PRODUCT ASSORTMENT UNDER CUSTOMER-DRIVEN DEMAND SUBSTITUTION IN RETAIL OPERATIONS

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1 PRODUCT ASSORTMENT UNDER CUSTOMER-DRIVEN DEMAND SUBSTITUTION IN RETAIL OPERATIONS Eda Yücel 1, Fikri Karaesmen 2, F. Sibel Salman 3, Metin Türkay 4 1,2,3,4 Department of Industrial Engineering, Koç University, 34450, Turkey Abstract: In this paper, we model the problem of product assortment and inventory planning under customer-driven demand substitution in a multi-period setting. Our model also takes into account other realistic issues in a retail context such as supplier selection, shelf space constraints, and poor quality procurement. First, the characteristics of optimal assortment for different substitution costs is examined. Next, the performance of the modified models, one of which neglects customers substitution behavior and the other excludes supplier selection decision, are analyzed separately. The results of the analysis demonstrate that neglecting customerdriven substitution or excluding supplier selection leads to inefficient assortments. Keyword: Product assortment, demand substitution, inventory management, supplier selection 1 Introduction This paper examines optimal product assortment and inventory stocking policies for a given product category and a set of suppliers under customer-driven substitution. Customer-driven substitution means that if a product type is unavailable, the customer might purchase a substitute product or might not purchase anything which leads to a lost sale. Thus, when certain products are not stocked or stocked out, substitution causes the demand for the remaining products to increase, affecting their optimal inventory levels. An important trade-off exists in retail operations in finding the right product assortment since increasing variety increases customer satisfaction but has a negative effect on operational costs. Therefore, product assortment decision is impacted by several closely-related issues such as category management, selection of suppliers, and inventory levels. Since each product corresponds to a brand, product assortment and selection of suppliers cannot be separated. In addition, as a result of product substitution behavior of customers, increasing assortment reduces demand per variant. Therefore, customerdriven substitution should not be neglected in operational decision making. Due to shelf space limitations, inventory management should also be incorporated in the decision process. Product assortment, demand substitution, supplier selection, and inventory management have been extensively studied separately. One can consider two forms of substitution: In assortment-based substitution, a consumer might substitute when her favorite product is not in the assortment carried by the store, whereas in stockout-based substitution, a consumer might substitute when her favorite product is stocked-out at the moment of purchasing. Van Ryzin and Mahajan (1999) developed a model to determine optimal assortment under a stochastic demand, single-period setting with the utility-based approach, namely the multinomial logit (MNL) choice decision model. Their model allows assortment-based substitution, but does not consider stockout-based substitution. Later, Mahajan and van Ryzin (2001) proposed a sto- Corresponding author: eyucel@ku.edu.tr

2 chastic sample path optimization method for the same model under both assortment-based and multiple rounds of stockout-based substitution. As an alternative to the MNL model, the probabilistic model of substitution is mostly used in inventory models. Smith and Agrawal (2000) studied the assortment planning problem using multi-period base-stock inventory models. They considered both assortment-based and stockout-based substitution, but allowed for one substitution attempt only. In their model, substitution is based on given substitution probability matrices and they presented an approximation to the objective function of the resulting integer program. Our study differs from the previous two studies in the fact that we work on multi-way demand substitution including both assortment-based and stockout-based substitution. Hsu and Bassok (1999) presented a single-period, multiproduct, downward substitution model. They determined the optimal production units to satisfy demand. In order to model random yield and random demand they used the technique of generating random scenarios. Rao et al. (2004) studied the same problem as Hsu and Bassok (1999) but they integrated setup costs in the system. These papers are relevant for production systems rather than retailer systems since substitution is not customer-driven. Kok and Fisher (2004) studied joint assortment selection and inventory planning problem under substitution. Their model of substitution is a probabilistic model with one substitution attempt only. They developed an estimation methodology for substitution rate and an iterative optimization heuristic for the assortment optimization problem which is a nonlinear, nonseperable, discrete knapsack problem. They also considered realistic constraints such as discrete maximum inventory levels and delivery lead times. In this paper, although we do not try to estimate substitution parameters, we also consider realistic constraints such as self space limitations and ordering quantity quotas for suppliers. In addition, our model does not neglect the supplier selection decision which is an important concern when maximizing total profit. Most of the existing literature on decision methods for supplier selection do not consider inventory management. Only some models incorporate the decision to schedule orders over time together with the supplier selection. In reality, the ordering policy and supplier choice affect one another. For instance, if frequent ordering is necessary due to inventory management reasons (e.g. perishable inventory), a supplier with low unit price but high ordering cost might generate a higher total cost than a supplier with a high unit price and low ordering cost. As another example, when suppliers offer quantity discounts, the trade-off between savings in purchasing and inventory holding costs should be considered. Bender, et al. (1985), Buffa and Jackson (1983), and Degraeve, et al. (2000) consider inventory management in supplier selection problems. They propose mathematical programming methods for the problem. In addition, most of the existing literature on supplier selection are relevant for production systems rather than retail operations (Chen and Munson, 2004; Goyal, et al., 2003; Bender, et al.,1985). To the best of our knowledge, multiple products, multi-way product substitution, inventory planning, and supplier selection are not considered in an integrated model in the literature. In this paper, we formulate multi-period, multi-product inventory, product assortment, and supplier selection problem with multi-way demand substitution in order to maximize the profits of a retail store. In that sense, our work provides the first step towards developing a guideline for retailers during their decision making process. The rest of the paper is organized as follows. In Section 2, the model is presented. In Section 3, computational experiments are performed and results and analysis of the results are given. Section 4 concludes the paper and provides possible extensions that might be performed. 2 The Model Our model aims to determine inventory levels that maximize profit in the existence of fixed ordering costs placed per order, fixed costs of supplier selection due to costs of establishing

