When to take an item out of the assortment- an optimal stopping problem.

(1)

0

When to take an item out of the

assortment-an optimal stopping problem.

Master Thesis Supply Chain Management

University of Groningen

Faculty of Economics and Business

June 23, 2017

B.C. Brounen

s2189062

b.c.brounen@student.rug.nl

Grote Beerstraat 185

9742RH Groningen

Supervisor: N.D. van Foreest

Second supervisor: W.H.M. Alsem

Acknowledgements:

(2)

1

Abstract

Companies dealing with seasonal demand, such as companies in the apparel industry, are experiencing decreasing demand for their items. They want to keep the inventory of these items as long as the cost of holding the inventory is lower than the projected revenue. The objective of this thesis is to determine when such a company should take an item out of the assortment and dispose of any remaining inventory. The problem is seen as an optimal stopping problem. Accordingly, this thesis developed an optimal stopping model. Doing multiple experiments using simulation, it is found that for the investigated items of the case company, it is better to keep them in the assortment on day t (28-03-2017). In addition, the optimal stopping policy has been compared with simple policies. The results show that in the current situation of the case company, the simple policy of taking the item out of the assortment after 4 years performs nearly the same as the optimal stopping policy. This is caused by the low holding cost of the inventory of the case company. Therefore, using this simple policy is sufficient for the case company. For companies subject to higher holding costs, using the optimal stopping policy is worth the time and effort.

(3)

2

1.Introduction

Companies dealing with seasonal demand, such as companies in the apparel industry, have to make a decision on when to take an item out of their assortment. Usually these companies want to have inventory of these items to meet potential demand. However, they only want to do so if the cost of holding the inventory is lower than the projected revenue. If this is not the case, the item should be taken out of assortment and the inventory should be disposed of. An example of a company dealing with this problem is Shop4. Shop4 is an online web shop which sells cellphone covers. They experience a decrease in demand for their cellphone covers each time a cellphone manufacturer introduces a new model. Despite this decline, Shop4 strives to keep inventory of the older models as long as the cost of holding the inventory is lower than the projected revenue. When this is not the case, the inventory is seen as financially obsolete. Hence, the question arises: “When to take an item out of the assortment

and dispose of any remaining inventory?”.

The problem of the timing of disposal is related to literature on disposal of excess inventory. Authors have proposed models to determine the economic retention quantity or the economic disposal quantity, when there is excess inventory (e.g. Rosenfield, 1989; Tersine and Toelle, 1984). However, the problem is not to determine the amount of inventory to dispose of, but when to take an item out of the assortment and dispose of all the inventory. Problems similar to the timing of the decision to take an item out of the assortment are optimal stopping problems. The theory of optimal stopping involves problems where an agent chooses a time or point to take a certain action to maximize profit and minimize costs (Clark & Kastellec, 2013). In this case this would mean that the agent has to monitor the inventory on a daily basis in order to determine the optimal moment to take an item out of the assortment.

When excess inventory is not disposed of in time, it can become obsolete (Van Jaarsveld & Dekker,2011). Pay (2010, p.1) stated “Obsolete inventory is one of the largest components of inventory cost and often is larger and more costly than executives are willing to admit”. Still, very little has been written about obsolete inventory. This is stressed by Teunter, Syntetos and Babai (2011, p.607), who state the following; “Despite its importance, obsolescence is ill-researched”. In addition, inventory models and policies dealing with the disposal of excess inventory cannot be used to determine when to take an item out of the assortment and dispose of any remaining inventory. Therefore, a new method should be developed.

This research identifies the timing problem of taking an item out of the assortment as an optimal stopping problem. Accordingly, this research will develop an optimal stopping model to determine the optimal timing for taking the item out of the assortment and disposing of any remaining inventory. Demand data from the case company will be used to forecast demand, after which the optimal stopping model and its policy will be tested by means of simulation. The results show that by making use of the optimal stopping model the decision of when to take an item out of the assortment and dispose of any remaining inventory can be determined.

(5)

4

2. Theoretical background

In this section related literature to the problem of taking an item out of the assortment and dispose of any remaining inventory will be discussed. First, models and policies for disposal of excess inventory available in literature will be reviewed. After which the problem will be related to the theory of optimal stopping.

2.1 Disposal models for excess inventory

A stream of literature related to the decision of taking an item out of the assortment is dealing with the disposal decision of excess inventory. When excess inventory is not disposed of in time, it can become obsolete (Van Jaarsveld & Dekker,2011). This raises the question how should a company deal with this (partial) obsolete inventory? Partial obsolescence occurs when the demand for a certain item has severely diminished, but has not necessarily vanished (Pince & Dekker, 2011).

Disposal creates benefits in at least two ways, namely: the salvage revenue from the disposed inventory and the saved holding cost (Willoughby, 2010). However, due to ongoing operational usage of the item, the company might face future repurchasing costs. This occurs when the company disposes too much of the inventory (Willoughby, 2010). This results in a tradeoff between salvage revenue and reduced inventory holding costs versus future repurchasing costs.

A simple decision rule regarding determining excess inventory is proposed by Silver, Pyke and Peterson (1998). They suggest to calculate for each item the expected time at which the inventory would be depleted. Managers can list the value of this so-called item’s coverage in a descending order. If the item’s value is also known, managers can quickly obtain an indication of the value of the excess inventory. Disposal of a part of this inventory can free up capital invested in inventory (Silver et al, 1998). Brown (1977) suggests an even simpler rule. He states that managers should use their intuition in setting two limits: the number of weeks of supply and the dollar value of supply. If inventory surpasses one of these limits, it needs to be regarded as excess inventory.

