Inventory Positioning in Supply Chain Network: A Service-Oriented Approach

This study develops a mixed-integer linear programming model based on a guaranteed service approach for an inventory positioning problem in a supply chain under the base stock inventory policy. Our proposed model aims to determine appropriate inventory positions and amounts and the optimal service level for the supply chain to minimize the total cost of safety inventory holding and shortage. Two demand scenarios, based on normal and empirical distributions, are investigated. An extensive numerical experiment is conducted to illustrate the applicability and effectiveness of our model, especially under empirical distribution. The experiment features a practical network structure and demand data from an industrial user. Moreover, to further validate the experimental results from the mathematical model, they are compared with the result from a simulation model, which is constructed to imitate the operations of the supply chain. The comparison result indicates that the model solution under the empirical demand distribution is close to the simulation regarding the difference in the total cost (less than 1%). This solution significantly outperforms the model solution under the normal demand, which results in a significant difference in total cost (more than 25%) compared to the simulation.

cial flows [1]. A supply chain's performance is influenced that the total inventory value in 2020 is around 46.5%, con-30 stituting the most significant portion of the total logistics cost 31 The associate editor coordinating the review of this manuscript and approving it for publication was Nikhil Padhi . structure [2]. Hence, optimizing inventories across the supply 32 chain is motivated by economic reasons. To substantially 33 reduce inventory cost in a supply chain, inventories of all 34 supply chain members are jointly, rather than separately, 35 considered as a target to improve. A supply chain usually 36 experiences demand uncertainty propagated downstream to 37 upstream. This uncertainty and operational constraints may 38 result in inventory shortages at many locations. These short-39 ages subsequently affect the supply chain's performance. 40 Therefore, keeping safety stocks at suitable locations is a 41 countermeasure that mitigates the impact of demand variabil-42 ity and maintains a desired customer service level [3], [4]. 43 Identifying the suitable locations and levels of safety stocks 44 of different materials (parts, components, semi-finished, and 45 finished goods) throughout the supply chain is essential and a 46 challenge for both supply chain practitioners and researchers. 47 This paper involves the problem of choosing the locations 48 and amount of safety stock at each location in a supply 49 Li and Chen [22] and Li et al. [23] are also explored by Chen 106 and Li [24] under various operating flexibilities. Despite its 107 popularity in research, the (R, Q) policy is not widely applied 108 in practice. The most common policy to handle inventory 109 systems like warehouses and distribution centers is the base 110 stock policy [25]. This policy is often implemented for a peri-111 odic review system, where ordering costs can be minimized 112 when orders are arranged and consolidated [26]. Therefore, 113 a majority of researchers consider the base stock policy in 114 their safety stock placement problems [11], [15], [16], [18], 115 [19], [20], [27], [28]. 116 In addition to inventory policy, another important assump-117 tion in the safety stock placement problem is the under-118 lying assumption of the distribution for customer demand. 119 To simplify the problem characteristic, most studies assume 120 that the demand follows either a theoretical distribution, 121 such as Normal [11], [14], [16], [18], [19], [20], [25], 122 [27], [28], Poisson distribution [22], [23], [24], or a 123 stochastic process, in which the demand still follows a 124 normal distribution with a dynamic variance [12], [15]. 125 Although these demand assumptions are widely applied 126 in inventory management literature, they are well-known 127 for the poor approximation of several real demand patterns, 128 which are uncertain, intermittent, and unpredictable [29]. 129 Given a demand distribution, the total demand during a 130 replenishment cycle of a supply chain is split into two unequal 131 parts. The larger part, referred to as bounded demand, is ful-132 filled by available inventory, while the smaller one, known 133 as unbounded demand, is handled by operating flexibili-134 ties, such as accelerated production [19], and subcontracting 135 [18], [20]. Under this demand-splitting scheme, the timing of 136 fulfillment (or lead time) is always guaranteed. In addition, 137 since the bounded demand represents the fraction of total 138 demand in a replenishment cycle that is satisfied by the 139 inventory system, it implicitly determines the cycle service 140 level of the supply chain. Therefore, many GSA studies 141 determine the size of bounded demand by specifying a cycle 142 service level, such as 90% [11], 95% [15], [16], [27], 97.5% 143 [19], [20], or 97.7% [28]. It is conventional wisdom that 144 service level is prescribed by either customers or managers. 145 Therefore, most GSA research studies treat the service level 146 as a given input and focus on minimizing the total inventory 147 cost without considering the impact of handling an additional 148 amount of unbounded demand. However, most managers and 149 customers usually indicate a service level based on experi-150 ence and preference rather than a comprehensive analysis 151 of trade-offs among factors such as operating flexibility and 152 inventory carrying costs [24]. Indeed, carefully evaluating 153 these factors would provide a better service level that min-154 imizes the total inventory cost [25]. This approach is demon-155 strated in the study of Aouam and Kumar [18]. The authors 156 show that optimizing service level in addition to the safety 157 stock placement decision by considering extra measures, 158 including subcontracting and overtime, results in a lower 159 total inventory cost than taking a service level as an input 160 parameter. 161 VOLUME 10, 2022 The remainder of this paper is organized as follows. The 187 description of the safety stock placement problem is pre-188 sented in Section 2. The mathematical model formulations 189 that consider multiple service levels are provided in Section 3. 190 Then, a simulation model that evaluates the performance of 191 the proposed models is presented in Section 4. A comparison 192 of results between the mathematical model and the simulation 193 is given in Section 5. Finally, the conclusions are made in 194 Section 6.

