University of Groningen
Master Thesis
Stochastic Production Routing Problem with service standards.
Author:
Piriyev Huseyn
Student Number:S3833607
Supervisor:
Dr. Onur Kilic
June 24, 2019
Abstract
This thesis reflects the stochastic production routing problem (SPRP) with a single item and vehicle, as well as the capacitated production. According to this problem, a supplier must decide every period how much of a certain product should be delivered to a customer, when the customer should be served and which routes should be followed. In general, the decisions based on the SPRP do not take into account service standards. In real-life situa- tions, the absence of the service standards leads to the occurrence of different inconveniences between the supplier and the customer. The sources of these inconveniences mainly involve vehicles with less loads and inconsistency in delivery quantities. To cover aforementioned inconveniences, in this paperwork we integrated two service standards into the existing for- mulation of SPRP that was presented in the literature. In addition, we proposed an efficient mixed integer model, which involves the service standards and is an exact method for PRP where a customer demand follows normal distribution. The model is based on the piecewise linearisation of the first order loss function. The results of numerical experiments revealed that proposed model is far efficient both for small and large instances. Moreover, performed experiments allowed to unfold potential effects of two service standards on the optimal total cost and to identify trade-offs.
Keywords: stochastic production routing problem, service standards, piecewise lineari-
sation, first order loss function, normal distribution.
Contents
1 Introduction 3
2 Background and Literature review 6
2.1 The Classical PRP . . . . 6
2.2 The Stochastic PRP . . . . 7
2.3 Service standards . . . . 8
2.4 A discussion of literature . . . . 10
3 Methodology 11 3.1 Stochastic PRP model . . . 11
3.2 Extended SPRP model . . . 14
4 Integration of the loss function in SPRP 16 5 Numerical Analysis 21 5.1 Settings for numerical experiments . . . . 21
5.1.1 Demand data . . . 22
5.2 Performance experiments . . . 22
5.2.1 Performance results . . . . 22
5.2.2 Difference in objective functions . . . . 24
5.2.3 SF-SPRP and line segments . . . . 25
5.3 Effects of service standards . . . . 26
5.3.1 Service standards - one at a time . . . . 26
5.3.2 Service standards - two at a time . . . . 28
5.3.3 Service Standards and Degree of Uncertainty . . . . 30
6 Implications, Limitations and Further Research 30 6.1 Implications . . . . 30
6.2 Limitations and Further Research . . . 31
7 Conclusion 32
Bibliography 33
1 Introduction
In traditional buyer-supplier relationships, the buyer manages his own inventories by reviewing inventories and placing replenishment orders when necessary, while the supplier fulfills these orders. This means that supplier and buyer are considered independent. They act on their own and concentrate on optimizing their objectives locally without taking into attention the other party (Tokta¸s-Palut and ¨ Ulengin, 2011). However, in such a relation- ship, potential benefits of collaboration between these parties in terms of cost-effectiveness cannot be realized. A well-known counterpart to such an independent relationship between manufacturer and customer is the concept of vendor management inventories (VMI), where suppliers serve as central decision makers instead of customer. That is, they decide when and how much product should be delivered to each customer and how logistics operations should be organized to carry out these deliveries. Coelho et al. (2012) noted that despite their decision power is delegated to supplier, the customers still benefit from VMI’s due to reduced ordering costs. The manufacturer’s quest for better decision making in terms of production and routing decisions under a VMI policy leads to an integrated operational planning approach which is often referred to as the production routing problem (PRP).
PRP is the problem of simultaneous optimization of several planning decisions to seize the benefits of coordination (Adulyasak et al., 2014). Owing to the VMI policy, applications of PRP are diverse and they can be found in many industries (see Andersson et al, 2010;
Dauzere-Peres et al.,2007).
The classical PRP introduced by Bard and Nananukul (2009a) can be described as follows. The supply chain network is made up of a production plant and several retailers.
The plant and a retailer have storage areas such as warehouses, where finished products are stored. In every period, the demand at every retailer must be met by a supplier. The plant, in each period, must decide whether the product should be produced or not, and then determine the size of the lot. If the production takes place, then a fixed setup cost and unit production costs are incurred. It is not possible for the lot size to be in excess of the production capacity. A limited number of capacitated vehicles make deliveries from the plant to the retailers while incurring routing costs. There are unit inventory holding costs, in case products are stored and hold for future use at the plant or at the retailers. When the setup and production costs are neglected, the problem reduces to inventory routing problem (IRP), which is the special case of PRP (Adulyasak, 2012).
Up to now, various PRP models have been introduced. They are subject to different
production, inventory and distribution characteristics that obviously affect the decision-
making as described above. Most of these models assume that the customer demand is
known in advance. In the literature, these are referred as deterministic PRPs. However,
such deterministic models can end up with high costs and cause wrong decision-making
due to the assumption of certainty in the customer demand that is not coinciding with the
reality. The uncertainty of the demand is of paramount importance and has to be taken into
consideration in the decision-making process. In that respect, this paperwork concentrates
on the counterpart of the deterministic PRP referred to as stochastic PRP (SPRP). As
in real-life, in the SPRP a customer demand appears unknown in each period, but it is
expressed by a specific probability distribution. Because of the uncertainty in the demand,
the supplier sometimes unable to fully satisfy a customer demand on time. In this case, the
customer demand is backlogged. The backlog can be considered as amount of demand that
is not satisfied within a specific time period. For each unit of unsatisfied demand the supplier
incurs backlog costs, which can also be deemed as penalty costs. In this respect, we will consider the SPRP with backlogs. Adulyasak et al. (2012) have proposed two formulations for a stochastic PRP (SPRP), the multi-stage SPRP (M-SPRP) and the two-stage SPRP (2-SPRP). The difference between these formulations is in the time at which the actual demand become known. This paper will focus on the two-stage SPRP of Adulyasak et al.
(2012), in which the vehicles are assigned to the customers and routing tours and production setups are set in the first stage and in the second stage decisions regarding an inventory, production as well as delivery quantities are made.
