. Overall, this paper motivates the integration between operational (inventory) and structural factors issues, since the integration of more variables was shown to improve overall system performance. References Bozer, Y.A. and J. Kim, 1996, Determining Transfer Batch Sizes in Trip-based Material Handling Systems, International Journal of Flexible Manufacturing Systems, 8(4), Chen, F., Echelon reorder points, installation reorder points, and the value of centralized demand information, Management Science, 44(12), S221-S234. Choi, S. and J. S. Noble, 2000a. Determination of economic order quantities (EO) in an integrated material flow system, International Journal of Production Research, 38(14), Choi, S. and J. S. Noble, 2000b. An integrated material flow system approach for determining the economic production quantity (EP), International Journal of Production Research, 38(15), Clark, A.J. and H. Scarf, Optimal policies for a multi-echelon inventory problem, Management Science, 6(4), Dowsland, K.A., Simulated Annealing, Modern Heuristic Techniques for Combinatorial Problems, John Wiley & Sons, pp u, W.W., J. H. Bookbinder and P. Iyogun, An integrated inventory-transportation system with modified periodic policy for multiple products, European Journal of Operational Research, 115, Sarmiento, A. and R. Nagi, A review of integrated analysis of production-distribution systems, IIE Transactions, 31, Silver, E.A., D.F. Pyke and R. Peterson, Inventory Management and Production Planning and Scheduling - 3rd edition, John Wiley & Sons, New York. Taha, H.A. and R.W. Skeith, The economic lot sizes in multistage production systems, AIIE Transaction, 2(2), Vickery, S. K. and R. E. Markland, Multi-stage lot sizing in a serial production system, International Journal of Production Research, 24(3),