196
This paper considers a multi-echelon inventory optimization 197 problem for a production company that coordinates a sup-198 ply chain network. The network consists of many stages 199 at different locations to manufacture a single product. The 200 final assembly of this product is commenced at the most 201 downstream stage, referred to as Stage 1, in the network. 202 In other words, end-customer orders are received and fulfilled 203 by Stage 1. Upon receiving an order, the end customer is 204 quoted a lead time for delivery. The delivery lead time is 205 also interchangeably referred to as committed service time 206 in the remaining parts of this paper. In the meantime, orders 207 for components and subassemblies required to produce the 208 finished product are sent to upstream stages. These stages 209 also quote different lead times for the received orders. Stage 210 1 must wait for all components and subassemblies to be deliv-211 ered to start its production. Most stages in the network are 212 centrally managed by the company, while some are operated 213 by suppliers and subcontractors. These stages are defined as 214 internal and external stages, respectively. Since the company 215 is uncertain about customer demand, inventories, especially 216 safety inventories, may be required at several stages to protect 217 the supply chain against any demand fluctuation and maintain 218 a consistent customer service level. These inventories can be 219  The supply chain network consists of N stages. Stage 1,249 which is closest to the end customers, performs the final 250 assembly operation. The other (N − 1) stages are associated 251 with each of the (N − 1) items, i.e., raw materials, compo-252 nents, or subassemblies, that the final product requires. Each 253 item is either produced by an internal manufacturing stage 254 or procured from an external supplier stage. The network is 255 modeled as a directed graph G (N , A), where N is the set 256 have safety stocks are delivered before its production is 296 commenced. This waiting time is defined as incoming service 297 time S in i of stage i and is equal to the largest total, among all 298 stages j, of outgoing service times, quoted to stage i, including 299 the required transportation time T j,i , i.e., S in i = S out j + T j,i . 300 The production time at stage i is assumed to take P i > 0 time 301 periods and is assumed to be independent of the production 302 quantity [3].

304
In the inventory positioning problem, the information about 305 the end customer demand is instantaneously passed to the 306 upstream stages of the supply chain through a series of 307 orders. In order words, all internal stages are under single 308 ownership and are assumed to share information so that every 309 stage observes the same demand pattern as Stage 1. If the 310 customer demand for the finished product of Stage 1 in a 311 period follows a probability distribution with a mean of µ D 312 and a standard deviation of σ D , the demand for the output item 313 of an upstream stage j follows the same distribution.

314
Modelling the demand in a period by using the normal 315 distribution is a common practice in some inventory manage-316 ment studies [11], [14], [16], [18], [19], [20], [25], [27], [28]. 317 This assumption allows the demand over multiple periods, 318 i.e., replenishment lead time, to be approximated using the 319 normal distribution. However, this may provide a poor esti-320 mation of the system behavior when the underlying shape 321 of the demand distribution is non-normal. This is especially 322 the case for many medium-and slow-moving items that 323 experience intermittent demand and, for some items, a few 324 outliers. Under this demand pattern, the normal distribution is 325 effective only when the lead time is extremely long such that 326 it can overcome the intermittency and presence of outliers, 327 which is rarely the case in practice. To properly model the 328 demand with such characteristics, empirical distribution is 329 used to obtain a better estimation. Description of the base 330 stock policy under the two demand modeling approaches 331 and a comparison of system performance measures between 332 them are provided in the subsequent section and a numerical 333 experiment.

