A Classiﬁcation Framework for Time Stamp Stochastic Assignment

Problems

Kees Kooiman

, Frank Phillipson

2 a

and Alex Sangers

Erasmus University, Rotterdam, The Netherlands

Netherlands Organisation for Applied Scientiﬁc Research (TNO), Den Haag, The Netherlands

Keywords:

Assignment, Time-stamp Stochastic Assignment Problem, Classiﬁcation Framework, Simulation Algorithms.

Abstract:

In this paper Time-stamp Stochastic Assignment Problems are studied. These problems occur in places where

an incoming request or task has to be connected to a resource immediately. A deﬁnition and framework for

these problems is given, in which the different Time-stamp Stochastic Assignment Problems can be cate-

gorised. An explicit notation is introduced to distinguish the different categories. Several solution methods for

Time-stamp Stochastic Assignment Problems are listed and their advantages and disadvantages are discussed.

1 INTRODUCTION

A problem that often occurs in manufacturing and lo-

gistics is assigning resources (machines, stock, em-

ployees, etc.) to tasks whose arrival time or time order

of the occurrence is uncertain. When the task appears,

an immediate choice has to be made which resource to

assign to that task, no queues or buffers are allowed.

The assignment restricts future assignments to up-

coming tasks whose arrival times and types of which

are unknown. This type of real-time assignment of

scarce resources to an ongoing stream of stochasti-

cally appearing tasks deﬁnes a so-called Stochastic

Assignment Problem (SAP). Areas of application are:

• Maintenance or upgrade of networks

• Mechanic work (electricity, printers, coffee ma-

chines)

• Real-time assigning of customers to (multi-

skilled) servants

• Real-time assignment of tasks to machines

• Assignment of available kidneys to patients on a

waiting list

Finding optimal or near optimal solutions to an SAP

can be hard in practical applications. Planners may

have a good feeling for short term effects of their

decisions, however the often complex long term ef-

fects are hard to grasp for a human mind. Indeed, the

hard part of optimal decision making in an SAP is to

https://orcid.org/0000-0003-4580-7521

properly take into account how current assignments

inﬂuence the opportunity set for the assignment of re-

sources to uncertain future tasks appearing. Sophisti-

cated mathematical tools might therefore be useful to

help making more efﬁcient real-time assignment deci-

sions. These have to take into account the probability

distribution of future demands on available resources.

A distinction between two types of Stochastic As-

signment Problems is the following:

• Sequential Stochastic Assignment Problem

(SSAP): For these problems only the order of

events matter, not their exact times.

• Time-stamp Stochastic Assignment Problem

(TSAP): For these problems the event times

are an essential part of the speciﬁcation of the

problem instance.

For Sequential Stochastic Assignment Problems

the exact event times are not important, but only the

order of occurrence of events. One can think of pa-

tients on a waiting list for kidneys, where the kidneys

sequentially become available. The assignment de-

pends on the patient and kidney type. The assignment

of a kidney to a patient is made immediately upon

arrival of the kidney and an assigned kidney never

re-enters the pool of available kidneys. Under these

assumptions, exact times are irrelevant.

In the early study of (Derman et al., 1972), the Se-

quential Stochastic Allocation Model is introduced,

which is used to solve a Sequential Stochastic Assign-

ment Problem. This model will be called the DLR

(named after Derman, Lieberman and Ross) model in

Kooiman, K., Phillipson, F. and Sangers, A.

A Classiﬁcation Framework for Time Stamp Stochastic Assignment Problems.

DOI: 10.5220/0008871701290137

In Proceedings of the 9th International Conference on Operations Research and Enterprise Systems (ICORES 2020), pages 129-137

ISBN: 978-989-758-396-4; ISSN: 2184-4372

129

the remainder of this section. The model addresses

the assignment of n available men (resources) to per-

form n jobs. The jobs arrive in a sequential order and

the job types are independent and distributed identi-

cally. The objective is to ﬁnd an allocation between

the men and the jobs that maximises the total reward.

