An MIP model to optimize a Colombian cash supply chain

Authors


Abstract

Financial institutions operate a supply chain where only one product is moved across the network. Although the cash is kept in several nodes to service the demand of final customers, availability causes opportunity cost related with its investment options. When planning the inventory of cash that should be maintained across the network, transportation decisions are also expected to be made. Cash transportation has a high cost, associated with the high risk of theft. Increasing the inventory can reduce the need for transportation, but the opportunity cost can be very high. Moreover, the inventory is also related with the service level perceived for final customers; therefore because of low inventory some people are not able to make some transactions. The aim of this work is to find optimal decisions related with cash inventory and transportation across the network, trying to balance the cost of the service and the quality perceived for final users.

1. Introduction

Financial institutions form a very particular supply chain with only one product moving across the network: cash. It seems that, given this simplicity, they should be easier to manage than those complex supply chains from the manufacturing world, where thousands of different products are involved. However, cash supply chains face some other issues. Cash is the asset with the highest liquidity, hence holding cash inventory is quite expensive and at the same time, not having enough cash would mean low service levels and will directly affect final customers; cash transportation involves high risk and is therefore expensive; just to mention a couple of big issues.

Several characteristics of a cash supply chain are similar to the those in the manufacturing world: offices are located over a wide geographic area; demand is stochastic but unlike most manufacturing firms, each financial office will have two demand series for its only product, a demand for incoming cash and a demand for outgoing cash; inventory decisions are to be made in terms of how much cash inventory to hold and where to hold it. Provided those similarities, it is possible to use the wide spectrum of optimization tools that have been developed to deal with manufacturing supply chains, and to apply them to the management of cash in a money supply chain. Although there is not a transformation stage in a cash supply chain, the transportation of the one product as well as holding cash are pretty much the same activities that a distribution supply chain has to face. The cost elements that can be mentioned as having the highest relative importance would be: opportunity cost, fixed and variable transportation cost, cash handling cost, insurance cost, stockout cost. Modern cash supply chains are not allowed to have backorders, so this last cost component is quite important. If a financial institution is asked to serve a customer with cash and they do not have all of it, they will have to use an extra money order from their central office or another institution, whose opportunity and transportation costs are usually higher.

The problem addressed by this research arose in the management of a Colombian financial institution that has 85 offices across the country. Some of those offices have ATMs whose demand was included in the analysis. A mixed integer programming (MIP) model was developed to represent the situation and it was solved within the context of multi-objective optimization by developing an iterative procedure that combines a standard MIP solver and historical simulation. Neural networks were used to forecast the demand, and historical simulation was used to account for the service level and stockout cost.

This paper includes several characteristics that were not taken into account in the previous research. Unlike most of the literature reviewed, both the walk-in transactions and ATM demands were included to analyze the demand series. Additionally, the measure of service level was carried out and was used as a way to design the cash-management policies for the system in contrast to the usual approach where the only objective was to minimize the total cost accepting the implicit service level associated with that cost-optimal decision. Furthermore, the usual approach makes the assumption that the demand series follows a normal distribution and exhibits a steady behaviour over the planning horizon and therefore the mean and variance are constant. However in this paper, the analysis of the demand series showed that its behaviour was different for different periods of time. The neural network approach used to do the forecast allowed a better representation of the demand series pattern.

2. Literature review

Although available literature regarding cash supply chain management is scarce, on the other hand there is a vast amount of papers dealing with manufacturing supply chain optimization and management. We briefly describe some of those references as it is not our intention to do a review. Escudero et al. (1999) developed a general modeling framework for the oil supply chain. Goetschalckx et al. (2002), Meixell and Gargeta (2005), Papageorgiou and Georgiadis (2008) as well as Klibi et al. (2010) present reviews on supply chain optimization. Villegas (2004) presents a supply chain model applied to a Colombian coffee supply chain.

Another body of literature that also has been active over the years is that related to the cash balance problem. It is a stochastic periodic review inventory problem faced by a firm in which the customer demands can be positive or negative. The objective used to approach this problem has always been to minimize the total expected cost. In this approach, the concept of supply chain has not been present. Whisler (1967) as well as Eppen and Fama (1969) studied the problem assuming that there were no fixed costs associated to the ordering or return. Girgis (1968) approached the problem including at most one fixed cost. Neave (1970) showed that the problem is much more complex when both fixed costs are present. However, he was able to partially characterize the optimal policies. More recently Feinberg and Lewis (2007) extended the results from Neave to the case of an infinite horizon, and Chen and Simchi-Levi (2009) presented a different approach than Neave getting similar result to characterize optimal policies. Its worth recalling that literature associated with the cash balance problem does not have the supply chain scope, it is only based on cost minimization and makes the assumption of a stochastic but steady behaviour for the demand series.