Next to simple decision rules in determining excess inventory, analytical models exist. These models try to determine the economic retention quantity or the economic disposal quantity, when there is excess inventory. Mohan and Garg (1961) and Hart (1973) investigated the disposal of excess inventory. In their research, they look at disposal of items with deterministic demand with the possibility of obsolescence. They both found that disposing of a certain amount of inventory can significantly increase profitability by reducing inventory costs. Tersine and Toelle (1984) proposed a model which calculates the optimal time-span of supply of inventory. If the inventory on hand exceeds this optimal level it has to be salvaged. However, they do not take into account the possibility of obsolescence. Their model assumes that demand for the item will keep on existing. A similar research has been done by Stulman (1989), but he assumed a continuous review system with stochastic demand. Rosenfield (1989) investigated the problem of disposal of slow-moving or obsolete inventory. He proposed concrete formulas with a trade-off between the cost of holding and disposing of slow-moving inventory. Silver and Willoughby (1999) investigated the disposal of excess inventory after the completion of a construction project. The inventory is used as spare parts during the ongoing phase of the facility. They developed an efficient procedure to determine the amount of inventory to dispose of in this decision-making environment, reducing inventory costs.

(6)

5 economic retention quantity or the economic disposal quantity, when there is excess inventory. The problem explained in the introduction, however, is not to determine the amount of inventory to dispose of, but when to take an item out of the assortment and dispose of all the inventory. Therefore, this thesis sees this problem as an optimal stopping problem. Literature described above uses different variables to determine the economic disposal quantity, when there is excess inventory. In table 2.1 these variables are depicted.

Variables Literature

Salvage revenue Tersine and Toelle (1984), Stulman (1989), Rosenfield (1989), Hart (1973) and Silver and Willoughby (1999)

Demand Tersine and Toelle (1984), Stulman (1989), Rosenfield (1989), Hart (1973) and Silver and Willoughby (1999)

Future repurchasing cost Tersine and Toelle (1984) and Silver and Willoughby (1999) Reorder costs Tersine and Toelle (1984) and Silver and Willoughby (1999) Inventory level Tersine and Toelle (1984), Stulman (1989), Rosenfield (1989), Hart

(1973) and Silver and Willoughby (1999)

Holding cost Tersine and Toelle (1984), Stulman (1989), Rosenfield (1989), Hart (1973) and Silver and Willoughby (1999)

Selling price Rosenfield (1989) and Hart (1973)

Discount rate Tersine and Toelle (1984), Stulman (1989) and Silver and Willoughby (1999)

Table 2.1 Variables related to the disposal decision of excess inventory identified in literature.

These variables will be used for the development of the optimal stopping model, except the variables future repurchasing cost and reorder costs. These are not applicable, as repurchasing is not possible as the item is in this case taken out of the assortment. In addition, in the experiment section it will be investigated, if it is necessary to work with a discount rate of future cash flows as done by Tersine and Toelle (1984), Stulman (1989) and Silver and Willoughby (1999).

2.2 Relating the problem to the theory of optimal stopping

Similar to the problem of when to take an item out of the assortment are optimal stopping problems. The theory of optimal stopping involves problems where an agent chooses a time or point to take a certain action to maximize profit and minimize costs (Clark & Kastellec, 2013). Optimal stopping problems are common in economics, finance, operations research and statistics (Riedel, 2009). To the best of the knowledge of the author, no literature exists using the theory of optimal stopping for inventory management. A famous optimal stopping problem in operations research is that of the house-selling problem (Ferguson, 2012). Here, offers come in daily for a house that you wish to sell. Each day you have to decide, whether you want to accept the bid and sell the house or wait for a better offer. When you wait, you have cost. You can see this cost as the cost of living in the house. You know that eventually a higher offer could appear, but does this outweigh the cost of waiting? The theory of optimal stopping can be applied to the problem of taking an item out of the assortment. To explain this, it will be related to the house-selling problem. Each day it can be decided to dispose of the inventory and gain the salvage revenue (the bid on the house that day) or to wait one more day, pay holding cost for the inventory (the cost of living) and meet potential future demand (a higher bid). This can be translated into two decisions:

A; Stop: Take the item out of the assortment and salvage all of the remaining inventory.

(7)

6 The optimal stopping problem in this case is one with a finite horizon, as it is assumed that the item is taken out of the assortment after four years due to the launch of new cellphone models. Due to this finite horizon the optimal expected revenue for the next day, when continuing, can be derived by using backward induction (Hill, 2009). How this works is explained in section 4.1.

This research aims to identify how companies dealing with seasonal demand, like companies in the apparel industry, can decide on when to take an item out of the assortment. Hence, the following research question:

When to take an item out of the assortment and dispose of any remaining inventory?

(8)

7

3.The optimal stopping model

The inventory system under consideration is a periodic-review single item system. As the problem is one with a finite horizon, the inventory will be held for no longer then n days. In addition, there will be no option for reordering. Therefore, the inventory we have at day t (It) is all the inventory we have to meet potential future demand. Within the model it is assumed that the past demand (D-1,D-2,D-3…….,Dt) is known. Using this past demand, demand sequences (D+1,D+2,D+3…….,Dn) will be simulated. How this is done is explained in section 4.3. In addition, it is known that a salvage value of a product decreases over time, therefore, the salvage value in this model decreases with a fixed percentage per year. Lastly, salvaging a part of the inventory is not an option.

Notations:

t = The day a decision has to be made D = Expected demand

S = Expected sales

I = Expected inventory level h = Holding cost per item per day R = Expected revenue

r = Revenue per sold item s = Salvage value per item n = Final day of the finite horizon

E = The decision of stopping and taking item out of assortment salvaging the remaining inventory C = The decision of continuing to keep the item in assortment

At the start of day t, two choices can be made on the state of the system, which is given by It-1 and D t-1;

A; Stop (E): Take the item out of the assortment and salvage all of the remaining inventory It-1.