As the classical PRP, the 2-SPRP does not consider some features that is crucial for ensuring proper routing service quality. The reason for disregarding the features is the VMI policy which the 2-SPRP is based upon. This policy at times may lead both customer and a supplier to certain inconveniences. For instance, consecutive small deliveries to the customer can be followed by very large ones, which could lead the customer to end up with not being visited by the supplier for a very long period of time. In real-life situation, this occasion is referred as inconsistency in the delivery quantities and it is negatively perceived by customers. The reason is, they can face with frequent and time-consuming visits when the deliveries are small and with congested warehouse when the deliveries are large. Another example appear to be the imbalanced loading of delivery vehicles that is regarded unfavorable by the supplier. Some delivery vehicles in this regard dispatched from the station nearly full while others nearly empty. As a result the drivers are annoyed due to the uneven distribution of loads for the vehicles.
To cover the aforementioned examples Coelho et al. (2012) have proposed several con- straints with capability to increase the routing service quality, while guaranteeing cost- effective solutions. In this respect, this paperwork aims to integrate the constraints of Coelho et al. (2012) into the 2-SPRP of Adulyasak et al. (2012) with a view to enhance the quality of the routing decisions. From a practical viewpoint, being in line with service standards is important, as it determines the quality of a service provided to customers by suppliers. The suppliers should not focus only on cost-optimal solutions, but rather con- sider high service standard as well and that could be implicitly achieved by incorporating the service constraints into the SPRP model. An influence of the service standards on pro- duction, delivery and routing decisions under demand uncertainty is not known yet, since they have not been implemented in the SPRP before. Moreover, the combination of several service standards have not been tested before in terms of their influence on the total cost.
Unfolding potential side-effects of them is important because in a dynamic environment, where customers demand can fluctuate sharply, ensuring two service standards at a time on a long-term basis can turn out to be too costly. In previous works the standards also have not been studied extensively. Trade-offs that can arise because of the standards remained unknown even in the deterministic context.
It is publicly known that integration of the service standards into the SPRP can dramat-
ically increase a problem size and thus computation time. The basic SPRP itself requires
a long time for computation of the cost-optimal solution mainly due to the scenario tree
approach behind it. As per this approach, various realizations of random demands are sepa-
rated into branches and corresponding probability is assigned to each branch. Each branch
represents a possible demand scenario. More demand scenarios allow obtaining better re-
sults, meanwhile negatively reflecting on the performance of models. In other words, the
performance of the SPRP becomes much worse if used instances are large in terms of num-
ber of time periods, scenarios or customers. Hence, a challenge is to ensure the extended
problem can be solved to optimality for both small and large instances and in a reasonable
amount of time. To achieve this, in this paper we proposed a different approach that allows to have SPRP model which is much more efficient in terms of performance due to sharp reduction of the problem complexity. The source of reduction is the absence of demand scenarios that were mentioned before. This approach also has been successfully integrated into various stochastic lot-sizing models and proven to be quite efficient in terms of better performance.
This paper mainly makes four contributions to the literature of SPRP and service stan- dards. First, we present a far efficient MIP model for the SPRP. It is based on the novel approach, which has not been implemented in the SPRP before. Moreover, the proposed MIP model is an exact method for a PRP where a customer demand follows normal dis- tribution, which is actually often the case in reality. Second, we show that our proposed MIP model can also be a good heuristic for the SPRP despite the fact they are based on different approaches. Third, we integrate two service standards such as vehicle filling rate and quantity consistency into our proposed MIP model. These standards have not been worked out in stochastic context in previous works. We demonstrate how the standards influence not only total cost but also total demand backlog cost. Moreover, we show how the standards are affected by different degrees of demand uncertainty. Fourth, we demon- strate trade-offs and effects of the standards when they are maintained together. To the best of our knowledge, in previous works, the combination of the standards have not been tested. Moreover, in those works trade-offs that can arise as a result of maintaining the service standards together and separately have not been discussed. The rest of this paper is structured as follows. In chapter 2, an extensive review on relevant literature is presented.
In chapter 3, the methodology is described, extended SPRP model is shown and explained
in details. In chapter 4, the approach behind our proposed MIP model is presented and
discussed. In chapter 5, the results of numerical analysis are shown and discussed. In chap-
ter 6, the limitations, further research and implications are demonstrated and followed by
a conclusion in chapter 7.
2 Background and Literature review
The aim of this chapter is to review t7e extant formulations of both stochastic and determin- istic PRP in order to show what have been explored in these fields and further to present the missed aspects. It is also necessary to link these missed facets with routing service standard features. Therefore, the literature review for proposed research is structured as follows. In subsection 2.1, the variety of deterministic PRPs in terms of characteristics are reviewed. In subsection 2.2, stochastic PRP under uncertain demand is given and work that have done in this field is explained in detail. In subsection 2.3, the background of the Coelho et al.
(2012) ideas is demonstrated. In subsection 2.4, the main insights from literature review are presented and linked with research.
2.1 The Classical PRP
As emphasized in the introduction, a PRP has variety of characteristics mainly related to the inventory policies, production and vehicles. Archetti et al. (2011) introduced an extensive discourse on the two common inventory policies that are commonly used both by supplier and customers. These policies provide specific guidelines and include rules for the proper control of inventory. To be more specific, Archetti et al. (2011) proposed the PRP under the “maximum level” (ML) as well as the“order up to” (OU) policies. Under the ML policy, the delivery quantities are allowed to be at any non-negative value, but upon a condition that they do not exceed maximum inventory level at each customer. On the other hand, under the OU policy, the customer inventory has to be replenished up to a specified stock level by the supplier.
The scope of PRP formulations is beyond the inventory policies. Several studies have been focused on the production characteristics such as number of plants, variety of prod- ucts and production limitations. Ruokokoski et al. (2010) have implemented different re-formulations of lot sizing for the PRP. They were characterized by uncapacitated pro- duction of a single product and one uncapacitated vehicle. That is, a supplier was able to produce a certain product as much as needed and a vehicle used for the distribution could handle deliveries at any quantity. Qiu et al. (2018) and Li et al. (2019) have extended the above formulation to multiple products, where the supplier could produce the different types of products unless a lot-size did not exceed the capacity. The focus of Archetti et al. (2011), on the other hand, was on the PRP using the uncapacitated production and one capacitated vehicle, while introducing a number of valid inequalities for the solution of the problem. The main difference between the formulations of Ruokokoski et al. (2010) and of Archetti et al. (2011) was in the vehicle capacity. In the latter one, a vehicle could be loaded up to the maximum loading capacity, whereas the former one did not pose any restrictions for the vehicle in terms of load level.