The first article to address the topic is Meng-Huai (1991), in which the main objective was to find the optimal amount of cash to be held in bank branches. Back in 1991, the author proposed a periodic revision of the cash inventory level to check the cash available only every R units of time. Given the facility of online operations and the technology available nowadays for tracking the inventory of cash in a continuous way, with actualization right after every single transaction, the model proposed by Meng-Huai seems outdated.

Rajamani et al. (2006) present a conceptual framework to analyze in detail all the elements present in a cash supply chain. The article does not present a particular model, however they do a good job in setting a baseline for the conceptual understanding of the whole system. This article is based on the US financial system, which is similar in its structure to the Colombian system: both have a central bank from which cash is delivered to and received from other banks and banks’ branches. In a recent paper by Dawande et al. (2010), a strategic approach to the problem of cash recirculation is presented. This problem has been addressed by the US Federal Reserve seeking to minimize the societal cost of providing cash to the public.

Most of the cash-management solutions developed in the previous literature have been developed for ATM networks. These networks differ from an office's network because most of the time ATMs can only handle outgoing flows whereas an office's network offers a wide variety of services including inflow and outflow transactions. Wagner (2007) used discrete event simulation as a framework to optimize an ATM network for a financial institution ranked among the world top 700 banks. He found that up to 28% cost saving can be achieved by improving the inventory policies and cash transportation decisions. Castro (2009) developed optimization models for an ATM network using a stochastic programming approach in which the demand at every ATM is modeled as a stochastic variable, several models and planning horizons were proposed, and some experiments were conducted to determine the relative importance of the different cost elements. Simutis et al. (2007, 2009) also did an analysis of an ATM network where neural networks have been used to forecast the demand. The inventory levels of cash as well as the cash flows were obtained as to minimize the total cost, and the optimization was carried out by applying simulated annealing. To the best of our knowledge, there is no abundant literature on cash supply chain. However, there are several software-based solutions for cash supply chain in the market. Table 1 summarizes some of these solutions.

Table 1. Software-based solutions for cash supply chain
CompanyProductWeb page
C-EquilibriumCSO- Cash Supply Optimizationhttp://www.c-quilibrium.com
FiservCash and logisticshttp://www.carreker.com/main/solutions/cash/icom.htm
SAPmySAP™ Supply Chain Managementhttp://www.sap.com/solutions/business-suite/scm/index.epx
ProyectosCash4uhttp://www.xmartlink.com/
Compartidos  

In Colombia, we did not find any approach to this problem coming from academia. However, the financial sector has implemented some of the solutions shown in Table 1. Most of them require a high initial investment as well as high maintenance and license costs. The organization involved in this research has calculated an investment of about $500 million COP (Colombian pesos), about US$250,000, for using one of those solutions. Besides, none of the companies selling software solutions provides details about the methodology that they used for managing cash.

3. Problem presentation

The supply chain analyzed has 85 offices and 47 of them have one or several ATMs. Among these offices a classification can be made based on the difference between the cash inflow and outflow: if the inflow is high the office is considered a collector (net cash recipients), and the excess of inventory over a period of time has to be send to a central office. On the other hand, if the outflow is high these offices are payers (net cash suppliers) and they have to determine how much cash to order, in order to have enough to service their customers.

In this system, it is pretty clear that the total cost should be minimized, but the cash inventory and cash flows also should be set as to have a good service level, meaning that the probability of a stockout should be as small as possible. If the cash demand were known in advance then it would be possible to determine an optimal pattern of inventory and flows, but the demand is in fact a stochastic variable. The total cost and the probability of stockout are conflicting objectives. Consider the case in which the inventory level at a payer office is allowed to be particularly high, so no extra cash orders are required to match demand, therefore expedite and transportation costs are lower, but opportunity and insurance costs would be high; a low inventory would reduce the opportunity and insurance costs but will cause more extra cash orders to be asked, and expedite and transportations costs will be higher.