B; Continue (C): Keep the item to meet the potential demand for day t, pay holding cost for the remaining inventory, and decide at the start of the next day t+1 to salvage the inventory or to continue. The question is how to make the optimal decision between these two alternatives. Therefore, we need to know what the revenue is, if we take the item out of assortment and salvage the inventory. Equation 1 calculates this option. Where st is the salvage value of the item on day t.

Rt(It-1, E) = st*It-1 (1)

If it is decided not to salvage the inventory at the start of day t, it could be that we run out of stock on day t. Here the problem also stops. This happens when Dt ≥ It-1. Therefore, the expected sales St on day t is the minimum of the forecasted demand or of the expected inventory level at the start of day t. Therefore, equation 2 gives the expected sales on day t.

St= min{D_t,I_t-1} (2) If it is decided to continue, the expected inventory level on day t is given by equation 3.

(9)

8 Assuming we know the optimal revenue for Rt+1(It) for day t+1, which starts with expected inventory It, and the demand for day t (Dt) is forecasted based on D1,……,Dt-1 , we can calculate the expected revenue for day t if we continue. Equation 4 gives this expected revenue.

Rt( It-1, C) = r*S t– h* It + Rt+1( It) (4)

Assuming we know Rt+1 (It) and we have Dt based on the forecasted demand, the best decision for that day is the decision which generates the highest reward. Equation 5 gives the optimal expected reward at the start of day t.

𝑅𝑡(It-1) = max { Rt(It-1, E), Rt( It-1, C)} (5)

Thus, the optimal choice for day t is

a

*

(t)= arg

max{ Rt(It-1, E), Rt( It-1, C)}

So how do we find the expected optimal reward for the next day (Rt+1 (It))? This is done by making use

of backward induction. To be able to do this we need to know the last day n. At the end of day n, we will salvage any remaining inventory. Equation 6 will give the expected reward for day n.

Rn= st*In (6)

The expected inventory level at In is given by the forecasted demand explained in section 4.3. Now that

we know the final expected reward of the last day, we can use backward induction with the dynamic programming equation (5). Using this equation, we can solve Rn-1, Rn-2, Rn-3 …. until we have reached day

t at which point we have to make the decision between stopping and salvaging the inventory or

(10)

9

4. Methodology

First the general approach of the problem will be discussed. In section 4.1 arguments on why simulation and dynamic programming are used, are elaborated upon. In addition, the case company will be introduced briefly. Then it is explained how items are selected, on which the optimal stopping model will be tested; this is done in section 4.2. After which the demand will be forecasted for the chosen items by using historical demand data from the case company. This is needed in order to solve the optimal stopping problem. How the forecasting will be done is explained in section 4.3. Finally, the inventory policies and the key performance indicator (KPI) will be stated in section 4.4.

4.1 Method used

For an optimal stopping problem with a finite horizon it is common to use dynamic programming (Adda & Cooper,2003). In order to solve the problem, a dynamic programming formula is needed which allows for backward induction. Backward induction can be applied, when it is known what the final optimal choice would be for the last period (Hill, 2009). In the case of the case company this would be the salvage revenue of the remaining inventory on the final day of the item. This value now contains information for the optimal decision for the day before. This allows for calculating the optimal decision for that day, disposing of the inventory or keeping the item in the assortment. This then can be repeated, until the optimal decision is calculated for the current day on which a decision has to be made.

The model can be analyzed by using backward induction via simulation (Clark & Kastellec,2013). Simulation is most appropriate for our situation, as it allows to predict system performance, to compare alternative system designs (scenarios) and to determine the effect of alternative policies on system performance (Robinson, 2004). Furthermore, simulation requires few, if any, restrictive assumptions. In addition, simulation allows for a higher face validity. This can give researchers and managers a better understanding of, and more confidence in the model (Robinson, 2004). Lastly, computing the dynamic programming formula manually for a large time horizon would be very time consuming. By using simulation this can be done in seconds. One of the disadvantages of simulation mentioned in literature is the fact that simulation software is often quite costly. This is not the case here, as the simulation of the inventory system is done in Excel 2016, which is a relatively cheap program. For more information about the advantages of simulation in comparison with other methods see Robinson (2004). Details about the simulation model are given in Appendix A.

The data is provided by the case company. The case company has been active in the smartphone accessory market since 2014 by selling products via their online web shop. They are experiencing a decrease in demand for their older cellphone covers due to the launch of new models. They want to keep inventory of these items as long as the cost of holding the inventory is lower than the projected revenue from potential demand. In addition, at this moment in time they have 21K stock keeping units (SKU’s). In 2014 their SKU’s consisted of 6K. This rapidly increasing number is causing difficulties for them to keep track of their inventory. Therefore, they seek a way to determine when to take an item out of their assortment and dispose of any remaining inventory.

4.2 Item selection

(11)

10 can be used to choose items subject to financial obsolescence. It would not be logical to determine whether an item should be taken out of the assortment, when it has only just been taken into the assortment. Due to time constraints, from each group one item is selected. These items are shown in table 4.1.

Year taken into assortment Categories

2014 2015

TUS Item 1 Item 2

US Item 3 Item 4

Table 4.1: Items selected.

The characteristics of these items are given in section 5.2.