335
For an internal stage i ∈ I that chooses to keep the safety 336 stock of its output item, a proper base stock level B is 337 determined. To maintain this level, a stage i always generates 338 orders for its input items and sends them to upstream stages 339 immediately after it receives a customer order. Typically, 340 it takes S in i periods for the orders from the upstream stages to 341 arrive and P i periods to manufacture item i for the customer. 342 Since stage i commits to fulfilling its customer demand after 343 S out i periods, stage i requires a net outgoing replenishment 344 time, l ≥ 0, to fulfill the customer order. In addition to 345 keeping the safety stock of the output item, each internal stage 346 i ∈ I may choose to keep some inventory of the input item 347 from an upstream stage j in storage as a buffer to shorten 348 the supply lead time, i.e., S out The net 349 92990 VOLUME 10, 2022 replenishment time of these two scenarios is expressed by the 350 following Equation.
As stage i faces the demand with a mean of µ D and SD of 353 σ D in every period, the total demand over l periods has the 354 mean of µ D l and SD of σ D √ l [11]. 355 In the case that the demand of stage i is assumed to follow 356 the normal distribution, the base stock level of an item, either 357 output from stage i or input item from stage j, is usually 358 specified as, is estimated as, Also, using the base stock level as specified above, the 369 expected inventory of an item, which is kept in storage at where the first term represents work-in-process or pipeline  (5). 394 The safety stock to minimize can be determined by sub- G (k|l) − µ D l. Similar to the case of the normal demand, the 397 expected amount of unsatisfied demand is estimated as, Generally, an inventory positioning problem aims to mini-401 mize the total safety stock cost for input and output items 402 for a given service level. Under the normal and the empiri-403 cal demands, the total safety stock cost can be respectively 404 expressed as, While Equation (9) contains non-linear terms of decision 410 variables, i.e., l, and Equation (10) contains G (k|l), which 411 is dependent on l. In this research, Equations (9) and (10) The objective function is to minimize the total annual cost 487 of the supply chain for both input and output items across 488 all stages. Constraints (14) force the outgoing service time 489 of each external stage in the supply network to be no shorter 490 than its minimum quoted service time. Constraints (15) and 491 (16) determine the net replenishment time for each inventory 492 of input and output items, respectively. Both constraints are 493 derived from Equation (1), which represents the net replen-494 ishment time. In addition, each constraint chooses a specific 495 cycle service level k among all possible levels and a value 496 of the net replenishment time. Constraint (17) ensures that 497 the outgoing service time at Stage 1 would not exceed the 498 minimum service time committed to the end customers. Con-499 straints (18) and (19) represent the correspondence between 500 the net replenishment time of either output or input items 501 and the selected net replenishment time. Constraints (20) 502 and (21) imply that not more than one net replenishment 503 time is selected for each inventory position. Constraints (22)    of shortage, the ending inventory eOH i,t is updated. When 558 the simulation reaches period n, it is terminated, and the 559 statistics for this inventory system of stage i are collected. 560 The total inventory cost of a stage TAC i is computed as 561 follows: where l = x ik (or l = y ijk ), EOH i and ES i are the expected 564 inventory on-hand and the expected shortage of stage i over a 565 cycle during n days of simulation, respectively.
For an internal stage i keeping input items from stage j in its 568 storage, the simulation of its inventory system is conducted 569 with the same logic. Since all stages in the supply chain 570 network receive the same demand information as it is sent 571 upstream from Stage 1 and the service times between stages 572 are guaranteed, each can be simulated as an independent 573 inventory system based on the same demand dataset and 574 simulation logic.  chain network is illustrated in this problem as a diagram, 580 including vertices and arcs. A vertex (or node) represents 581 a manufacturing stage, while an arc represents the flow of 582 materials from one vertex to another (see Figure 4).   Intel Core i7-10710U processor and RAM of 32.0 GB 64-bit. 642 Optimal positions of safety stocks in the network are then 643 derived from the optimal quoted service times from the 644 model. If an outgoing service time exceeds an incoming 645 service time of a stage, then that stage requires a safety stock 646 placement.