It is worth noticing that the problem addressed leads naturally to a stochastic optimization problem. The decisions that need to be made are how much cash inventory to hold as well as the size and timing of the cash orders. Those decisions should achieve the minimum cost without negatively affecting the service level. In spite of the stochastic nature of the problem, the approach used to solve it was deterministic. A forecast technique was used to estimate the future cash balance for each office and therefore the stochastic variable turns into a parameter for the model. Nonetheless given the fact that any forecast is subject to error, an iterative procedure solution that uses historical simulation was suggested to deal with the implicit variability.

3.1. Demand forecast

The planning horizon used is one month and decisions about flows are made on a daily basis, therefore forecast should be daily. Different days during the week and month exhibit different behaviour, so the forecast technique should be able to deal with those characteristics.

Neural networks were developed, trained, and used to forecast the demand at every service node, following the suggestion of Zhang (2004) and Simutis et al. (2008). The SPSS Modeler software package was used to do all the tasks necessary to use the neural networks in this research. The resulting neural network has 21 input neurons, corresponding to 20 past daily historical data, and 1 neuron to handle the qualitative information pertaining to any specific day (like pay day, end of the month, and so on). There is one output neuron corresponding to the forecast of a particular day. One neural network was trained for every office.

3.2. Mathematical model

The mathematical model of the system is developed following some assumptions. Demand is known based on the forecast obtained from neural networks; the cash flows do not include bill denomination, cash flows between offices are allowed, all the offices are inside a single country and therefore the model is regional, the planning horizon is of one-month length and demand should be satisfied all the time, even incurring in expedite costs. The transportation is provided by a third-party company and have fixed and variable costs. This last fact is quite important because transportation decisions pertaining specific routing are therefore not included. The model simply decides if an order is required or not and a third-party company serves those orders. In other words, this is not an integrated inventory-routing problem.

Parameters definition and notation

CFEC

= fixed cost of sending cash to the central office [$/order]

CFEO

= fixed cost of sending cash between offices [$/order]

CVEC

= variable cost of sending cash to the central office [$/million]

CVEO

= variable cost of sending cash between offices [$/million]

CFPC

= fixed cost of receiving cash from the central office [$/order]

CVPC

= variable cost of receiving money from the central office [$/million]

CCO

= cash handling cost [$/million]

T

= central office incoming tax [%]

C

= opportunity cost of the company (daily) [%]

Bdi

= net cash flow forecasted for the office i during day d [$]

I0i

= initial cash inventory at office i [$]

Decision variables definition and notation

Xdi

= cash flow from the office i to the central office on day d [$millions]

Wdi

= 1, if Xdi> 0, 0 otherwise

Ydi

= cash flow from the central office during day d to office i [$millions]

Zdi

= 1 if Ydi>0, 0 otherwise

Udii*

= cash flow on day d between offices i and i* [$millions]

Vdii*

= 1 if Udii* >0, 0 otherwise

Idi

= cash inventory at the end of day d in office i [$millions]

TOUi

= cash inventory maximum limit at office i [$millions]

TOLi

= cash inventory minimum level at office i [$millions]

Objective function

display math
display math
display math

The objective function accounts for fixed and variable transportation costs (the first six terms), tax transactions costs, handling costs, and opportunity costs (last three terms respectively). “O” refers to the set of offices and “D” stands for the set of days. Constraints are as follows.

Cash flow balance

display math

Inventory level limits

display math

Logical constraints

display math

Domain constraints

display math

Note that the mathematical model basically handles a situation in which there is a deterministic time-varying demand. In this particular case, the deterministic time-varying demand pattern is provided by the forecast stage after using neural networks. The model determines the timing and sizing of replenishment so as to minimize the total cost function. The model also considers inventory decisions. For a typical manufacturing supply chain, the inventory level of a product is usually constrained by an upper limit related to the physical capacity of a warehouse, and also constrained by a lower limit, usually related to a safety stock, an administrative policy developed to deal with demand and lead time variability.

In a system like this where backorders are not allowed, the minimum inventory will be zero if no safety stock is maintained. Also note that if the upper limit is not imposed, one possible decision from the model will be to order only once at the beginning of the planning horizon, and then to pay the inventory holding cost and opportunity costs for the rest of the time horizon. That would be just a feasible solution but the optimal solution will depend on the relationship and trade-offs between the different cost elements. The variables TOL and TOU in this model have the same spirit as the safety stock and the maximum capacity, respectively, for a typical supply chain. In every branch, the TOL variable sets the minimum level of cash that should be available at the beginning of every period of time whereas the TOU variable sets the maximum cash level that should be allowed at the end of every period.