4.3 Forecasting demand

To be able to solve the optimal stopping problem with a finite horizon, backward induction is needed. In order to be able to apply backward induction, the demand sequence needs to be known. Therefore, the demand will be forecasted using simulation. Using other forecasting methods like the moving average method does not give this sequence but only the forecasted demand for a certain time period. By analyzing the historical demand data, probabilities can be assigned to the size of the order (e.g. 0,1,2 etc). For every forecasted day, the simulator generates a number between 0 and 1. This number will be compared with range values assigned to different order sizes. This determines if an order arrives and, if so, how large the order is.

An example will illustrate how it works: after analyzing the historical data it is found that item 1 was

726 days in the assortment, of which on 37 days there was 1 demand and on 3 days a demand of 2. So, for the rest of the 686 days there was 0 demand. Calculating the probabilities this means that the probability of 0 demand is 94%, 1 demand is 5% and 2 demand is 1%. Now by expressing this in ranges the chance of 0 demand is 0-0,94, of 1 demand 0,94-0,99 and of 2 demand 0,99-1. Now for each forecasted day Excel will pick a random number between 0 and 1. For instance, it picks 0,55. This lies between the 0-0,94 range, so the forecasted demand for that day is 0.

To simulate a decrease in demand over time, the decrease in demand in the historical data is calculated. For instance, for item 1 the demand decreased with 28% between the first and second year. With this information, the demand probabilities after 4 years can be calculated. The difference between the probabilities, divided by the number of days to be forecasted, is the amount the probabilities have to increase or decrease per day. This can be seen in table 4.2.

Demand size Time period

0 1 2

Historical demand probabilities 94% 5% 1%

Demand probabilities after 4 years 98,3% 1,2% 0,5% Chance per forecasted day +(4,3%/730) -(3,8%/730) -(0,5%/730) Table 4.2: Probabilities of forecasted demand item 1.

(12)

11

4.4 Inventory policies and KPI

The inventory policy of the optimal stopping model is that, if the expected revenue of continuing is lower than the revenue of stopping, the item is taken out of the assortment and the remaining inventory is disposed of. To investigate if it is worth the effort and time to make use of the optimal stopping policy or if a simple policy is sufficient, four simple policies determining when an item should be taken out of the assortment are suggested.

1. Take item out of assortment after 3 years in assortment. 2. Take item out of assortment after 30 days of no demand. 3. Take item out of assortment after 90 days of no demand. 4. Take item out of assortment after 180 days of no demand. These policies will be compared based on the expected revenue.

To answer the research question, experiments are done in the next section. Firstly, the optimal stopping model is tested to see if it is valid and reliable. Next the characteristics of the selected items are given. Moreover, the scenarios in which the optimal stopping model will be tested are described. Then, multiple demand sequences are simulated for each item, to use as input for the optimal stopping model. Subsequently, this input is used to see whether the items should be taken out of the assortment or kept in the assortment on day t (28-03-2017). In addition, a sensitivity analysis is performed to investigate the influence of the different parameters on the continuation and stopping value. Lastly, the optimal stopping policy will be compared with the simpler policies described above to see if it is worth the time and effort to make use of the optimal stopping model or if it is sufficient for the case company to make use of a simpler policy.

(13)

12

5.Experiments

In this section, the optimal stopping model will be applied to the selected items shown in section 4.2. The forecast, as explained in section 4.3, will be used to generate the demand sequences for the various items. Using the simulation method, 10 different demand sequences are simulated for each item. Before applying the optimal stopping model, the validity and the reliability will be tested. This is done in section 5.1. After which the characteristics of the selected items and the different scenarios which are going to be tested are given in section 5.2. Finally, in section 5.3 the outcomes of the model and the other simple policies for the different scenarios and different items are shown.

5.1 Testing the validity and reliability of the model

To test the validity and reliability of the optimal stopping model, a test suite is made. This is a collection of tests to see if the model behaves in the appropriate way. Within these test cases, values are inserted in a certain order to see if the optimal stopping model provides expected outcomes. In addition, calculations are done manually to see if the outcomes given by the optimal stopping model are correct. This is the case for all the test suite cases, thus it can be concluded that the optimal stopping model is valid and reliable to a certain extent. The test suite cases and the results can be seen in Appendix B.

5.2 Items and scenarios

The optimal stopping model is used for the selected items to investigate if they should be taken out of the assortment or not. The item selection is explained in section 4.2. The characteristics of the selected items are depicted in table 5.1.

Items Characteristics 1 2 3 4 Selling price 9,97€ 10,36€ 9,92€ 9,97€ Inventory 70 43 71 55 Historical demand until 28-3-2017 43 148 136 55

Date taken into assortment

28-3-2015 2-7-2014 4-7-2014 18-6-2015

Finite horizon 28-3-2019 2-7-2018 4-7-2018 18-6-2019 Table 5.1: Selected items and their characteristics.

Four scenarios are used to investigate if, under different conditions, the decision of taking an item out of the assortment or keeping it in the assortment changes. In addition, these scenarios are used to investigate if the optimal stopping policy outperforms the simpler policies under different circumstances. By changing one parameter and holding the other parameters constant it can be tested how these changing conditions influence the optimal decision given by the optimal stopping model (Clark & Kastellec, 2013). Parameters which can be changed within the optimal stopping model are: starting inventory level, holding cost per item per day and the starting salvage revenue per item. Hence, the following scenarios are tested:

1. The first scenario is the current situation, which is based on the real situation at the case company. 2. The second scenario is a scenario in which the inventory is doubled.

3. The third scenario is a scenario in which the holding cost per item per day is multiplied by a factor of 4.

4. The fourth scenario is one in which the starting salvage revenue is doubled.

(14)

13 Item Scenarios Parameters Current Double inventory Higher holding cost Higher Salvage revenue Item 1 Inventory 70 140 70 70

Holding cost per item per day 0,0014€ 0,0014€ 0,0056€ 0,0014€ Starting salvage revenue per

item

1€ 1€ 1€ 2€

Item 2 Inventory 43 86 43 43

item

1€ 1€ 1€ 2€

item

1€ 1€ 1€ 2€

item

1€ 1€ 1€ 2€

Table 5.2: Different scenarios for each item.

In addition, for each item 10 different demand sequences are simulated. These 10 different demand sequences will be used as input for the optimal stopping model. Just simulating one demand sequence would not be sufficient, as this would not represent reality. By simulating multiple demand sequences and comparing the outcomes, a more reliable and better outcome can be given by the optimal stopping model.

5.3 Outcomes of the optimal stopping model

In this section the outcomes of the optimal stopping model for each item and different scenario are given. The 10 different simulated demand sequences for each item are used as input for the optimal stopping model. Moreover, a sensitivity analysis is conducted to further investigate the influence of the different parameters on the optimal decision given by the optimal stopping model. In addition, the optimal stopping policy will be compared with the simple policies proposed in section 4.4 on their expected revenue. This is done to investigate if making use of the optimal stopping policy is worth the time and effort, or if using a simple policy is sufficient.

5.3.1 Testing different scenarios

(15)

14

Discount rate Decision Average expected reward

N/A Continue €211,14 Stop €70 3% Continue €206,04 Stop €70 6% Continue €204,10 Stop €70 12% Continue €200,33 Stop €70

Table 5.3: Outcome optimal stopping model for item 1 for day t (28-03-2017) using different discount rates. The value for stopping is the inventory level times the salvage value per item. The value for continuing is the expected sales minus the holding cost plus the expected reward of the optimal decision the next day. The values are the average values using the 10 different forecasted demand sequences as input for the optimal stopping model.

It can be observed that the influence of making use of discounted cash flows is small. If there is a possibility to invest the money in a project with a 12% rate of return, this only decreases the continuation value with approximately 5% in comparison with no option to invest the money in a project. The stopping value remains the same as there are no future cashflows within this value. Thus, making use of discounted cash flows does not influence the stopping and continuation value significantly. Therefore, no use will be made of discounted cash flows.

On day t (28-03-2017) the manager of the case company is considering whether to take the items out of the assortment or not. To determine this, the expected reward of the two decisions is computed using the optimal stopping model. This is done for the four scenarios described in section 5.2. For each item, ten different demand sequences are simulated and used as input for the optimal stopping model. Using the outcomes of the ten different demand sequences, the average expected reward is computed. In addition, the standard deviation of the outcomes using the 10 demand sequences are given. The results are shown in table 5.4, 5.5, 5.6 and 5.7.

Item Decision Average expected reward σ Optimal decision

Item 1 Continue €211,14 €67,92 Continue

Stop €70 €0

Stop €34,40 €0

Stop €56,80 €0

Stop €55 €0

Table 5.4: Outcome optimal stopping model for the current situation of the case company.

Table 5.5: Outcome optimal stopping model for the scenario in which the starting inventory is doubled.

Stop €140 €0

Stop €68,80 €0

Stop €113,60 €0

(16)

15 Table 5.6: Outcome optimal stopping model for the scenario in which the holding cost per day is multiplied by a factor of 4.

Table 5.7: Outcome optimal stopping model for the scenario in which the salvage revenue per item is doubled.

It is observed that in each scenario, for each item, the optimal decision given by the optimal stopping model is to continue to keep the items in the assortment on day t (28-03-2017). This is most likely caused by the low holding cost. It can be seen in table 5.6, that if the holding cost is multiplied by a factor of 4, the average expected reward of continuing for the four items decreases significantly. For instance, for item 1 this value drops from €211,14 to €106,99. Still, the optimal decision remains to continue to keep the items in the assortment on day t (28-03-2017). In addition, the standard deviations of the average expected reward of the two decisions are given. It can be seen that the standard deviation for stopping is €0. The reason for this is the fact that the demand does not influence the expected reward after stopping. Therefore, there is no difference in expected reward for the different demand sequences. The standard deviation of the average expected reward for continuing is for all the items in all the scenarios quite high. This is most likely caused by the fact that only 10 demand sequences are used as input. The differences between these demand sequences are quite substantial due to the fact they are simulated using probabilities. Resulting in different outcomes and thus a high standard deviation.

To determine when to take the item out of the assortment using the optimal stopping model, we have to move forward in time. Using the optimal stopping model each day to compute the stopping and continuing value until you reach the day where the optimal decision is to stop and take the item out of the assortment. As it would not be logical to determine this beforehand. However, the differences between the continuation value and stopping value are so big for the current situation of the case company due to the low holding cost, it is most likely close to the optimal decision to keep the item in the assortment until the finite horizon. Therefore, a simple policy which takes the item out of the assortment after the finite horizon of 4 years, will be added to the simple policies and compared with the optimal stopping policy in section 5.3.3 to investigate if this is indeed the case.

Stop €70 €0

Stop €34,40 €0

Stop €56,80 €0

Stop €55 €0

Stop €140 €0

Stop €68,80 €0

Stop €113,60 €0

(17)

16

5.3.2 Sensitivity analysis

To further investigate the influence of different values for the parameters on the average expected reward of stopping and continuing on day t (28-03-2017), a sensitivity analysis is performed. A sensitivity analysis can be done to investigate the relationship between the input variables and output variables (Pannell, 1997). This is done for item 1. The values are the average values found when using the 10 different demand sequences as input for the optimal stopping model. In figure 5.1, 5.2 and 5.3 the sensitivity analysis for the parameters holding cost, starting inventory level and salvage revenue per item is shown. Each parameter is multiplied with a factor 1,2,3,4 and 5.

Figure 5.1: Sensitivity analysis: Different outcomes for the continue value and stopping value for item 1 when the holding cost is multiplied by a certain factor.

(18)

17 Figure 5.3: Sensitivity analysis: Different outcomes for the continue value and stopping value for item 1 when the salvage revenue per item is multiplied by a certain factor.

When looking at the figures, it becomes clear that the holding cost is the most important factor influencing the stopping or continuing decision. When the holding cost is increased, the difference between the stopping and continuation value becomes smaller. When the holding cost is multiplied by a factor of 5, the average expected reward of stopping and continuing is nearly the same. The stopping value remains the same, as the holding cost has no influence on this value. When it is decided to stop, no holding cost has to be paid anymore. When the parameters starting inventory level and salvage revenue per item are increased, the difference between the stopping and continuation value becomes smaller. The lines of the stopping and continuation value are moving towards each other, but in a smaller rate in comparison to when the holding cost is increased. In addition, when the parameters starting inventory level or salvage revenue per item is multiplied by a factor of 5, the average expected reward for continuing is still higher than that of stopping.

5.3.3 Comparing the optimal stopping policy with simple policies

In order to investigate if making use of the optimal stopping policy is worth the time and effort, or if using a simple policy is sufficient, the performance of the optimal stopping policy will be compared with the simple policies described in section 4.4. In addition, the simple policy of taking the item out of the assortment after the finite horizon of 4 years is added to investigate if it is indeed nearly optimal to keep the items in the assortment until the end, for the current situation of the case company. The simple policies are:

1. Take item out of assortment after 3 years in assortment. 2. Take item out of assortment after 4 years in assortment. 3. Take item out of assortment after 30 days of no demand. 4. Take item out of assortment after 90 days of no demand. 5. Take item out of assortment after 180 days of no demand.

(19)

18 Figure 5.4: Comparison simple policies which take the item out of assortment after certain criteria with the optimal stopping policy. The values of the simpler policies are the expected revenue expressed in percentages compared with the optimal expected revenue, for the current scenario of the case company.

Figure 5.5: Comparison simple policies which take the item out of assortment after certain criteria with the optimal stopping policy. The values of the simpler policies are the expected revenue expressed in percentages compared with the optimal expected revenue, for the scenario in which the starting inventory is doubled.

0 10 20 30 40 50 60 70 80 90 100

Item 1 Item 2 Item 3 Item 4

Perf o rm an ce in p erce n ta ge s Items

After 3 years in assortment After 4 years in assortment After 30 days of no demand After 60 days of no demand After 180 days of no demand

0 10 20 30 40 50 60 70 80 90 100

(20)

19 Figure 5.6: Comparison simple policies which take the item out of assortment after certain criteria with the optimal stopping policy. The values of the simpler policies are the expected revenue expressed in percentages compared with the optimal expected revenue, for the scenario in which the holding cost per day is multiplied by a factor of 4.

Figure 5.7: Comparison simple policies which take the item out of assortment after certain criteria with the optimal stopping policy. The values of the simpler policies are the expected revenue expressed in percentages compared with the optimal expected revenue, for the scenario in which the salvage revenue per item is doubled.

It can be observed that the optimal stopping policy outperforms the simple policies substantially in nearly all scenarios. Only in the scenario of the current situation of the case company, the simple policy which takes the item out of the assortment after its finite horizon, which is 4 years, comes very close to the performance of the optimal stopping policy. This is due to the low holding cost, which makes keeping the item in the assortment until the end nearly as profitable as using the optimal stopping policy. When the holding cost per day is multiplied by a factor of 4, the performance of this policy deteriorates. In this scenario, it is more profitable to take the item out of the assortment earlier due to higher holding cost paid. The optimal stopping policy takes this into account, and takes the item out of the assortment when the expected revenue from meeting potential demand is lower than the saved holding costs plus the salvage revenue of the remaining inventory. While the policy of keeping the item in the assortment for 4 years just keeps the item in the assortment until the end.

0 10 20 30 40 50 60 70 80 90 100

After 3 years in assortment After 4 years in assortment After 30 days of no demand After 60 days of no demand After 180 days of no demand

0 10 20 30 40 50 60 70 80 90 100

Perf o rm n ce in p erce n ta ge s Items

(21)

20

6.Discussion

In the experiment section the optimal stopping model was used for four different items to determine whether it is more profitable to keep them in the assortment or to take them out of the assortment on day t (28-3-2017). Multiple scenarios were tested to investigate whether the optimal decision changes under different circumstances. By changing one parameter and holding the other parameters constant it can be tested how these changing conditions influence the optimal decision given by the optimal stopping model (Clark & Kastellec, 2013). In all the scenarios, for all the items, the optimal decision is to keep the items in the assortment on day t (28-03-2017). The main reason for this is that the holding costs of the items are lower than the projected revenue of meeting potential demand. This is supported by the results of the scenario in which the holding cost per item per day is increased. Here, the continuation value for all items decreased significantly. In addition, this is also supported by the sensitivity analysis done for item 1. The results show that when the holding cost is multiplied by a factor 5, the continuation value and stopping value are nearly the same. When the parameters, starting inventory level and salvage revenue per item are multiplied by a factor 5, this is not the case. Here, the difference between the stopping value and continuation value are still substantial. In which continuing to keep the item in the assortment is the optimal decision.

Authors who have written about the disposal of excess inventory found that disposing a part of the inventory can increase profitability by reducing inventory costs (e.g. Mohan and Garg, 1961; Hart, 1973; Tersine and Toelle, 1984). Disposing of inventory namely results in salvage revenue and reduced holding cost (Willoughby, 2010). They investigated problems in which the item is kept in assortment. Thus, their models propose to salvage a part of the inventory and keep a part of the inventory to meet potential future demand. This thesis found that in a situation in which salvaging a part of the inventory is not an option, salvaging all the inventory does not increase profitability for the case company. This is caused by the fact that the holding cost paid by the case company for the items are so small that it does not outweigh the risk of not meeting potential demand. So instead of disposing of all the remaining inventory, it is more profitable to keep the inventory until the end of the finite horizon to meet potential demand.

In addition, the optimal stopping policy has been compared with more simple policies. These policies are:

1. Take item out of assortment after 3 years in assortment. 2. Take item out of assortment after 4 years in assortment. 3. Take item out of assortment after 30 days of no demand. 4. Take item out of assortment after 90 days of no demand. 5. Take item out of assortment after 180 days of no demand.

(22)

21 their products, using the optimal stopping policy could be worth the time and effort. In addition, if the situation at Shop4 changes (e.g. holding cost increases) they might consider to use the optimal stopping policy instead of the simple policy of taking the item out of the assortment after 4 years. So how can Shop4 then use the optimal stopping model to determine when to take an item out of the assortment? With the experiments done in the experiments section it became clear that for all the investigated items on day t (28-3-2017) it is better to keep them in the assortment. Doing these calculations manually for all the items every day would be too time consuming. Therefore, an example of how Shop4 could implement the optimal stopping model is presented. Shop4 should incorporate the optimal stopping model within their online inventory system, Shop4raad. Within their inventory system they have the historical demand data for all the items. Using this historical demand data, the inventory system should simulate multiple demand sequences for each item, for which the managers of Shop4 assigned the TUS or US status. The simulation of these demand sequences can be done as it is done in this thesis, which is explained in section 4.3. Now the inventory system can use these multiple demand sequences as input for the optimal stopping model to calculate the stopping and continuing values for these items. The system could do this periodically, reporting the items to the management of Shop4 on a weekly basis, for which the optimal decision is to stop. The management of Shop4 then can use this list to determine whether they want to take the item out of the assortment or not. If they do not want to spend time on this, they can indicate to the inventory system that these items should be taken out of the assortment automatically. The inventory system then takes the items out of the web shop and automatically sends a message to the wholesale dealer that they can buy the remaining inventory of the item. If the wholesale dealer agrees, the inventory system then sends a message to the third-party logistics provider of Shop4 to ship the remaining inventory to the wholesaler.

(23)

22

7.Conclusion

The case company Shop4 is dealing with decreasing demand for their older cellphone covers. Still, they keep these items in their assortment. They want to do this as long as the cost of holding the inventory is lower than the projected revenue from potential demand. Not only Shop4, but also other companies in the apparel industry are dealing with this problem. Therefore, the aim of this research was to determine when to take an item out of the assortment and dispose of any remaining inventory. To be able to answer the research question, this thesis developed an optimal stopping model. The results showed that by using the optimal stopping model it can be determined when to take an item out of the assortment and dispose of any remaining inventory. The item has to be taken out of the assortment when the stopping value is higher than the continuing value. In addition, it was found that for the current situation of Shop4, using the simple policy of taking the item out of assortment after its finite horizon of 4 years, performs nearly the same as the optimal stopping policy. This is due to the low holding cost. Therefore, using this policy is sufficient for Shop4 at this moment in time. However, the results showed that for companies which deal with higher holding cost for their inventory, higher inventory levels or higher salvage values for their products, the optimal stopping policy performs substantially better than the simple policies and thus is worth the time and effort.

This thesis contributes to the literature on disposal of excess inventory. Current scientific literature focuses on determining the economic retention quantity or the economic disposal quantity, when there is excess inventory. To the best of the knowledge of the author no research has investigated when to take an item out of the assortment and dispose of all of the remaining inventory. This study fills this gap by developing an optimal stopping model which can determine when to take an item out of the assortment and dispose of all the remaining inventory. In addition, in literature, optimal stopping problems have been solved within several fields of research, such as finance, economics and statistics. This thesis showed that optimal stopping problems also exist in inventory management, and showed that methods to solve these problems can also be used here.

In addition to the theoretical implications the results can also be interesting from a managerial point of view. The results show that in most cases a simple policy determining when to take an item out of the assortment performs substantially worse than the optimal stopping policy with regard to the expected revenue. So, if a company uses one of these simple policies instead of the optimal stopping policy, it can cost the company a lot of money. Therefore, using the optimal stopping policy can help managers to determine the optimal moment to take an item out of the assortment, minimizing the cost of this decision.

Limitations of this research regarding the generalizability and validity exist. Firstly, only one case company was researched and used as data source. This harms the generalizability of the results. Secondly, due to time constraints only 10 demand sequences were simulated for each item. It could well be that this not the optimal amount of demand sequences, which harms the validity of the outcome of the optimal stopping model. Lastly, this thesis made the assumption in forecasting the demand that the decrease in demand is constant, while it could be possible that this decrease is exponential. This harms the validity of the results.

(24)

(25)

24

8. References

Adda, J., & Cooper, R. W. (2003). Dynamic economics: quantitative methods and applications. MIT press.

Brown, R. G. (1977). Materials management systems. New York: Wiley.

Clark, T. S., & Kastellec, J. P. (2013). The Supreme Court and percolation in the lower courts: an optimal stopping model. The Journal of Politics, 75(1), 150-168.

Ferguson, T. S. (2012). Optimal stopping and applications. Retrieved from https://www.math.ucla.edu/~tom/Stopping/Contents.html

Hart, A. (1973). Determination of Excess Stock Quantities. Management Science, 19(12), 1444–1451. Hill, T. P. (2009). Knowing when to stop: how to gamble if you must—the mathematics of optimal stopping. American Scientist, 97(2), 126-133.

Mohan, C., & Garg, R. C. (1961). Decision on Retention of Excess Stock. Operations Research, 9(4), 496–499.

Pannell, D. J. (1997). Sensitivity analysis of normative economic models: theoretical framework and practical strategies. Agricultural economics, 16(2), 139-152.

Pay, R. (2010). Avoiding obsolete inventory: Possession is 9/10ths of the problem. As Published in

IndustryWeek.

Pinçe, Ç., & Dekker, R. (2011). An inventory model for slow moving items subject to obsolescence.

European Journal of Operational Research, 213(1), 83-95.

Riedel, F. (2009). Optimal stopping with multiple priors. Econometrica, 77(3), 857-908.

Robinson, S. (2004). Simulation: the practice of model development and use. Palgrave Macmillan. Rosenfield, D. B. (1989). Disposal of Excess Inventory. Operations Research, 37(3), 404–409.

Silver, E. A., Pyke, D. F., & Peterson, R. (1998). Inventory management and production planning and

scheduling (Vol. 3, p. 30). New York: Wiley.

Silver, E. A., & Willoughby, K. A. (1999). Disposal of excess stock at the end of a project when facing on-going operational usage. International Journal of Production Economics, 59(1), 189-194.

Stulman, A. (1989). Excess Inventory with Stochastic Demand: Continuous Reporting Model. The

Journal of the Operational Research Society, 40(11), 1041.

Tersine, R, Toelle, R. (1984). Optimum stock levels for excess inventory items. Journal of Operations

Management, 4(3), 245–258.

Teunter, R. H., Syntetos, A. A., & Babai, M. Z. (2011). Production, Manufacturing and Logistics

Intermittent demand: Linking forecasting to inventory obsolescence. European Journal of Operational

(26)

25 Van Jaarsveld, W., & Dekker, R. (2011). Spare parts stock control for redundant systems using

(27)

26

Appendix A: Simulation model in Excel 2016

The simulation of the optimal stopping model represents a periodic-review single item system. In image A1 the simulation in excel is depicted.

Image A1 Simulation of optimal stopping model in excel

-In the column Day (n) are the amount of days until the finite horizon of 4 years. For instance, for Item 1 it is 730 days away from its finite horizon of 4 years.

-In the column Demand, the forecasted demand sequences are given as input. These are generated by the demand simulator explained in section 4.2.

-In the column Inventory, the inventory level for each day is given. This is the inventory level of the previous day minus the demand on the current day.

-In the column Sales, the expected revenue of that day is calculated. This is done by multiplying the demand on that day times the price of the item. Which is given in column P.

-In the column Holding cost, the holding cost for that day is calculated. This is done by multiplying the inventory level times the holding cost per day. Which is given in column P.

-In the column distribution of salvage reward, the decrease in percentages of the salvage reward is given. For example, after 60 days the salvage value of the item decreases with 10%. Therefore, distribution salvage reward is 0.9.

-In the column Salvage reward, the salvage revenue for each item is given. This is done by multiplying the salvage reward in column P times the value in the column of the distribution of salvage reward. -In the column Stop, the stopping value is calculated. This is done by multiplying the inventory level on that day given in column C times the salvage reward for each item given in column G.

-In the column Continue, the continuation value is calculated. This is done by adding the sales of that day and the optimal choice of the next day (given in column K) minus in the holding cost given in column E.

-In the column Optimal, the optimal expected reward is given. This is done by giving the MAX value of column H and column J.

(28)

27

Appendix B: Test suite

Different test cases are used to determine if the optimal stopping model behaves in an appropriate way. The item on which these test cases are done is item 1.

Case 1

In this case logical reasoning can be used. When the parameter of the cost of holding an item (h) is set higher, it would be expected that the continuation value for day t would be lower. The results can be seen in table B1.

Scenarios Continuation value

p= 9,97€, h= 0,001429€, s= 1€, i= 70 181,38€

p= 9,97€, h= 0,005463€, s= 1€, i= 70 69,79€ Table B1 The value of continuation for a higher holding cost per item per day

It is observed that the optimal stopping model behaves in the expected way. Case 2

In this case also logical reasoning can be used. When the parameter salvage revenue per item (s) is set higher it would be expected that the stopping value for day t would be higher. The results can be seen in table B2.

Scenarios Stopping value

p= 9,97€, h= 0,001429€, s= 1€, i= 70 70€

p= 9,97€, h= 0,001429€, s= 2€, i= 70 140€ Table B2 The value of stopping for day t for a higher salvage value per item

It is observed that the optimal stopping model behaves in the expected way. Case 3

At the end of the finite horizon the model should stop, as the problem stops here. In the image B1 it can be seen that the optimal stopping model indeed stops.

(29)

28

Case 4

To see if the optimal stopping model gives the right values, the calculations can be done by hand. The

stopping value and the continuation value will be calculated for the day before the last day, Rn-1. This

will be done by using the equations from the optimal stopping model for continuing;

Rt(It-1, C) = (r*Dt – h*It + Rt+1(It))

and for stopping;

Rt(It-1, S) = s*It-1

Continuing: (9,97*0)-(0,00143*51)+51= 50,92714286 Stopping: 1*51= 51

Thus the optimal choice is to stop, given: a∗(t)= max{50,92714286,51}

When looking at figure B1, it can be seen that the optimal stopping model gives the same values and the same optimal choice.

Case 5

In a scenario in which the total forecasted demand until the end of the finite horizon is 0, the optimal stopping model should suggest to stop and take the item out of the assortment. It can be seen in figure B2 that this is the case and thus the optimal stopping model behaves in the appropriate way.

When to take an item out of the assortment- an optimal stopping problem.