As one can see, the deliveries to the customers are done by one or multiple vehicles
which are either capacitated or uncapacitated. These kind of vehicle setups are referred to
the vehicle characteristics. They are mainly differentiated by a vehicle fleet that could be
either homogeneous or heterogeneous. In all aforementioned studies, the vehicle fleet was
treated as homogeneous. In other words, all the delivery vehicles have the same features
in terms of the capacity and the transportation costs. However, the Lei et al. (2006) have
developed the PRP with limited heterogeneous fleet, where the certain number of vehicles
had own capacity as well as operating costs. Moreover, unlike other formulations, number
of production plants were multiple capable to produce only one type product. Likewise,
the PRP with heterogeneous vehicle fleet was recently studied and extended in terms of inventory characteristics by Ji et al. (2018) and Miranda et al. (2018).
The main issue of the PRP appears to be its complexity as it extended upon the vehicle routing problem (VRP), proven to be regarded as a NP-hard problem (see Dror ,2008).
Formally, this implies that there does not exist any polynomial time algorithm to solve the model. The number of steps required to address the problem increases as its size upgrades. In other words, the computation time can grow exponentially with respect to the size of problem. In this respect, in order to solve the problem in a reasonable time several solution methods had been employed. These methods range from the various metaheuristics (see Boudia et al., 2007; Bard and Nananukul, 2009a) to the heuristics (see Bard and Nananukul,2009b; Archetti et al. 2011; Qiu et al. ,2018) ones.
2.2 The Stochastic PRP
Unlike the PRP under deterministic demand, the stochastic one has not been addressed ex- tensively. It was only introduced by Adulyasak et al. (2012), where the two-stage (2-SPRP) and the multi-stage SPRP (M-SPRP) under demand uncertainty have been formulated. The difference between M-SPRP and 2-SPRP formulations is, in the latter one actual demand for the whole planning horizon becomes known after routing decisions are obtained. In other words, the decisions of production, inventory and delivery quantities for each time period are done at one go. On the other hand, in the M-SPRP recognition of the actual demand for a certain stage relies on the decisions that had been made in the previous stages. These models have been extended on the basis of deterministic PRP’s formulations that were stud- ied by Archetti et al. (2011) and Solyali and Sural (2011). They are characterized with the capacitated production, the ML inventory policy and the homogeneous vehicle fleet size.
Moreover, unlike the others, in the 2-SPRP and the M-SPRP the stock-out cost has been incorporated due to presence of variability in demand and possibility of not satisfying it. In both formulations, the first stage seeks to determine routing decisions including assigning vehicles to customers and the routes to be followed by the vehicles. These decisions also regarded as “here and now” decisions. The reason is that these decisions are made before information regarding the stochastic demand data has been realized. In successive stage(s), demand is realized and the decisions related to inventory, lot sizes and delivery quantities are done. In the 2-SPRP and the M-SPRP, a customer demand follows scenarios approach.
Each scenario represents the possible realization of demand. More scenarios allow to obtain better results but at same time they dramatically increase complexity of the problems.
Despite the fact that the SPRP has not been studied sufficiently, lots of work undertaken to address the specific case of it which is known as the stochastic IRP (SIRP). It is worth to review this literature in order to gain more insights about the SIRP, that holds similarity to the SPRP. Federgun and Zipkin (1984) have first developed the SIRP, where a planning horizon was set as single period. Jaillet et al. (2002) have extended this model to the multiple periods, where each customer has been charged with a fixed delivery cost and the same pattern of distribution over the whole planning horizon has been used. On the other hand, Kleywegt et al. (2002) implemented the SIRP with discrete time periods over the infinite planning horizon and direct delivery policy where a single customer was visited appearing to be on the same route. This formulation was extended by Kleywegt et al.
(2004) by approach that several customers could be visited in each vehicle route. The SIRP
formulations with a finite-horizon and discrete time have been tackled upon by Bertazzi
et al. (2011) and Solyali et al. (2012). Both formulations were characterized with the OU
inventory policy and a single capacitated vehicle. Moreover, when a customer demand could not be satisfied, a penalty cost was incurred. The difference between them can be outlined as follows. In formulation of Solyali et al. (2012) the backlogs were allowed, that is an unsatisfied demand has been compensated in later periods. In contrast, in formulation of Bertazzi et al. (2011) the stock-outs were allowed. That is, an unplanned and an extra delivery to the customer is arranged to compensate the unmet demand. The SIRP with stock-outs was also studied by Bertazzi et al. (2013), where a single capacitated vehicle visits all the customer in each period.
Hvattum and Lokketangen (2009) noted that problems with stochastic parameter(s) appear to be more complex and more time-consuming. To handle the 2-SPRP and the M-SPRP, Adulyasak et al. (2012) has implemented both the Benders decomposition (BD) approach and a branch-and-cut (BC) algorithm. They illustrated that for the large number of scenarios the BD approach performs better than the BC algorithm in terms of perfor- mance and computation time. Moreover, almost in all numerical tests the optimality gap in case of BD approach was lower than the BC algorithm. There is also another approach to handle stochastic problems with uncertain demand, which allows to have scenario free model, unlike the previous one. It is based on piecewise linear approximation of a non-linear function which represents expected backlog. Rossi et al. (2014) had a comprehensive dis- cussion on the piecewise linear approximation of standard normal first order loss function, where a continuous variable followed normal distribution. The outcome of their study was several constant parameters, which allow to linearize the loss function independently of pa- rameter of Normal distribution and to integrate it into MIP models. This approach has been successfully applied to resolve different types of stochastic lot-sizing problems. The performance of it turned to be efficient for both small and large instances in terms of com- putation time (see Rossi et al., 2015). We will go in details of this approach in the chapter 4.
2.3 Service standards
Coelho et al. (2012) have firstly introduced a number of features that constituted the concept of consistency to represent some generic service standards in terms of quality. As Francis et al. (2007) and Coelho et al. (2013) noted, these standards reflect number of issues frequently arising around manpower and service management. All features of Coelho et al. (2012) are introduced and summarized in the table 1.