4. Solution approach

The formulation of the MIP model includes TOL and TOU as decision variables because in fact it is necessary to determine the lower and upper cash inventory limits for every office. However, solving the model for the single scenario coming from the forecast would be dangerous because it is only one possible realization of the demand series, the best realization that was possible to estimate following the forecast methodology. Nevertheless, the demand is still a stochastic variable. Therefore, iterative solution procedures are developed to set values for TOL and TOU outside the MIP model, and the resulting MIP is solved. Historical simulation (Linsmeier and Pearson, 1996) is used to check if the particular values used for TOL and TOU lead to stockouts so that stockout cost could be taken into account.

The calculation of TOU and TOL is performed for every office in a separate way. Note that the mathematical model presented corresponds to the whole network of offices, however when setting up the TOL and TOU variables for a particular office a reduced version of the model is solved, in particular not considering the flows between offices (or setting the variables Udii* = 0). The reduced version of the model analyses the timing and size of orders between the central office and the office branch being analyzed. Once the values for TOL and TOU have been obtained for all the offices then the general model is run. The model is then allowed to have cash flows between offices, so that if for a particular planning period an office has an excess of cash and another have a shortage, the cash can be transported between them. The transportation costs are different when transporting cash between branches than from or to the central office. Therefore, the model might be able to find a cheaper transportation strategy by programming shipments between offices if the cash balance allows it.

It is worth noticing that the decision about TOL and TOU affects both the total cost and the service level. Consider for instance the case of a net cash supplier office (inflow lower than outflow) for which a low inventory value of TOL is set. In such a scenario, it is likely that the office will require extra cash orders to satisfy the demand and those extra cash orders will increase transportation and handling costs. On the other hand, the same scenario is expected to have a low opportunity cost (related to the investment opportunities lost by the company given that a decision has been made to keep the cash in the office instead of available for alternative investments). The low inventory will also cause a decrease in the service level, reflected in higher probability of a stockout.

Recall that the branches of the financial institution can be classified in terms of their typical net cash balance behaviour: net cash collector and net cash supplier. For the former, the main concern is about how much of that excess of cash at the end of a period should be kept at the office while for the latter the concern is about how much inventory should be available at the beginning of each period to support the demand, given that the cash inflow from the same period is not going to be enough. This classification allows for focusing on one or the other inventory limit first, taking into account the nature of the office. If the office is a cash collector, then the first limit to set up will be TOU and then TOL, if the office is a cash supplier then the order is reversed.

4.1. Setting up the TOU variable

For offices where the inflow is usually higher than the outflow (net cash collectors), the main concern with the inventory is to have more than the maximum level (increasing opportunity cost). In the same sense, note that in these offices the occurrence of stockouts is not likely to become an issue. To determine the TOU variable an iterative procedure is suggested. The reduced MIP model (where all Udii* = 0) is solved for several predetermined values of TOU, doing fixed marginal increments. The TOL variable is set to zero (0) based on the natural limit for a system in which backorders are not allowed. This limit will be revisited when setting up the TOL variable, following the procedure explained later. Figure 1 is a flow diagram explaining the methodology to set up TOUTOL when the first is more important. MinTOU and MaxTOU refer to the minimum and maximum trial values. TOU* refers to the optimum value for TOU. Figure 2 show the results obtained in the case study after varying the TOU between 50 and 250 million COP ($Col) with a step (Δ) of 1 million ($Col).

Figure 1.

Setting up TOUTOL for a net cash collector

Figure 2.

Costs elements versus TOU

It is easy to see that all the cost elements, except the opportunity cost, decrease along the increments on the TOU variable. This is coherent since increasing the TOU would allow the model to increase the inventory and hence the opportunity cost would be higher. Since the total cost decreases with every increase in the value of TOU, some other considerations besides cost minimization should be taken into account to determine the final value of TOU.

Given the conflicting nature of opportunity cost (last term in the objective function) and logistics cost (transportation, transactions, and consolidation costs—first 8 terms in the objective function), it might be worth doing some comparisons between those two elements. Both costs should be finally paid for by the organization but let us take a look at their behaviour, as depicted in Figure 3.