647
Stochastic demand for finished goods is based on one-year 648 historical data of an actual product. In addition, it should 649 be noted that the demand data are not normally distributed. 650 Instead, it is intermittent by nature. As previously mentioned, 651 demand and shortage during net replenishment time are usu-652 ally approximated by a normal distribution in some studies. 653 However, there are situations where the normal distribu-654 tion provides a poor approximation in practice, specifically 655 for medium-and slow-moving items, when the demand is 656 intermittent, in which case the empirical distribution pro-657 vides a better estimation. Therefore, the two distributions are 658 experimented with in our numerical study to evaluate their 659 performance in estimating demand and shortage during the 660 net replenishment time under such demand characteristics.

661
The average daily demand, µ D , and standard deviation, 662 σ D , are computed from the actual demand data for the nor-663 mal demand scenario. The amount of safety stock and the 664 expected shortage are determined by the second term of Eq. 665 (2) and Eq. (3), respectively. For the empirical distribution, 666 the amount of safety stock and the expected shortage are com-667 puted by the second term of Eq. (7) and Eq. (8). In addition, 668 the maximum possible quoted service time between any two 669 stages is determined based on the critical path, which is the 670 longest path through the network. This network's critical path 671 is the path from stages producing the following components, 672 R6→C10.1→C10.2→FG. In other words, lead times can 673 vary between 0 and 86 days in this problem instance. Other 674 model parameters are given in the input data column in 675 Table 2.

676
In the table, stages indexed with the letter R are external 677 stages (R1 to R9), while the others are internal stages. Stages 678 at the beginning of the network have no predecessor, while 679 the other stages may have one or more upstream stages. 680 Each stage has only one downstream stage, except for the 681 final assembly stage, which has no successor. The proposed 682 model aims to determine the safety stock locations and their 683 quantities that minimize the total inventory cost, including 684 inventory holding cost and shortage cost. The holding and 685 shortage costs are computed as 10% and 30% of the unit 686 product values, respectively. The service level is defined 687 in 19 scenarios from 90%, 91%, . . . , 99%, 99.1%, 99.2%, 688 . . . 99.9%. Tables 2 and 3 show the optimal solutions at the 689 service level of 98%, where demand is under normal and 690 empirical distributions, respectively.

691
In this problem instance, the safety factor associated with 692 the service level of 98% is −1 (0.98) = 2.0537. From the 693 table, we can identify supply chain stages that should use a 694 VOLUME 10, 2022 make-to-stock production strategy and keep the safety stock.
On the contrary, the remaining stages without safety stock fulfill its downstream stage request since its optimal outgoing 704 service time is shorter than its optimal incoming service time.

705
For example, from    would increase stock-out, which leads to an increase in the 751 shortage cost. On the other hand, a high service level indicates 752 more safety stock to be kept, which reduces shortage but leads 753 to a higher inventory holding cost. The result shows that the 754 minimum total cost is achieved at the service level of 98% 755 from both approaches, i.e., solving k CSLs at once and one at 756 a time. This level of customer service suggests keeping more 757 safety stock rather than experiencing shortage.

758
From Table 5, when the demand is modeled using empir-759 ical distribution, the total inventory costs are higher than 760 the normal demand for all the CSLs. This result is because 761 empirical distribution can accurately capture the uncertainty 762 of actual demand, while normal distribution fails to capture 763 this skewness. Therefore, if we compute the safety stock 764 using the normal distribution, the estimation of inventory 765 cost may not be accurate. The next section will show the 766 accuracy in estimating the total inventory cost by comparing 767 the results from the MILP model under each of these two 768 demand distributions with results from a simulation model. 769

770
The preliminary results during the simulation model valida-771 tion process give an estimate of the standard error of the 772 key system measure of performance. Based on the standard 773 error, the required number of replications for the simulation 774    applied for the case between the MILP solution under the 800 normal distribution and the simulation results. The test results 801 are presented in Table 8. Thailand. His research interests include logistics 998 and supply chain systems in the problem areas 999 of supply chain network design, vehicle routing problems, and inventory 1000 optimization; data analysis using machine learning, computer visions, and 1001 statistical analysis; and system modeling using discrete-event system simu-1002 lation, Monte Carlo simulation, and system dynamics.