Feature Description
1. Quantity consistency This feature states that the size of all deliver-
ies undertaken to a customer must lie within
certain intervals in order to avoid large fluctua-
tions. Presence of the intervals regarding the de-
livery quantities allows to prevent frequent visits
to the customer which is usually caused by the
too small deliveries. It also allows to avoid too
large deliveries which can result in a congested
warehouse on the customer side.
2. Vehicle filling rate The possibility of using a vehicle takes place when the filling rate falls within a certain inter- val. This feature also can allow to get rid of small loads associated with vehicles.
3. Order-up-to (OU) policy This policy states that whenever a customer is paid a visit, the inventory capacity of the cus- tomer must be filled up to a predefined stock level
4. Driver consistency: This requirement is an indication that one driver is assigned to each customer.
5. Driver partial consistency This feature also implies that a customer should be visited by the same driver. However, it some- times allows for some drivers not to visit the cus- tomer which was assigned to them.
6. Visit spacing There is imposition of a minimum and a maxi- mum interval between two successive visits has to be made to the same customer.
Table 1: Consistency Features.
The features 2,4,5 and 6 are concerned with routing standards, whereas 1 and 3 are related to inventory policies and delivery quantity decisions. For instance, the quantity consistency feature can prevent large fluctuations in the delivery quantities. The amount of products supplied to customers will be stable as they will not receive too large or too small deliveries on successive time periods. We can illustrate this feature as follows. Assume that in one period a supplier delivers 30 units and in the next period 10 units. In total over two periods supplier will deliver 40 units. If customer imposes a restriction on minimum delivery quantity as 20 units and maximum 25, this would lead the supplier to deliver every period at least 20 and at most 25 units. To achieve this supplier for instance can deliver 21 units in the first period and 24 units in the second. As one can see, the supplier still delivers 40 units, but this time the difference between quantities of consecutive deliveries is 3 units which is much less than 20. Consequently, the issues such as a congested warehouse or stock-out will not be experienced by the customers. Another example is the vehicle filling rate, capable to prevent the unbalanced load amongst the delivery vehicles. As a result, a supplier will able to have the vehicles for the delivery with roughly equal loads and the drivers will not be irritated due to the imbalanced loading. Likewise, on the occasion that the supplier has single vehicle for deliveries, this feature will ensure the vehicle is not weakly loaded. We can illustrate this feature for two delivery vehicles as follows: assume that one vehicle is loaded 90% of its capacity, whereas another one 20% of its capacity. By initializing the vehicle filling rate standard and setting the vehicle load to minimum as 50%, the supplier will have, for instance, two vehicles where each of them is loaded 55% of their capacity.
Up to now these features, except for the OU policy studied by Archetti et al. (2011)
and Solyali and Sural (2009) have not been used and studied for the classical PRP. In
case of the SPRP formulation, even the OU policy has not been incorporated. The only
formulated SPRP was based on the ML policy. However, they have been incorporated into
the IRP formulations of Archetti et al. (2007) and Bertazzi et al. (2002), where a single
vehicle made visits to all customers. It was shown that all these features could not be used
at the same time due to their inherent nature. For instance, the features 1 and 6 require
demand stability. Worth remembering that these two features are about delivery and visit time consistencies. When the demand has large variations, a customer will expect that visit times will be variable as well. Consequently, the consistency in deliveries and visit times will make less sense. Coelho et al. (2012) also noted that for a certain parameter values, the combination of 1 and 2 can yield an infeasible solution. Moreover, the OU policy cannot be enforced if one of the features like 1,2 or 6 is incorporated into the model. The last example is related to the features 4 and 5, where the former one is much stronger than the latter one and the use of their combination is meaningless. This implies that incorporation of these features rely not only on the scope of the research, but also require compatibility with each other.
2.4 A discussion of literature
Based on the review of existing formulations of the PRP, it seems clear that the contributions in this field have been primarily focused on addressing the different inventory policies, such as OU and ML, and vehicle characteristics. This explicitly points that the aim of these studies was to capture the essential aspects and elements of real-world supply chains. The literature review on SPRP, evidences that PRP under demand uncertainty has not been extensively studied. The cause of it could be that the SPRP has only been explored by Adulyasak et al. (2012). In contrast, the SIRP has been studied more widely. Nonetheless, a direction of the works undertaken in relation to SIRP have turned out to be the same as in the aforementioned PRPs. The reason is, in these works also the inventory policies and vehicle characteristics have been addressed.
The two SPRP formulations of Adulyasak et al. (2012) have allowed to increase the accuracy and quality of the production decisions in comparison to the deterministic PRPs.
However, the quality of routing decisions still has not been improved in both formulations.
The reason is that the 2-SPRP and the M-SPRP formulations have the same routing aspect.
Consequently, this paperwork will focus on the 2-SPRP due to the absence of difference on the routing aspect. This model lacks the high routing service quality as all the other models emphasized in subsection 2.1 and 2.2. In real-life, decisions based on the 2-SPRP may cause inconveniences between a supplier and customers. These inconveniences can shift up due to the large fluctuations in the deliveries and irregularity of services.
Nevertheless, from subsection 2.3, it appears that the features proposed by Coelho et
al. (2012) can indeed enhance the quality of routing decisions. The description of these
features unveils that only two features, the vehicle filling rate and the quantity consistency
can be incorporated. The reason is, both of them cover possible inconveniences between the
supplier and customers described above. Furthermore, they can be incorporated together in
the SPRP model without raising compatibility issues, which is often the case for the other
combination of features. Despite the fact that these service standards have been addressed
in the IRPs, it is still interesting to integrate them in the SPRP for 2 reasons. The first
is that these service standards have never been tested in stochastic context and thereby
their effect on the amount of unmet demand and total cost still remains unknown in this
context. It would be interesting to see how degree of demand uncertainty affects them and
to identify which of the standards can be maintained effectively for particular degree of
the uncertainty. The second reason is that, these service standards have not been tested
together and thereby effects of such combination of standards on total costs is not known
yet. It is important to investigate because by maintaining two standard at the same time
the effect on total cost may turn to be very large. Lastly, in the previous works trade-offs
associated with the service standards were not shown.
It is obvious that incorporation of the service standards will increase the computational complexity. In subsection 2.2 it was shown that Benders Decomposition approach is efficient in the reduction of the computational complexity of SPRP with large number of scenarios.
Another approach was also introduced that is based on approximation of the first order loss function by means of some constant parameters. It has not been adapted in SPRP before, but it was proven to be quite efficient in the stochastic lot-sizing problems. It can allow to model SPRP without presence of a scenario tree, which is one of the main sources of high complexity. In this respect, in this paperwork we are going to extend the SPRP and to implement the linear approximation approach. We would like to remove the scenario tree from the SPRP and ensure the extended problem could be solved in a reasonable amount of time.
3 Methodology
The paper strives to extend the stochastic production routing model of Adulyasak et al.
(2012) by means of the service constraints of Coelho et al. (2012). Moreover, we intend to reformulate our extended SPRP model by incorporating the loss function into it and removing scenario tree concept from the SPRP. The extended model will be compared with the existing formulation in order to demonstrate changes in computational performances.
Moreover, we will perform several experiments to show changes in total costs due to the enhancements in quality of routing decisions. To achieve this aim, the mixed integer pro- gramming (MIP) model will be developed. The methodology section is structured as follows.
In subsection 3.1, the formulation of 2-SPRP of Adulyasak et al. (2012) is demonstrated.
In subsection 3.2, the extended SPRP model is presented. In subsection 3.3, the extended and scenario free SPRP model is presented.
3.1 Stochastic PRP model
In the stochastic PRP the demand can take a value with certain probability from the limited
set of demand scenarios. These scenarios follow the concept of scenario tree, which separates
various realizations of random demand into branches and assigns corresponding probability
to each branch. The simple scenario tree with 3 branches can be illustrated as follows.
Demand of a customer
Low 0.25
Medium 0.5
High
0. 25
Scenario 2
Scenario 3 Scenario 1
Figure 1: Scenario Tree with 3 branches
Let the index ω denotes possible scenario from the Ω which represents the set of all demand scenarios. Then, the probability of each scenario is p
w.
The 2-SPRP network is defined on the undirected graph G = (N, E), where E{(i, j) : i, j ∈ N, i < j} represents a set of vertices and N a set of supplier and customers indexed by i ∈ 1, .., n. To be more specific, the supplier is denoted as N
0, whereas a set of customers as N
c= N/{0}. Let S ⊆ N to be a given set of nodes and E (i, j ∈ S) set of vertices.
Likewise, let δ{S} to be a set of vertices adjacent to before defined set of nodes S.
Let also denote T = {1, . . . , t} as a finite set of time periods over which single product is produced at the supplier and delivered to a customer by the set of homogeneous vehicles K = {1, . . . , k} with capacity Q. If a vehicle travels from node i to j then the transportation cost c
i,jis incurred. The demand of customer i in time period t under scenario ω is defined as d
i,t,w.If the supplier has not sufficient inventory to satisfy all demand of customer i then a unit cost θ
iis incurred. At the beginning of planning horizon the inventory available at the customer i and the supplier is defined as I
i,0and I
0,0respectively. Likewise, note that they equal to I
i,0,wfor each ω ∈ Ω and i ∈ N , because the information about the level of initial inventory at each customer and the supplier is known in advance. The production cannot exceed capacity of C in every period. When the production takes place, a unit production cost u and a fixed setup cost f are incurred. Inventory cannot be more than inventory capacity L
iboth at the supplier and the customers. The supplier and the customer incur a unit inventory holding cost h
iper period. Lastly, let M
t,w= minC, P
Tt=1P
i∈Nc
d
i,t,ωand M
i,t,w0= minL
i, Q, P
Tt=1
d
i,t,ωfor every time period and scenario.
In order to formulate the 2-SPRP the following decision variables have been used:
1. x
i,j,k,t: binary variable equals to one if vehicle k travels from node i to j in period t.
2. z
i,k,t: binary variable equals to one if in period t vehicle k visits node i.
3. y
t: binary variable equals to one when production takes place in period t.
4. P
t,w: the quantity of product that is produced by the supplier in period t under scenario ω.
5. I
i,t,ω: inventory level at node i at the end of period t.
6. q
i,k,t,ω: the amount of product that should be delivered to customer i in period t by vehicle k under scenario ω.
7. e
i,t,ω: the amount of unmet demand in period t at customer i under scenario ω.
min X
t∈T
X
ω∈Ω
ρ
ω(uP
tω+ X
i∈N
h
iI
itω+ X
i∈Nc
θ
ie
itω) + f y
t+ X
(i,j)∈E
X
k∈K
c
ijx
ijkt(1)
subject to
I
0,t−1,ω+ P
t,ω= X
i∈Nc
X
k∈K
q
i,k,t,ω+ I
0,t,ω∀t ∈ T, ∀ω ∈ Ω (2)
I
i,t−1,ω+ X
k∈K
q
i,k,t,ω+ e
i,t,ω= d
i,t,ω+ I
i,t,ω∀i ∈ N
c, ∀t ∈ T, ∀ω ∈ Ω (3)
I
0,t,ω≤ L
0∀t ∈ T, ∀ω ∈ Ω (4)
I
i,t,ω+ d
i,t,ω≤ L
i∀i ∈ N
c, ∀t ∈ T, ∀ω ∈ Ω (5)
P
t,ω≤ M
t,ωy
t∀t ∈ T, ∀ω ∈ Ω (6)
X
i∈Nc
q
i,k,t,ω≤ Qz
0,k,t∀k ∈ K, ∀t ∈ T, ∀ω ∈ Ω (7)
q
i,k,t,ω≤ M
i,t,ω0z
i,k,t∀i ∈ N
c, ∀k ∈ K, ∀t ∈ T, (8)
∀ω ∈ Ω X
k∈K
z
i,k,t≤ 1 ∀i ∈ N
c, ∀t ∈ T (9)
X
(j,j0)∈δ(i)
x
j,j0,k,t= 2z
i,k,t∀i ∈ N, ∀t ∈ T, ∀k ∈ K (10)
X
(i,j)∈E(S)
x
j,j,k,t= X
i∈S
z
i,k,t− 2z
i,k,t∀S ⊆ N
c: |S| ≥ 2,
∀e ∈ S, ∀t ∈ T, ∀k ∈ K
(11)
e
i,t,ω, p
t,ω, I
i,t,ω, q
i,k,t,ω≥ 0 ∀i ∈ N, ∀ω ∈ Ω, ∀t ∈ T, ∀k ∈ K (12)
y
t, z
i,k,t∈ {0, 1} ∀i ∈ N, ∀t ∈ T, ∀k ∈ K (13)
x
i,j,k,t∈ {0, 1} ∀(i, j) ∈ E : i 6= 0, ∀t ∈ T, ∀k ∈ K (14)
x
0,j,k,t∈ {0, 1, 2} ∀j ∈ N
c, ∀t ∈ T, ∀k ∈ K (15)
The equation (1) represents the objective function which seeks to minimize total transporta-
tion costs of the first stage decisions and expected total setup, production and inventory
costs of the second stage decisions. Constraints (2)-(6) correspond to the lot-sizing part.
For each scenario the inventory flow balance is enforced by constraints (2) as well as (3).
Constraints (4) and (5) ensures that inventory level at the supplier and customers does not exceed capacity. Constraint (6) ensures non-negativity of production quantity when pro- duction setup is made. Constraints (7-12) impose routing tours and restrictions regarding the loading of vehicle. Constraint (7) ensures that vehicle capacity is not violated, whereas Constraint (8) allows non-zero delivery quantities only when customer i is visited by vehicle k in period. Constraint (9) makes sure that in each period t customer i is not visited with the same vehicle more than once. Constraint (10) ensures that number of adjacent edges equal to 2 if the customer i is visited by vehicle k in period t. Constraint (11) is applied in order to eliminate sub tours. The remaining constraints (12)-(15) enforce integrality and ensure that decision variables are not negative. Lastly, it should be mentioned that con- straints (2-8) corresponds to the second stage decisions, whereas constraints (9-11) to the first stage.
3.2 Extended SPRP model
The extend model is based on two particular ideas of Coelho et al. (2012), which related to the vehicle filling rate and the quantity consistency. In order to integrate the former one, it is necessary to define a new parameter which will serve as minimum vehicle loading level.
Let denote β as filling rate. A vehicle will be dispatched only if it is filled at least β percent of the vehicle capacity.To ensure that the above condition holds for all delivery vehicles and loads among them are balanced every period, the following constraint need to be added in the model.
X
i∈Nc
q
i,k,t,ω≥ βQz
0,k,t∀k ∈ K, ∀t ∈ T, ∀ω ∈ Ω
On the other hand, an interval [g
l, g
u] should be defined to incorporate the quantity con- sistency into the model. This interval can be considered as a lower and upper limit for the delivery quantities. Likewise, it is necessary to determine the average demand of a customer over the whole planning horizon to ensure that the delivery quantities are always within the intervals. Thus, let denote p and ω as a total number of time periods and scenarios respectively. Consequently, the following constraints have to be added:
q
i,k,t,ω≥ g
lz
i,k,tP
ω∈Ω
P
t∈T
d
i,t,wpω q
i,k,t,ω≤ g
uz
i,k,tP
ω∈Ω
P
t∈T
d
i,t,wpω
However, the lower bound for delivery quantities sometimes can be violated in stochastic
models if the value of g
lis high. In this case, we assume that a supplier has two options
either to not satisfy minimum delivery quantity of customer i and incur penalty cost κ
ior to initialize outsourcing by incurring subcontract cost ν. The upper-bound for delivery
quantities also sometimes can be violated. In this case, the supplier incurs also penalty cost
κ
i. Moreover, when a vehicle is not filled at least β percent of its capacity, then the supplier
incurs cost α. To model before mentioned cases, it is necessary to define five additional
decisions variables which are as follows.
8. RL
i,t,ω: the amount of unmet minimum delivery quantity in period t at customer i under scenario ω
9. RU
i,t,ω: the amount of delivery quantity that exceeded g
uin period t at customer i under scenario ω
10. S
i,t,ω: the extra amount of product produced by the supplier for a customer i in period t under scenarios ω
11. v
i,t,ω: binary variable equals to one if extra production takes place for customer i in period t under scenarios ω
12. B
k,t,ω: the amount of unmet minimum load for vehicle k in period t under scenario ω .
The integration of before mentioned decisions variables into the base model will yield the following extended model:
min X
t∈T
f y
t+ X
(i,j)∈E
X
k∈K
c
ijx
ijkt+ X
ω∈Ω
ρ
ω(uP
tω+ X
i∈Nc
κ
i(RL
i,t,ω+ RU
i,t,ω)
X
i∈Nc
νS
itω+ X
i∈Nc
X
k∈Nc
αB
k,t,ωX
i∈N
h
iI
itω+ X
i∈Nc
θ
ie
itω) (16)
subject to (2),(4),(6),(8)-(15) I
i,t−1,ω+ X
k∈K
q
i,k,t,ω+ e
i,t,ω= d
i,t,ω+ I
i,t,ω− S
i,t,ω∀i ∈ N
c, ∀t ∈ T (17)
∀ω ∈ Ω
I
i,t,ω+ d
i,t,ω+ S
i,t,ω≤ L
i∀i ∈ N
c, ∀t ∈ T, ∀ω ∈ Ω (18) S
i,t,ω≤ M v
i,t,ω∀i ∈ N
c, ∀t ∈ T, ∀ω ∈ Ω (19) S
i,t,ω≤ M z
i,k,t∀i ∈ N
c, ∀t ∈ T, ∀ω ∈ Ω, (20)
∀k ∈ K X
i∈Nc
(q
i,k,t,ω+ S
i,t,ω) ≤ Qz
0,k,t∀k ∈ K, ∀t ∈ T, ∀ω ∈ Ω (21) X
i∈Nc
(q
i,k,t,ω+ S
i,t,ω) + B
k,t,ω≥ βQz
0,k,t∀k ∈ K, ∀t ∈ T, ∀ω ∈ Ω (22)
q
i,k,t,ω+ RL
i,t,ω+ v
i,t,ωM ≥ g
lz
i,k,tP
ω∈Ω
P
t∈T
d
i,t,wpω ∀i ∈ N
c(23)
∀k ∈ K, ∀t ∈ T, ∀ω ∈ Ω q
i,k,t,ω+ S
i,t,ω+ (1 − v
i,t,ω)M ≥ g
lz
i,k,tP
ω∈Ω
P
t∈T
d
i,t,wpω ∀i ∈ N
c(24)
∀k ∈ K, ∀t ∈ T, ∀ω ∈ Ω
q
i,k,t,ω+ S
i,t,ω− RU
i,t,ω≤ g
uz
i,k,tP
ω∈Ω
P
t∈T
d
i,t,wpω ∀i ∈ N
c(25)
∀k ∈ K, ∀t ∈ T, ∀ω ∈ Ω
S
i,t,ω, RL
i,t,ω, RU
i,t,ω, B
k,t,ω≥ 0 ∀i ∈ N, ∀ω ∈ Ω, ∀t ∈ T, ∀k ∈ K (26)
v
i,t,ω∈ {0, 1} ∀i ∈ N, ∀t ∈ T, ∀ω ∈ Ω (27)
The equation (1) and constraints (2),(4),(6),(8)-(15) have the same purpose as it was men- tioned in the previous subsection. The constraints (17), (18) and (21) are reformulated versions of the inventory flow balance, inventory and vehicle capacity constraints respec- tively. The constraint (19) and (20) ensures non-negativity of subcontracted quantity if a supplier chooses to subcontract some amount of units to satisfy the minimum delivery requirement and if the supplier visits the customer. The constraint (22) makes sure that a vehicle are sent off to the customers if they are filled at least β percent of its capacity.