3 relations with suppliers, purchasing costs, inventory holding costs, costs incurred as a result of poor quality products received, and substitution costs that seem fictitious in the short-run, however, become real costs in the long-run. Constraints of the model include shelf space limitations and ordering quantity quotas of the suppliers. The multi-period planning model considers one product category, such as shampoos or laundry detergents, consisting of a set of products denoted by P and a set of suppliers denoted by S. 2.1 Assumptions As discussed in Section 1, it is possible to consider different levels and types of substitution. While Mahajan and van Ryzin (2001) allow multiple attempts of stockout-based substitution, Kok and Fisher (2004), Smith and Agrawal (2000) and Netessine and Rudi (2003) allow one substitution attempt only. In our model, we allow at most M levels of substitution. We assume deterministic demand and deterministic substitution behavior for customers. That is, customers are expected to substitute from one product to another with deterministic proportions. These proportions might be obtained by market research or by the methodology proposed by Kok and Fisher (2004). This is a reasonable deterministic approximation for probabilistic models of substitution. We assume each product is supplied by exactly one supplier, whereas a supplier may supply more than one product. 2.2 Decision Variables Decision variables of the model are as follows: z it : inventory position of product i at the end of period t. x it : quantity of product i to be ordered per order in period t. y it : 1, if product i is ordered in period t; 0, otherwise. ofs j : number of orders placed with supplier j over all periods t = 1,..,T. o jt : 1, if an order is placed with supplier j in period t; 0, otherwise. ss j : 1, if supplier j is selected (that is, if any order is placed with supplier j in any period); 0, otherwise. x0 it : the amount of satisfied demand for product i in period t. xs mikt : the amount of product i used to satisfy m th substitution from product k in period t. (m = 1, 2,...,M) We use Figure 1 to explain the role of the substitution variables, indicating first choice demands and substituted demands to the products in any period t. For each product, there are three sets of arcs in the figure: i) First choice demand arc, denoted by x0 it, incoming to product i; ii) Substitution demand arc set, one for each level of substitution denoted by j xs mijt, incoming to product i; iii) Substituted demand arc set, one for each level of substitution denoted by k xs mkit, outgoing from product i. 2.3 Parameters The model has the following parameters: w ik : the proportion of customers whose preference is product k that substitute product k with product i. (w kk = 0) [W : Substitution probability matrix]

4 Figure 1: Demand Substitution c i : unit cost of purchasing including unit cost of transportation for product i. oc j : cost of ordering per order placed with supplier j. ssc j : cost of selecting supplier j as a supplier. d it : forecasted demand for product i in period t. OQ i : order quantity quota for product i. SS i : shelf space limitation quantity for product i. a ij : 1, if product i can be supplied by supplier j; 0, otherwise. [A: Availability matrix] h i : inventory holding cost per unit of product i for one period. pq i : unit cost due to receiving poor quality products of type i. q i : percentage of defective products of type i. p it : unit price of product i in period t. s mi : penalty cost of m th substitution from product i. (Lost sales are assumed to be substitution to a dummy product.) z0 i : initial inventory position for product i. 2.4 Objective Function The objective of the model is to maximize the total profit over the time horizon. Max. Total Profit [TP]= Total Revenue [TR] - Total Cost of Ordering [TCO] - Total Cost of Supplier Selection [TCSS] - Total Cost of Substitution [TCS] - Total Cost of Purchasing [TCP] - Total Cost of Inventory Holding [TCI] - Total Cost of Poor Quality Products [TCPQ]