Figure 3.

Logistic and opportunity cost versus TOU

It is possible to see the trade-off between these cost elements as well as the inflexion point for both curves, around $188 million. After that value the decreasing cost changed its tendency and the other one does the same in the opposite direction. This might give some insights about setting the final value for the variable TOU. It is good to remember that the TOU variable makes sense for those offices that exhibit a cash collector behaviour (net cash recipients). If by the end of the day an office has more than the amount given by its TOU variable, then the office is required to send the extra amount to the central office. Setting a low value for TOU will cause that almost every day some cash needs to be sent to the central office, but the inventory cost will be low. A high value for TOU will prevent the occurrence of cash shipments to the central office at the end of the day, but will create opportunity cost as well as higher insurance premiums. It is also important to point out that having more money than allowed by the policy does not affect negatively the service level to the final customer (in terms of demand satisfaction), only causes some extra cost.

4.2. Setting up the TOL variable

For net cash suppliers, the TOL variable is quite important because of its direct impact over the minimum inventory (safety stock), and what it means to service costumers’ withdrawals. TOL can be seen as cash safety stock, against demand variations as well as forecast errors. In terms of service level, there is a probability of a stockout associated with every particular value for TOL. An iterative procedure is suggested such that the reduced MIP model is run for several values of TOL. Once the optimal result is obtained, the system behaviour is analyzed against real data occurring for a given period of time. In order to do this, some historical data were used to train the neural network, and to finally obtain a forecast, but the historical data were old enough to allow having a forecast of demand values that actually already occurred. In this way, it was possible to see when a stockout would have occurred and therefore to estimate stockout probability and cost. Note that by using historical simulation, it is not necessary to fit a probability distribution to the demand series (Linsmeier and Pearson, 1996).

The following steps are a detailed version of the procedure. Figure 4 shows the flow diagram:

  1. A set of testing values is defined for the TOL variable. These values can be related to storage capacity, insurance policy, and risk level. In this case, the values tested were between $0 and $190 million, with step of $1 million ($Col).
  2. Cash demand is forecasted for a past period. The real demand of that past period will be used as an input for the historical simulation procedure.
  3. A single value for the TOL variable is selected. TOU is set to infinity but this will be revisited once the final value for TOL has been selected.
  4. The model is run to obtain the optimal decisions for inventory and cash flows.
  5. Historical simulation: The solution obtained from the MIP model is applied over the system, and therefore it is possible to see what would have been the real behaviour. In particular, it is possible to see if any of the decisions made by the model based on demand forecast and certain value for the TOL variable lead to a stockout.
  6. When stockout occurs, this information is saved to finally calculate the stockout probability, or, in other words, to account for the service level.
  7. The probability of a stockout is calculated as the ratio between the number of days when a stockout occurs and the total number of days analyzed.
  8. Steps 3 through 7 are repeated for every possible value of TOL defined in step 1.
  9. With the optimal value of TOL (TOL*), perform the procedure to set TOU.
Figure 4.

Setting up TOLTOU for a net cash recipient

Figure 5 shows the trade-off between TOL and probability of a stockout. Whenever a stockout occurs, the office should request an extra order from the central office. These last-minute orders are usually more expensive. Figure 5 shows the total cost curve for the real system after adding the expedite cost associated with the occurrence of a stockout. In fact the MIP model includes only the opportunity cost (alternative investments not made) for the case of holding inventory. The case of a negative inventory value (stockout) is handled by the simulation step of the solution procedure and the stockout (backorder) costs are accounted accordingly. It has been assumed that when a stockout occurs an extra cash order is used for exactly the size of the stockout. An emergency cash order causes fixed and variable transportation costs higher than a regular order. The total cost depicted in Figure 6 is then the cost from the MIP model plus the extra cost associated to stockouts when they occur.

Figure 5.

Stockout probability versus TOL

Figure 6.

Total Cost versus TOL (historical simulation)

It is possible to see that the minimum cost for this case is around $119 million. For this value of TOL, the associated stockout probability would be 0.16. In the real system, however, the stockouts are not going to occur because the office will cover the shortage by making an extra order of cash to the central office. This minimum cost is actually taking into account those stockouts cost, by using the historical simulation.