The constraint (23) ensures that the delivery quantities to each customer over the whole planning horizon are at least g
lof the average demand if the subcontract option is not triggered. The constraint (24) has the same purpose as constraint (21), but it is switched on if the supplier opts for subcontracting some amount of a product. The constraint (25) ensures that the delivery quantities to each customer over the whole planning horizon are at most g
uof the average demand. The constraints (26) and (27) enforce integrality and ensure that new decision variables are not negative. Note that constraint (22) corresponds to the vehicle filling rate feature, whereas constraints (23) -(25) corresponds to the quantity consistency feature of Coelho et al. (2012).
4 Integration of the loss function in SPRP
In previous section, the presented SPRP was based on the scenario tree concept. In this case, a customer demand is split into several branches and to each branch corresponding probability is assigned. In other words, the customer demand follows the scenario approach.
An issue with this approach is that it makes the problem very complex and thereby leads
to long computation time. One of the ways to reduce complexity of the SPRP is to de-
velop a MIP model where the customer demand follows certain probability distribution
(e.g. Normal, Uniform and so forth) rather than large amount of scenarios. Bear in mind
that this MIP model and the SPRP model presented before are completely different prob-
lems. We will refer to this MIP model as a scenario free SPRP. In this paper we will assume
the customer demand follows normal distribution with parameters µ and σ. As Hopp and
Spearman (2008) noted, the assumption of normally distributed demand for products is
convenient in most cases and frequently used in practice by many organizations. To model
the scenario free SPRP we will use an approach which was extensively discussed by Rossi et
al. (2014). General idea behind the approach is to approximate the standard normal first
order loss function by several linear line segments. According to the Edward et al.(1998),
in inventory management the first order loss function is used to express expected inventory
holding or backlog costs. This approach has been extensively implemented in the stochastic
lot sizing problems, where a customer demand followed normal distribution. Remember, the
SPRP comprises both the lot sizing and vehicle routing problems. The Rossi et al. (2015)
applied this approach in the non-stationary stochastic lot sizing problem. They showed that
piecewise approximation approach allows to model various types of the stochastic lot siz-
ing problems, where non-stockout probability, fill rate constraints or backlog penalty costs capture quality of service. They also demonstrated that the piecewise approximation ap- proach has more pros compared to other methods existing in the literature. Moreover, all instances used in their computational study have been solved within a few seconds by this approach. Tarim and Kingsman (2006) also applied piecewise linear approximation in their work to deal with complexity of a non-linear cost function in the stochastic lot sizing prob- lem under non-stationary (R,S) policy. The outcome of their study was a MIP model which is equivalent to the non-linear model. Likewise, Tunc et al. (2016) applied this approach and extended stochastic lot sizing problem under (R,S) policy to capture piecewise linear concave ordering costs. Based on aforementioned it seems clear that the piecewise linear approximation approach has well-established literature and has been applied in different stochastic lot sizing problems.
Now, we will go in details of the approach to implement it in the SPRP. Instead of considering large amount of scenarios as it was the case in the aforementioned SPRP models, here we assume that a customer demand follows normal distribution. Remember, here we will focus on the expected value of holding and backlog costs. Recall, I
tvariable which we defined as inventory level of a customer at the end of period t. In the SPRP with scenario tree concept, this variable could take only positive real value, because it was bounded from below by zero. However, here I
tis unbounded from below and can take either positive or negative real value. When on-hand inventory at a customer more than demand, then I
ttakes the positive value I
t+, i.e. max{0,I
t}. In this case, we expect inventory on the customer side. When on-hand inventory is not enough to satisfy demand, then I
ttakes the negative value I
t−, i.e. max{0,−I
t}. Given normally distributed customer demand with parameters µ
tand σ
t, the expected backlog and holding costs in period t can be computed as follows (Tunc et al., 2016).
h
iI
t++ θ
iI
t−+ (h
i+ θ
i)σ
tL I
t++ I
t−σ
t(4.1)
In function (4.1), L(·) corresponds to the standard loss function. By means of probability density function φ(x) and cumulative distribution function Φ(x) of normal distribution, the L(·) can be expressed as follows.
L(υ) = −υ + φ(υ) − (υ)Φ(υ) (4.2)
Note that, L(·) is a non-linear function and thereby it cannot be directly embedded into our MIP model. However, this function can be linearized by the approach which was used in before discussed stochastic lot sizing problems. According to this approach, the L(·) can be piecewise approximated by a set of linear line segments LS
ajbj. This is possible because the L(·) is a continuous convex function. By constructing several line segments tangent to L(·), it can be easily approximated. Each of the line segments is characterised with a slope
−b
jand an intercept a
j. The Rossi et al. (2014) has a comprehensive discussion regarding piecewise linear approximation of the L(·). They also estimated values of the slope and the intercept up to 11 line segments. These two constant parameters can be used in linearisation of L(·) independently of the mean and standard deviation of the normal distribution. Keep in mind that more line segments lead to better approximation of the loss function. The linearized standard normal loss function ˆ L(υ) can be formulated as follows.