5 2.5 Constraints The model tries to achieve the objective under the following constraints: Total revenue equation is expressed in Equation 1. TR = p i1 (z0 i + x it z i1 ) + i P T p it (z i,t 1 + x it z it ) (1) t=2 i P Total cost of ordering is expressed in Equations 2 to 4, where ordering frequencies for suppliers are calculated by using the availability matrix A. TCO = j S oc j ofs j (2) T ofs j = o jt, j S (3) t=1 o jt a ij y it, t = 1,..,T, i P, j S (4) Total cost of supplier selection is expressed in Equations 5 and 6, where selecting a supplier means at least one order is placed with that supplier. Total cost of purchasing is expressed in Equation 7. TCSS = j S ssc j ss j (5) ss j o jt, t = 1,..,T, j S (6) T TCP = c i x it (7) t=1 i P Total cost of inventory is expressed in Equation 8, where expected inventory is calculated as the average of entering and leaving inventory levels. TCI = i P z0 i + x i1 + z i1 h i + 2 T t=2 i P Total cost of poor quality products is expressed in Equation 9. ( ) zi,t 1 + x it + z it h i 2 (8) T TCPQ = pq i q i x it (9) t=1 i P Total cost of substitution is expressed in Equation 10, where demand for a product is the sum of satisfied demand and all substitutions from that product including lost sales, which is expressed as substitution to dummy product. T M TCS = s mi xs mikt (10) t=1 m=1 i P k P M x0 i1 + xs mik1 + z i1 = z0 i + x i1, i P (11) m=1 k P M x0 it + xs mikt + z it = z i,t 1 + x it, i P, t = 2,..,T (12) m=1 k P

6 The substitution behavior of a customer is expressed in Equations 13 and 14, which use the substitution probability matrix W. The amount of any level of substitution from product k to product i is less than or equal to a certain proportion of the satisfied demand for product k either by the stock of product k or by a substitution. This proportion is equal to the proportion obtained by multiplying the substitution probabilities in matrix W, which exist in that substitution chain. Substitution chain includes all the products that the customer tries to substitute from product k up to product i. Substitution inequalities are written for each of M levels of substitution. Because of the complexity associated with higher substitution levels, we provide only the first two of them. xs 1ikt (d kt x0 kt )w ik, i,k P, t = 1,..,T (13) xs 2ikt (d kt x0 kt xs 1rkt ) w rk w ir, i P, k P \{i}, t = 1,..,T (14) r P r P The shelf space limitations and ordering quantity quotas for suppliers are also considered. 3 Analysis z0 i + x i1 SS i, i P (15) z i,t 1 + x it SS i, t = 2,..,T, i P (16) 0 x it OQ i y it, t = 1,..,T, i P (17) y it {0, 1}, i P, t = 1,..,T (18) First, the model is analyzed to observe the effects of substitution cost in optimal order quantities. Next, the performance of a model, which neglects customers substitution behavior is examined. Finally, the analysis on the performance of another model, which excludes the supplier selection decision, is performed. 3.1 Experimental Data During the analysis, 4 time periods (T = 4), 10 products (with the 11 th product representing lost sales) and 5 suppliers are considered. We assumed that customers perform at most 3 levels of substitution (M = 3). This is a reasonable value for the maximum substitution level since we can both consider multiple-way of substitution and also get rid of the complexity caused by extremely small numerals of substitution probabilities of higher substitution levels. In addition to this, for many customers, substitution becomes meaningless after the third attempt of substitution. The availability matrix, A, is given in Table 1. Note that, according to this matrix, products correspond to brands so that a product cannot be supplied by more than one supplier. However, a supplier can supply more than one product/brand. - Product Supplier Table 1: The availability matrix, A The substitution cost is assumed to be a linear function of substitution level; s mi = SC i m where SC i denotes the first level substitution cost from product i. Also, SC i = θ mg i,