An interesting analysis of this approach is the comparison with the so-called (s, S) policy, related to a canonical optimal inventory policy for a general inventory problem. The policy states that the inventory should be checked continuously and whenever it is equal or below to “s” then an order should be made to raise it to the level “S.” Although the TOL and TOU variables seem to suggest that spirit, a simulation experiment conducted with the final results for TOL and TOU showed that the system does not actually behave following the ideas of an (s, S) policy. In fact the size and timing of the orders suggested by the model carefully take into account the future expected demands and not only the inventory limits. If TOL and TOU were to be used as an (s, S) policy, the total cost would have been at least 30% higher, according to the simulation study. In this work, a canonical (s,S) policy was not obtained. The historical simulation applied the values of TOL and TOU as if they were s and S, respectively, and the general results for the cost were observed. It is worth noting that a typical (s,S) policy assumes that the demand is normal with a steady behaviour, and in this case the demand exhibits a time-varying demand pattern.

An important operational indicator is the number of services contracted during a period with the cash-carrier transportation. It is worth noting, however, that the aim of the model is not only to reduce the number of services contracted. Instead, the idea is to reduce the cost associated with the operation of the system as a whole. Anyway, since the fixed cost per service is a large component of the total cost, the model tries to reduce the number of services. Table 2 illustrates these results.

Table 2. Decrease in transportation services required after optimization
Sending servicesReceiving servicesTotal servicesActual services% decrease
49639535101347%

The organization has been measuring an indicator called “cash transportation cost” in which they add up all the associated fixed and variable costs. This indicator does not include the opportunity cost or the transaction tax cost. The model developed actually accounts for those cost elements thus is more robust. The comparison shown in Table 3 is based only on the cost elements that are included in both, the actual and the proposed, methodologies. The variable cost per million is approximately 2.3% and the handling cost is approximately 3.2%, both related to the fixed cost. The opportunity cost estimated for the company was 15% annually. For the case of transshipments between offices, both the fixed and variable cost are almost double because those are not planned orders and therefore the transportation company charges higher rates.

Table 3. Cost reduction after optimization
New planned logistic cost ($)Actual cost($)Percent decrease
52,745,84499,738,56247%

5. Conclusions and further research

This research has used the robust knowledge on supply chain design developed over recent years by the academic community and that has been applied to several real manufacturing systems, to propose a framework to study cash supply chains, recognizing that there are many similarities between these supply chains and the typical ones for physical products. The good results obtained in improving the cash supply chain are an indicator of the functionality of the general concepts in both the economic sectors. Previous works have addressed the cash supply chain for ATM networks but in this case regular bank branches were included as well.

The problem that has been addressed is a complex problem in which there are a lot of options for setting up specific policy values. It is almost impossible to analyze systematically those endless options without using some sort of model to represent the system outcomes given a policy. In this case, an MIP model has been used and previous works dealing with the optimization of supply chains have shown that it is possible to deal with real systems where the amount of variables are hundreds of thousands.

In the case studied, it is worth noting that it is possible to achieve big savings by making better decisions regarding cash management. Such big savings can also be seen as a proof of the lack of a robust cash-management tool in the actual system. Poor supply cash management causes both high costs and low service level. Some hidden costs become explicit when formulating a formal model and that is something worth enough by itself.

The financial sector is a quite visible member in the services sector of the economy. Some other service sectors, such as the health care sector or postal services to mention just two, have used optimization applied to their supply chains so it seems like some other efforts should be made to apply what we know about typical supply chains to other systems.

One important and immediate extension of this work is the consideration of the demand as a random variable. In this case, a forecast technique as well as historical simulation has been used to analyze the real behaviour of the system. By using the same model proposed in conjunction with stochastic optimization, allowing the demand to be a random variable with a known probability distribution function, it would be possible to analyze the system closer to its real behaviour. The extension can also include the study of the TOL and TOU variables for every period, instead of keeping the same inventory limits over the whole planning horizon.

In the proposed model, the central office is modeled as having infinite capacity so that any amount of cash can be ordered from or sent to it. Relaxing this assumption would lead to a problem in which the inventory level at the central office should be studied as well in order to minimize opportunity cost, but at the same time maintaining some service level. This is another possible extension of this research. Besides, there is the possibility of including investment decisions as part of the model, since the only product is cash and there is always a need for cash in the financial sector. The model was designed for a closed-loop system, but relaxing this assumption would make the model even more interesting because in some specific periods it can be more profitable to give the cash to other companies instead of keeping it for doing the regular operations of the company.

Ancillary