L(υ) = ˆ max
j=1,...,LSaj bj
h
a
j− b
jυ i
(4.3)
Suppose that in one period a customer has Normally distributed demand with µ = 15 and σ=5. The expected backlog for the L(υ) and ˆ L(υ) with 11 line segments can be graphically illustrated as follows.
0 5 10 15 20 25 30
−20
−15
−10
−5 0 5 10 15 20
Delivery Quantity
Exp ected bac klog
Figure 2. E[Backlog] for the L(υ) with 11 line segments and for the non-linear function L(υ) ˆ Linearized function
Non-Linear function
In order to integrate the ˆ L(υ) into our SPRP model, it is necessary to define a new decision variable which is as follows.
13. E
i,t: approximated amount of unmet demand in period at customer i in period t Note that, the value of E
i,tcan be negative when the inventory level at the end of a period more than a customer demand. We have to ensure the value of E
i,tis non-negative for each line segment in the set of LS
ajbj. In other words, we have to bound the E
i,tfrom below by zero. We can write υ in equation (4.3) in terms of delivery and subcontract amount in period t, inventory level in period t − 1 and a customer demand in period t. Recall that cumulative demand of the customer in period t is P
tj=1
µ
tand its standard deviation is q P
tj=1
σ
t2. Consequently, the constraint for the approximated standard loss function can be formulated as follows.
E
i,t≥ a
jv u u t
t
X
j=1
σ
2i,t− b
jh I
i,t−1+ X
k∈K
q
i,k,t+ S
i,t− µ
i,ti ∀i ∈ N
c, ∀t ∈ T, ∀j ∈ LS
ajbjNote that we can also rewrite E
i,tin more compact way in terms of I
i,t+and I
i,t−as it was the case in Tunc et al. (2016). Based on aforementioned, the constraint for approximated standard loss function in a compact version can been formulated as follows.
E
i,t≥ a
jv u u t
t
X
j=1
σ
i,t2− b
jh
I
i,t++ I
i,t−i
∀i ∈ N
c, ∀t ∈ T, ∀j ∈ LS
ajbjApart from adding aforementioned constraint into the model, we have to make sure that expected inventory at the end of period t is either positive or negative real value. In other words, I
t+and I
t−are not allowed to be non-zero at the same time. To model this, we add the following constraints that were also used in a stochastic lot sizing problem (Tunc et al., 2016).
I
i,t= I
i,t+− I
i,t−∀i ∈ N
c, ∀t ∈ T I
i,t+≥ I
i,t∀i ∈ N
c, ∀t ∈ T I
i,t−≥ −I
i,t∀i ∈ N
c, ∀t ∈ T I
i,t+, I
i,t−≥ 0 ∀i ∈ N
c, ∀t ∈ T
Remember, here we consider expected holding and backlog costs for each customer. Hence, we have to slightly change the objective function (1) that was presented in subsection 3.1.
Based on the function (4.1), we have to add in the objective function the following equation.
X
t∈T
X
i∈Nc
h
iI
i,t++ θ
iI
i,t−+ (h
i+ θ
i)E
i,tAll aforementioned steps characterizes our MIP model. The entire scenario free SPRP model with the service standards is presented as follows.
min X
t∈T
f y
t+ X
(i,j)∈E
X
k∈K
c
ijx
ijkt+ (uP
t+ X
i∈Nc
h
iI
i,t++ θ
iI
i,t−+ (θ
i+ h
i)E
i,t+
h
0I
0t+ X
i∈Nc
κ
i(RL
i,t+ RU
i,t) + X
i∈Nc
νS
i,t+ X
i∈Nc
X
k∈Nc
αB
k,t! (28)
subject to (9)-(11),(13)-(15) I
0,t−1+ P
t= X
i∈Nc
X
k∈K
q
i,k,t+ I
0,t∀t ∈ T (29)
I
i,t= I
i,t−1+ X
k∈K
q
i,k,t+ S
i,t− µ
i,t∀i ∈ N
c, ∀t ∈ T (30)
E
i,t≥ a
jv u u t
t
X
j=1
σ
i,t2− b
j(I
i,t++ I
i,t−) ∀i ∈ N
c, ∀t ∈ T, ∀j ∈ LS
ajbj(31) I
i,t= I
i,t+− I
i,t−∀i ∈ N
c, ∀t ∈ T (32)
I
i,t+≥ I
i,t∀i ∈ N
c, ∀t ∈ T (33)
I
i,t−≥ −I
i,t∀i ∈ N
c, ∀t ∈ T (34)
I
0,t≤ L
0∀t ∈ T (35)
I
i,t++ µ
i,t+ S
i,t≤ L
i∀i ∈ N
c, ∀t ∈ T (36)
S
i,t≤ M v
i,t∀i ∈ N
c, ∀t ∈ T (37)
S
i,t≤ M z
i,k,t∀i ∈ N
c, ∀t ∈ T (38)
∀k ∈ K
P
t≤ M
ty
t∀t ∈ T (39)
q
i,k,t≤ M
i,t0z
i,k,t∀i ∈ N
c, ∀k ∈ K, ∀t ∈ T (40)
X
i∈Nc
(q
i,k,t+ S
i,t) ≤ Qz
0,k,t∀k ∈ K, ∀t ∈ T (41)
X
i∈Nc
(q
i,k,t+ S
i,t) + B
k,t≥ βQz
0,k,t∀k ∈ K, ∀t ∈ T (42)
q
i,k,t+ RL
i,t+ v
i,tM ≥ g
lz
i,k,tP
t∈T
µ
i,tp ∀i ∈ N
c, ∀k ∈ K, ∀t ∈ T (43) q
i,k,t+ S
i,t+ (1 − v
i,t)M ≥ g
lz
i,k,tP
t∈T
µ
i,tp ∀i ∈ N
c, ∀k ∈ K, ∀t ∈ T (44) q
i,k,t,+ S
i,t− RU
i,t≤ g
uz
i,k,tP
t∈T