7 Parameter w ik c i oc j ssc j 1 d it OQ i SS i h i pq i q i 2 p i Distribution Uniform distribution where w ik = 1 and 0 w ik 1, i, k P k P Uniform distribution, where 5 c i 10, i P Uniform distribution, where 30 oc j 50, j S Uniform distribution, where 15, 000 ssc j 50, 000, j S d it = 10, 000 α i, where i d it = and 0 α i 1, i P, t = 1,.., T Uniform distribution, where 1, 000 OQ i 8, 500, i P Uniform distribution, where 2, 000 SS i 10, 000, i P Uniform distribution, where 0.3 h i 1, i P Uniform distribution, where 2 pq i 4, i P Uniform distribution, where 0.0 q i 0.15, i P 3 p i = c i + mg i, where mg i has a normal distribution with mean 6 and deviation 2, i P. Table 2: Experimental Data where θ takes different values for different experiments. That is, the substitution cost for a product is a linear function of its margin. The parameter values are generated according to Table 2. We generate 100 random data sets according to the provided distributions. Average of investigated values over these 100 data sets are provided as test results. GAMS is used as the computational environment. 3.2 Impact of change in substitution costs The model is solved for different θ values that result in different substitution costs. In these experiments it is observed that all types of costs, revenue, and percentage of demand satisfied are sensitive to the substitution cost changes. Table 3 shows the results obtained for 0.0 θ 1. Although greater θ values such as 1, 0.9, and etc. may not be reasonable in most cases, we provide complete test results. θ TP 4 %ds 5 %ls 6 %f 7 %s %t TR 8 TCO 9 TCSS 10 TCP 11 TCI 12 TCPQ 13 TCS , , , ,845 33,009 12, , , , ,178 34,502 13,029 11, , , , ,286 35,548 13,315 15, , , , ,910 36,105 13,443 18, , , , ,437 36,784 13,678 18, , , , ,921 37,415 13,871 18, , , , ,070 37,744 13,970 17, , , , ,116 38,023 14,006 16, , , , ,923 38,176 14,048 15, , , , ,555 38,185 14,012 16, , , , ,132 38,495 14,146 15,325 Table 3: The effect of substitution cost for deterministic demand multi-period case According to these results, with θ = 0, the optimal system has limited assortment, selecting a subset of suppliers and favors substitution since it generates no cost. During the analysis, we set high supplier selection costs in order to observe this effect. When θ 0.1, the optimal 1 Total demand for products is assumed to be 10,000 for each period. 2 Price for products is assumed to be constant over all periods. 3 mg i can be considered as the margin of product i 1 TP: the optimal total profit 2 %DS: the average percentage of demand satisfied with the original product 3 %LS: the average percentage of lost sales 4 %f, %s, %t: the average percentage of substituted demand to all other products in the first, second and third level of substitutions, respectively 5 TR: total revenue 6 TCO: total cost of ordering 7 TCSS: total cost of supplier selection 8 TCP: total cost of purchasing 9 TCI: total cost of inventory 10 TCPQ: total cost of poor quality parts 11 TCS: total cost of substitution

8 system extends its assortment, increasing the number of selected suppliers. This shows that substitution costs significantly affect ordering and supplier selection decisions. As it is expected, the increase in the substitution-lost sales costs results in satisfying more demand and decreasing substitutions, especially for higher levels of substitution. In order to achieve this, the system purchases more, resulting in increased inventory. This shows that substitution costs affect purchasing and inventory decisions, as well. Hence, it is important to estimate substitution cost accurately. 3.3 Importance of substitution behavior In order to analyze the importance of considering customers substitution behavior in this problem, total cost of substitution (TCS) is excluded from the objective function. In order to obtain total profit, total cost of ordering of substitution is subtracted from the optimal objective value. In this way, we compared the optimal total profit of the model without substitution costs with that of the original model. Since the optimal solution of the original model is also feasible for the model without substitution costs, the optimal total profit of the former is at least as good as the latter. Then, clearly excluding substitution costs cannot provide better results. The results show that when substitution costs increase, neglecting customers substitution behavior results in increased profit loses which cannot be neglected. Table 4 shows the results obtained with the same experimental data used in Section 3.1. θ [TP] 15 [TP w/o sc] 16 [Difference] 17 [ % Difference] , , , ,406 7, , ,144 20, , ,881 35, , ,619 52, , ,357 70, , ,094 89, ,197 83, , ,984 61, , ,223 39, , ,330 17, , Table 4: The effect of substitution behavior 3.4 The Importance of Supplier Selection Decision In order to analyze the importance of supplier selection in this problem, total cost of supplier selection (TCSS), which is equal to the sum of supplier selection costs for each selected supplier, is excluded from the objective function. Similar to the previous case, in order to obtain the total profit, the total cost of ordering of selected suppliers is subtracted from the optimal objective value. We compared the optimal total profit of the model without supplier selection decision with that of the original model. As before, clearly, excluding supplier selection decision cannot provide better results. When the same experimental data of Section 3.1 is used, we observe that excluding supplier selection decision results in full assortment, that is, all of the suppliers are selected, hence, profit decreases. Table 5 shows these results. 12 [TP]: the optimal total profit of the original model 13 [TP w/o sc]: the optimal total profit of the model without substitution cost 14 Difference: [TP] - [TP w/o sc] 15 %Difference: ([TP] [TPw/osc]) [TP] [TP w/o ss]: the optimal total profit of the model without supplier selection 17 Difference: [TP] - [TP w/o ss] 18 %Difference: ([TP] [TPw/oss]) 100 [TP]