An assignment of the jobs to the available men is done

by creating intervals based on the distribution func-

tion of the type of the incoming jobs. The type of

the new incoming job falls in a certain interval and

is assigned to the corresponding man. After each as-

signment the intervals are recalculated. In (Derman

et al., 1975) the DLR model is used for an investment

problem.

Next, (Albright, 1974) investigates different ar-

rival distributions and discount functions for the DLR

model. In (Baharian Khoshkhou, 2014) practical vari-

ations and extensions of the DLR model are given. It

introduces the TSSAP, a target-dependent SSAP un-

der the threshold criterion, which attempts to min-

imise the probability of the total reward failing to

achieve a speciﬁed target value. A practical applica-

tion of the DLR model is the Aviation Security Prob-

lem described in (Nikolaev et al., 2007) and (McLay

et al., 2009). In the second stage of their sequen-

tial stochastic security design problem (SSSDP) they

model a policy of screening passengers that arrive at

a security station.

Another well known SSAP is the kidney alloca-

tion problem. This is described using the DLR model

by (David and Yechiali, 1995) and (Su and Zenios,

2005). A match has to be made between available or-

gans (e.g., a kidney) that will arrive sequentially and

the transplant patients on the waiting list. The type of

the patients are known, the type of the kidney is re-

vealed upon arrival. The reward of assigning a kidney

to a particular patient depends on both the type of the

kidney and the type of the patient.

The other category is the Time-stamp SAP, where

time plays an important role. The planning of nurses

in maternity care by (Phillipson, 2015) is an exam-

ple in this category. Here a maternity care agency

(MCA) has to plan nurses to help the mother when

a baby is born. The challenge is to assign the right

nurse to each upcoming demand, while taking into

account what the impact is for future decisions, un-

der uncertainty. To illustrate the difference between a

sequential and time-stamp problem we compare this

problem with the kidney allocation problem. In the

maternity care problem time plays an important role.

Here a time interval of a day is used. The delivery

dates follow a probability distribution based on days

and the mother needs care during the ﬁrst 10 days af-

ter birth. Then the resource (here nurse) re-enters the

pool or resources.

Other examples are the planning of known elec-

tive patients and unexpected emergency patients to

Surgery Rooms (SRs) as done in (Lamiri et al., 2009)

and assigning nurses to patients under stochastic sce-

narios as discussed in (Punnakitikashem et al., 2008).

Another example is the Online Ambulance Dispatch-

ing problem in, e.g., (Jagtenberg et al., 2015) and (Ji

et al., 2019), where it has to be decided which ambu-

lance to send when an incident occurs. (Jagtenberg

et al., 2015) look at a policy of sending an ambu-

lance, out of the subset of available ambulances, that

are within a target time of the incident. The core of

the Ambulance dispatching problem can be seen as

a Truckload Motor Carriers problem as discussed in

(Powell, 1996). It concerns the problem of assign-

ing drivers to pick up a load while the location and

magnitude of the demand is random. Other types

of TSAPs are weapon target assignment (Murphey,

2000), multi-robot task-allocation (Liu and Shell,

2011; Amato et al., 2015; Buckman et al., 2019),

and project portfolio management (Gutjahr and Re-

iter, 2010). Also the stochastic task assignment prob-

lems that are a special case of the dynamic vehicle

routing problem (DVRP) can fall under the Time-

stamp SAP, if no buffering occurs and the allocation

has to be done immediately on arrival. Examples here

are ride sharing (Simonetto et al., 2019), dial-and-

ride problems (Ho et al., 2018; Schilde et al., 2011)

and task assignment in spatial crowdsourcing services

(like Uber) (Chen et al., 2019; Cheng et al., 2017).

A slightly different type of problem is the

Stochastic Dynamic Generalised Assignment Prob-

lem (SDGAP) as described in (Kogan et al., 2005).

An example is the assignment problem in a copy cen-

tre at a university bookstore, having multiple ma-

chines of different types that can copy predeﬁned

ranges of page sizes. Each machine can carry out only

one task at a time so the agent resource is equal to

one. The task resources are deﬁned in the problem in-

stance. The SDGAP can be extended by dealing with

a multi-resource constraint. It differs from the gener-

alised assignment problem in that an agent consumes

not just one but a variety of resources in performing

the tasks assigned to him, as shown in (Gavish and

Pirkul, 1991). An extension to this Multi Resource

problem occurs when the resources are not being indi-

vidually capacitated per agent but collectively for all

agents, as can be found in (Toktas et al., 2004). As de-

scribed by (Albareda-Sambola et al., 2006) Stochastic

Generalised Assignment Problems can also be mod-

elled with a two stage recourse model. In stage 1 a

planning is made without stochastic information. In

stage 2 resources are reassigned to jobs when tasks

ICORES 2020 - 9th International Conference on Operations Research and Enterprise Systems

130

come in stochastically and incur penalty costs for cer-

tain reassignments.

In this paper we will focus on Time-stamp

Stochastic Assignment Problems. From the brief lit-

erature review we notice that SAPs can have many

different forms, but a general representation for var-

ious problems is lacking. To ﬁnd out what solution

method to apply, it is useful to have a categorised

framework that contains the different types of prob-

lems.

In the remainder of this paper we ﬁrst give a more

precise description of the TSAP. Then in Section 3

we zoom in on one of those two forms and propose a

framework for TSAP. Next, in literature used solution

methods are presented in Section 4. We ﬁnish with

some conclusions.

2 DEFINITION OF TSAP

The Time-stamp Stochastic Assignment Problem is a

stochastic assignment problem in which time is im-

portant. Because of this, not only the incoming se-

quence of demands is important but also the time

lapse between two incoming demand items. Next to

the time of arrival the demand type can also be known

or unknown. TSAPs can have a new incoming de-

mand item while one or multiple previous demand

items are still being processed and are still active in

the problem, i.e., parallel processing of demand items.

The serving time and/or arrival time of the items could

be stochastic and unknown. A good prediction of the

arrival time of incoming demand items and serving

time of serving items is desirable. Because stochas-

tic assignment problems have a very broad scope we

redeﬁne TSAPs as used in this paper. A Time-stamp

Stochastic Assignment Problem as elaborated in this

paper has the following deﬁnition. Let demand items

i be items with different types o

, which can be known

beforehand or revealed on arrival with a stochastic ar-

rival time τ

and let serving items j be items with

different known types q

with a deterministic serv-

ing time s

i j

(with i a demand item), where at least

one type of serving item has ﬁnite availability. Then

a Time-stamp Stochastic Assignment Problem is an

assignment starting at t = 0 and ending at t = T of

demand items to serving items.

The following assumptions are made:

Assumption 1. Finite time horizon 0 ≤ t ≤ T

Although this may not be entirely realistic we as-

sume that demand items arriving after T have negligi-

ble inﬂuence on the optimal allocation at t = 0.

Assumption 2. Assignments can not be retaken.

Choices made in the past cannot be undone. How-

ever after having completed its task, the serving item

may return in the pool of available serving items if

this is allowed in the problem. This re-entering pro-

cess is described in section 3.1.4.

Assumption 3. At least one serving item needs to be

available to serve an incoming demand.

The assignment of demand items to serving items

will be hard if there are insufﬁcient serving items

to fulﬁl the demand. The demand will get stacked

and has to wait until serving items become available.

When this happens the serving items need to be as-

signed to the demand items. This comes down to a

queueing system and as mentioned in the introduction

this is outside the scope of this study and that is why

the following collateral assumption is made.

Assumption 4. Inﬁnite serving items are available at

a penalty cost.

In a stochastic assignment problem the main prob-

lem is to make an assignment such that it is optimal in

the future; using/assigning a serving item now should

be optimal afterwards. If a serving item is assigned, it

can not be used during its serving time. A newly arriv-

ing demand item, that would have been a better match

with an already assigned serving item, may result in

a suboptimal solution. An important requirement for

the stochastic assignment problem is that the cardi-

nality of at least one of the types of serving items is

ﬁnite. If this is not the case, and the number of avail-

able serving items per type are inﬁnite the problem is

trivial and for each demand item the best correspond-

ing type of serving item can be chosen. This is trans-

lated in the following:

Assumption 5. The cardinality of at least one of the

types of serving items is ﬁnite.

A change in serving time has a minor inﬂuence on

the assignment problem. In a TSAP there is a low oc-

cupancy of serving items and there are no queues, de-

terministic serving times is a fair assumption. This is

why we assume the serving time of the serving items

are known and predeﬁned in the problem instance.

Assumption 6. The serving time is deterministic.

Another important aspect concerns the possibility

of a match between a speciﬁc type of serving item and

a speciﬁc type of demand item. Let M be a binary ma-

trix with rows corresponding to the different types of

serving items and the columns corresponding to the

different types of demand items, where M

i j

= 1 if de-

mand item type j can be served by serving item type

i. Matrix M should satisfy:

A Classiﬁcation Framework for Time Stamp Stochastic Assignment Problems

131

Assumption 7. Each column and row of M should

contain at least one 1.

The demand item type should at least be able to

be served by one of the serving item types. On the

opposite each serving item type should at least be able

to serve one of the demand item types.

Assumption 8.

∑

i j

≥ 2, for at least one i

If this is not the case each type of serving item

only has one possible type of demand item to serve

and the problem becomes trivial.

3 A CLASSIFICATION

FRAMEWORK

In this section, we derive a generic framework to clas-

sify TSAP applications in a uniform way. We intro-

duce an explicit notation to distinguish the different

categories. This not only allows us to classify TSAPs

unambiguously, but it also makes clear which char-

acteristics are common, where two problems look to-

tally different at ﬁrst.

The framework consists of four building blocks:

1. Demand arrival time;

2. Demand type;

3. Resource type;

4. Serving item feature.

Each block has a number of different options; in the

framework a speciﬁc TSAP is characterised by listing

the values these options take within each of the four

blocks.

First we discuss the different options observed in

practical TSAPs for each of the four building blocks.

Subsequently we introduce notation for these options

and present the framework.

3.1 Framework Elements

In this section a detailed description is given of the

four components that deﬁne the framework. The dif-

ferent options that can occur within these components

are deﬁned.

3.1.1 Demand Arrival Time

A TSAP consists of tasks, jobs, orders or other de-

mand items. Demand items arrive sequentially and

need to be assigned immediately to a serving item.

The arrival process is deterministic when the arrival

times are known beforehand. Deterministic arrival

times are scarce in TSAP applications but we do ad-

dress them in our framework. If the arrivals are un-

known or uncertain we have a stochastic arrival pro-

cess. In this case we need an arrival distribution and

its corresponding parameters (neglecting non para-

metric statistics). The distribution and its parameters

can be based on theory or estimated from historical

data. Ideally parameters could be updated during the

process by using Bayesian methods of parameter es-

timation, see, e.g., (Zellner, 1996).

3.1.2 Demand Type

The demand item type can either be known before-

hand or can be unknown or uncertain, in which case

it is treated as stochastic.

3.1.3 Resource Type

In a classical assignment problem an assignment is a

one-on-one assignment between a demand item and a

serving item. Based on the type of the demand item

and the type of the serving item an optimal allocation

is made. Another common assignment problem is

the generalised assignment problem. The difference

with the classical assignment problem is that the de-

mand item needs a speciﬁc number of resources and

the serving item has a speciﬁc number of resources

to contribute. If, for example, a demand item needs

8 pieces to be served, serving item a can provide 5

and serving item b the other 3. This is called an as-

signment problem with single item resource. It is also

possible the demand items need different items to be

served. As an example a demand item needs 8 blue

pieces and 6 red pieces, serving item a can provide 4

blue pieces and 4 red pieces and serving item b pro-

vides the other 4 blue pieces and 2 red pieces. This

is called a assignment problem with multi-item re-

source. In these generalised assignment problems the

serving items can individually or collectively be ca-

pacitated. In the latter example it could be the case

that serving item a is capacitated to only supply a

maximum of 4 red and 4 blue pieces. It could also

be that the serving items are collectively capacitated

meaning that all serving items together can supply a

capacitated number of resources of a speciﬁc type. In

our previous example this can mean that both serving

item a and b together have 8 blue pieces to supply.

Although both a or b could supply 8 blue pieces, a

combination is still possible.

3.1.4 Serving Item Feature

In a classic assignment problem a serving item can be

used once, and after the assignment it is no longer

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132

available for further assignments. Another serv-

ing item feature that is often part of a Time-stamp

Stochastic Assignment Problem is that after serving,

the serving item returns in the pool of ready to use

serving items and is again ready to be allocated to

a demand item. In these problems the serving time

plays an important role. In a few problems a serv-

ing item can switch to the new incoming demand item

while already serving another demand item as in the

maternity care problem, another serving item should

then take over from the switched serving item.

3.2 Classiﬁcation Scheme

In this section a classiﬁcation framework is provided

TSAPs. The basis of the framework is taken from

the classiﬁcation scheme of (Graham et al., 1979)

for scheduling problems. The framework established

in this paper uses the characteristics as the building

blocks deﬁned in the previous sections. Given an ap-

plication of a TSAP the components of the modelling

system specify the problem in the framework. To in-

dicate the qualitative characteristics of a Time-stamp

Stochastic Assignment Problem we propose a four-

position framework of the form α | β | γ | δ. The cor-

responding options are given below:

Demand Arrival Time Distribution. Options for α

are:

• det for deterministic arrival time

• stoch for stochastic arrival time

Demand Type Distribution. Options for β are:

• det for deterministic demand type

• stoch for stochastic demand type

Resource Speciﬁcation. Options for γ are:

• one for one on one allocation

• si-in for single item and individually capacitated

• si-co for single item and collectively capacitated

• mu-in for multi item and individually capacitated

• mu-co for multi item and collectively capacitated

Serving Item Feature. Options for ε are:

• once for when a serving item can only be used

once

• reass for serving items being re-assignable again

after service

• swit for when a serving item can switch to another

demand item during service

Part of the TSAP literature studied and summarised

earlier can be framed in the four position scheme as

in Table 1.

4 SOLUTION METHODS

In this section several solution methods for TSAPs

are elaborated and their advantages and disadvantages

are discussed. First we introduce the meaning of on-

line and ofﬂine optimisation and their relevance to the

study of TSAPs.

4.1 Optimisation Paradigms

In the ﬁeld of operations research a well-known and

commonly used distinction is made between ofﬂine

and online optimisation. In ofﬂine optimisation all

relevant information is known when solving the opti-

misation problem involved, i.e. there is no uncertainty

about any input data relevant to the problem. In on-

line optimisation the input data come in sequentially;

decisions have to be taken while part of the relevant

information is still lacking, since it will only become

available after the decision has been made. So in on-

line optimisation uncertainty about (part of) the rele-

vant parameters of the optimisation problem is essen-

tial. Only ex post, when all information has become

available, the truly optimal solution can be computed

ofﬂine. This solution forms a bound for the online so-

lution, based on the incomplete information available

when the decision had to be taken.

To account for the missing information, online

optimisation is based on an assessment of the pos-

sible future outcomes of the missing items. Actual

outcomes might deviate from these imputed expected

outcomes, rendering the online optimisation solution

sub optimal ex post. Indeed, only in the unlikely case

that all future realisations of the missing input items

happen to coincide with the imputed expectations, the

online solution will match the ofﬂine solution.

TSAP is a good example of an online optimisation

problem. The associated ex post ofﬂine optimisation

solution serves as a benchmark to check how well dif-

ferent solution methods of the TSAP perform.

4.2 Online Solution Methods

The literature review (summarised earlier) made clear

that a lot of different methods have been used in ﬁnd-

ing a solution for TSAPs. Because of the different

characteristics of TSAPs involved different solution

methods were suitable for solving these problems. In

this section an overview is given of the most com-

mon solution methods used. We start with (two stage)

stochastic programming followed by Markov chain

optimisation. Then a simulation method is explained

and ﬁnally a description is given of rule based deci-

sion making.

A Classiﬁcation Framework for Time Stamp Stochastic Assignment Problems

133

Table 1: Classiﬁcation of different TSAP applications.

Reference α β γ δ

Project portfolio (Gutjahr and Reiter, 2010) det stoch si-in once

Maternity care (Phillipson, 2015) stoch det one reass/swit

Job Shop Scheduling (Weber, 1982) stoch det one reass

Truckload Motor Carriers (Powell, 1996) stoch det one reass

Airport Gate Assignment (Yan and Tang, 2007) stoch det one reass

Pre-media printing (Kogan et al., 2005) stoch det mu-in reass

Container Shipping (Braekers et al., 2011) stoch det si-in once

Container Shipping (Kooiman et al., 2016) stoch det si-in once

Call Center (Tica et al., 2011) stoch stoch one reass

Surgery Planning (Lamiri et al., 2009) stoch stoch one reass

Ambulance dispatching (Jagtenberg et al., 2015) stoch stoch one reass

Nurse Assignment (Punnakitikashem et al., 2008) stoch stoch one reass

Location based mobile advertising stoch stoch one once

(Spentzouris and Koutsopoulos, 2017)

Bin Packing (Coffman et al., 1980) stoch stoch si-in once

4.2.1 (Two Stage) Stochastic Programming

Mathematical programming with uncertain data in the

objective function or constraints is called stochastic

programming. The uncertainty is usually translated

into a probability distribution on the parameters. The

uncertainty can in practice be characterised by a pre-

cisely deﬁned probability distribution or just a few

scenarios (possible outcomes of the data with cor-

responding probability). An obvious way of deal-

ing with this problem is using a recourse model, as

in (Kools and Phillipson, 2016) and (Leenman and

Phillipson, 2015). The recourse model requires that

a decision is made now such that it optimises the ex-

pected objective value of the consequences of that de-

cision. As an example we take x to be a vector of

decisions that we must take now, and y(ω) is a vector

of (later) decisions that correspond to the reactions

to the decisions of x or the consequences of x. This

way we can model a problem with the following Two

Stage formulation.

The optimisation problem is:

max f (x) + E[G(y(ω), ω)], (1)

subject to

(x) ≤ 0, i = 1, ..., m, (2)

(x, y(ω)) ≤ 0 for ω ∈ Ω j = 1, ..., k, (3)

x ∈ X, y ∈ Y. (4)

The constraints (3) are the links between the decisions

x for the ﬁrst stage and the decisions y(ω) in the sec-

ond stage. It is required that all constraints hold for

each possible ω ∈ Ω. The model can be extended

in numerous ways, e.g. by making it a multistage

problem, in effect make one decision now, wait till

new data is observed, and make a next decision. The

implementation of a recourse model is already com-

plicated for two stages, but the method described by

(Haneveld et al., 2006) can be used to solve the opti-

misation problem.

In stochastic programming used for TSAPs, the

optimisation solution is based on a probability aver-

age to deal with the uncertainty. Because a high num-

ber of demand items become observable one by one

in a TSAP, a large number of decisions need to be

made. This requires multistage programming with

a large number of stages, which is even more com-

plex to solve than the two-stage case. Because of

this characteristic of TSAPs, stochastic programming

does not work well. Stochastic programming is more

apt for stochastics with infrequent decision making

rather than frequent decision making as in TSAPs.

An advantage of stochastic programming is that af-

ter each stage a future allocation or planning is made.

Although in the future there may be a deviation from

the planning, it could be useful as an indication of fu-

ture assignments.

4.2.2 Markov Chain Optimisation

A commonly used way of tackling uncertainty in op-

erations research problems is using Markov Chains

and Markov Decision Processes. Markov Chains are

widely used in queueing theory, what is a form of a

TSAP, but due to earlier mentioned characteristics is

outside the scope of this study. In a Markov chain all

states of the problem/system are considered and their

corresponding transition probabilities. With known

mathematics an optimal allocation of demand and

serving items can be obtained, see (Tijms, 2003). The

downside of Markov chains is the computational bur-

den of the process. If there is a rather large number of

demand items and/or serving items and there is a rea-

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134

sonable large time window, then the number of states

can get very large as this grows exponentially in these

parameters. Here Machine Learning is used as solu-

tion method more and more, see for examples (Rahili

et al., 2018; Spentzouris and Koutsopoulos, 2017).

4.2.3 Simulation

By repeatedly simulating the uncertain factors differ-

ent scenarios are considered. When the decision is

based on these simulations a robust choice can be

made. With the following steps, derived from the sim-

ulation method used in (Phillipson, 2015), a simula-

tion algorithm for a TSAP can be designed for making

an online assignment between a demand item and one

of the available serving items.

• Step 1. Observe demand item i.

• Step 2. Consider each serving item j available that

is able to serve demand item i.

• Step 3. For each considered serving item j draw

a set of dates (and types) for all future demand

items within the time horizon, based on the corre-

sponding distribution.

• Step 4. For each draw ﬁnd the impact of assign-

ing serving item j to demand item i by assigning

the remaining simulated demand items to the re-

maining serving items, the impact is translated in

a score for serving item j.

• Step 5. Repeat step 3 and 4 a predeﬁned number

of times for each considered serving item j.

• Step 6. The demand item i is assigned to the serv-

ing item j with the best average score.

With this method possible scenarios of yet to come

demand items are considered. For each draw of fu-

ture demand items the problem becomes an ofﬂine

optimisation problem and a(n) (near) optimal assign-

ment can be made. Repeating the simulation a large

number of times results in a large set of possible sce-

narios being examined and a robust assignment can

be made. Because the simulation is repeated multiple

times a fast but accurate ofﬂine assignment method is

needed. The simulation algorithm is a good solution

method for TSAPs as it recalculates the current situ-

ation each time a demand item becomes observable.

This simulation approach can be seen as a part of a

Digital Twin environment for the system (Boschert

and Rosen, 2016).

4.2.4 Rule based Decision Making

An easy to use but rather simple solution method is

using a decision rule. With this rule the uncertainty in

future events is neglected and a decision is made upon

the current state of the problem. To assign an incom-

ing demand item a rule is used based on the character-

istics of this demand item, the available serving items

and assignments made in the past. This method ne-

glects uncertain information about future demands all

together and is likely to be sub-optimal therefor. It

can be implemented fairly easily and this is why it is

used often in practice. It can be used as a benchmark

to assess the gain in performance when using more

sophisticated solution methods. Note the also more

complex (greedy) heuristic approaches that are only

based on the current, known, situation, can be placed

under this category of rule based solutions.

5 SUMMARY AND

CONCLUSIONS

The topic of Stochastic Assignment Problems is ad-

dressed in this paper with the intention of deﬁning and

classifying SAPs and examining its solution methods.

From the literature review we concluded that there

exist a lot of different applications of SAPs but

that there is no good representation of the problem

characteristics. First we introduced the Time-stamp

Stochastic Assignment Problem.Starting from a liter-

ature review of Stochastic Assignment Problem we

have developed a comprehensive and versatile frame-

work for classifying Time-stamp Stochastic Assign-

ment Problems by assessing four characteristics of the

problem at hand: demand type distribution, demand

arrival time distribution, resource speciﬁcation and

serving item feature. The framework distinguishes

the different types of TSAPs and their applications in

a clear cut way by typing them with an explicit nota-

tion.

As to the available solution methods, four meth-

ods were identiﬁed: (two stage) stochastic program-

ming, Markov chain optimisation, simulation, and

rule based decision making. Which of these solution

methods is advisable depends strongly on the char-

acteristics of TSAPs involved. We concluded that

Markov chain methods are only feasible when the

number of states is small, whereas simulation meth-

ods seem to work rather well even in large problem

instances, provided there is useful stochastic informa-

tion about future events to be exploited.

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