9 θ [TP] [TP w/o ss] 19 [Difference] 20 [ % Difference] , ,035 27, , ,511 26, , ,477 24, , ,844 21, , ,655 19, , ,458 17, , ,862 15, , ,646 14, , ,699 13, , ,015 12, , ,475 10, Table 5: The effect of supplier selection decision as substitution costs change I[ssc] 22 [TP] [N(S)] 23 [TP w/o ss] [N(S) w/o ss] 24 15,000-50, , , , , , , , , , , , , , , , , , , , , , ,156 5 Table 6: The effect of supplier selection decision as supplier selection costs change In addition, the effect of including supplier selection decision in the model is analyzed with increasing supplier selection costs. Table 6 shows the results obtained with the same experimental data used in Section 3.1 except that the interval for uniform distribution of supplier selection cost is changed during the analysis and θ is considered to be 0.3. This time, the number of selected suppliers for both problems are provided in the table for comparison purposes. The results show that when supplier selection costs increase, excluding supplier selection decision results in increased profit loss since neglecting supplier selection cost during decision making results in selecting more suppliers. 4 Conclusion In this paper, we developed a model for the multi-period, multi-product inventory, product assortment and supplier selection problem with multi-way demand substitution and analyzed the behavior of the solution provided by the model in case of different substitution costs with deterministic demand. According to the results of the analysis, it is observed that substitution affects all kinds of decisions: purchasing, ordering, inventory management, and supplier selection. In addition, it is concluded that either neglecting customers substitution behavior or excluding supplier selection decision is not efficient. As a challenging extension, demand can be considered to be stochastic and the characteristics of the optimal assortment might be analyzed in this setting. 5 Biography EDA YÜCEL is a graduate student in the Department of Industrial Engineering at Koç University. She is mainly interested in optimization of supply chains in retailing operations. FİKRİ KARAESMEN is an associate professor in the Department of Industrial Engineering at Koç University. He is mainly interested in stochastic models of inventory and service systems. F. SİBEL SALMAN is an assistant professor in the Department of Industrial Engineering at 19 I[ssc]: supplier selection cost (ssc) interval 20 [N(S)]: the number of selected suppliers 21 [N(S) w/o ss]: the number of selected suppliers in the model without supplier selection

10 Koç University. Her current research interests are network models, optimization in telecommunication and logistics, and disaster management. METİN TÜRKAY is an assistant professor in the Department of Industrial Engineering at Koç University. His main research interests are mathematical programming, supply chain management, logistics, and systems biology. 6 Acknowledgement Financial support from KUMPEM-Koç University MIGROS Retail Education Center is gratefully acknowledged. References Bender, P. S., Brown, R. W., Issac, M. H. Shapiro, J. F Improving purchasing productivity at ibm with a normative decision support system, Interfaces 15: Buffa, F. P. Jackson, W. M A goal programming model for purchase planning, Journal of Purchasing and Materials Management 19: Chen, B. T. Munson, C. L Resource allocation with lumpy demand: To speed or not to speed?, Naval Research Logistics 51: Degraeve, Z., Labro, E. Roodhooft, F An evaluation of vendor selection models from a total cost of ownership perspective, European Journal of Operational Research 125: Goyal, S. K., Huang, C. K. Chen, K. C A simple integrated production policy of an imperfect item for vendor and buyer, Production Planning and Control 14: Hsu, A. Bassok, Y Random yield and random demand in a production system with downward substitution, Operations Research 47: Mahajan, S. van Ryzin, G Stocking retail assortments under dynamic consumer substitution, Operations Research 49: Rao, U., Swaminathan, J. Zhang, J Multi-product inventory planning with downward substitution, stochastic demand and setup costs, IIE Transactions 36: Smith, S. A. Agrawal, N Management of multi-item retail inventory systems with demand substitution, Operations Research 48: van Ryzin, G. Mahajan, S On the relationship between inventory costs and variety benefits in retail assortments, Management Science 45: