Systems Thinking Term Project: Scheduling a Fleet of Road Tankers

7/31/2019 Systems Thinking Term Project: Scheduling a Fleet of Road Tankers

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MIDDLE EAST TECHNICAL UNIVERSITY

I E 3 9 8 - S Y S T E M S T H I N K I N G

Term Project:

Scheduling a Fleet of Road

Tankers

May 2011, Ankara

Burcu Yzak - 1627884

Fato lbi - 1535459

Onur Ylmaz - 1627868


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Table of Contents

PAGE

Table of Contents .............................................................................................................. 1

1. INTRODUCTION .. 2

1.1. Project .............................................................................................................. 2

1.2. Related Issues ........................................................................................................ 2

1.3. Approach ............................................................................................................ 2

1.4. Recommendations ................................................................................................ 2

2. REPORT .... 3

2.1. Statement of the Problem....................................................................................... 3

2.2. Analysis of Specific System...................................................................................... 5

2.3. Scale Decisions and Critiques.................................................................................. 8

2.4. Major Steps of Analysis and Findings...................................................................... 9

2.5. Alternative Actions.................................................................................................. 14

2.6. Recommendations................................................................................................... 16

3. CONCLUSION .................................................................................................................. 17

4. GLOSSARY . 18

5. APPENDIX .... I


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1. INTRODUCTION

1.1. Project

This project is related to planning the fleet size and its content; and scheduling the

fleet of Transport Department of Asit Kimya in order to meet the next periods demand. This

project is not only based on managerial trade-offs and decisions but also includes engineering

approaches like planning and scheduling considering the limitations and objectives.

1.2. Related IssuesIn this project, not all the operations of the Asit Kimya or Transport Department are

considered. Scheduling the current fleet, changing the size and its content are the main

concerns of this study; on the other hand, maintenance operations, purchasing operations and

limitations of these operations are not considered in the concept of this study. In addition,

assumptions are made on the unknown or unspecified details of the operations while

considering the main structure of the problem situation.

1.3. Approach

The manager is not willing to clean the tankers since it is hazardous to cleaning

workers. He finds the cleaning decision reasonable only when he will need to buy a new

tanker in order to meet the demand of the next year.Thus, his approach to cleaning and

purchasing decisions will be reflected to the solution approach by using step by step

approach. At each step, cleaning or buying one more additional tanker options will be

compared. In addition to these, alternative actions will also be introduced.

1.4. Recommendations

It is recommended that Transport Department of Asit Kimya should proceed with a

preliminary analysis of the transportation operations in detail by its costs and critical

requirements. The analysis would develop a model for finding optimal scheduling scheme as

well as optimal fleet size, i.e. exact numbers specifically for every type of tanker, via

indicating number of cleanings at the beginning of the year and new tankers to be bought.


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According to this model, reliable estimates of the potential savings in operating costs should

be computed in order to justifying the results of the model.

In the next parts of the report, firstly, problem situation is described and then specific

system which is going to be studied is given. Following these, boundary of the system is

determined by declaring assumptions on the system. Then, major steps of analyzing this

system are shown in detail and findings and comments on these findings given in the

subsequent section. Considering these findings, alternative actions that can be taken are

mentioned and the final recommendations are made in the last section.

2. REPORT

2.1. Statement of the Problem

In this part, problem which will be solved in the next parts will be defined with

presenting different aspects and different roles of people which are involved in the problem.

Firstly, dividing the problem into smaller subsets will make it easier to analyze andrealize their effects on each other and the whole.

Since this is a business environment, whether to implement the solutions provided will

be the decision of the manager of the Transport Department of Asit Kimya and thereafter he

will be mentioned shortly as manager. Since this project aims to solve a problem, there will

be an objective or some objectives. In this manner, the managers objective is to operate the

Transport Department successfully by the means of its measurable and immeasurable aspects.

In order to control how well the main objective is reached, this one objective can be

divided into more specific goals on different areas which would provide rather small

environment to consider for a more focused study. First of these goals is the minimizing the

total costs which is the sum of transportation costs, procurement costs and operating costs.

Second is to minimize number of cleanings and damage of this cleanings on the cleaning

workers. As the third goal, it is aimed to minimize the spare time of trucks throughout the

year. For the last, it is aimed to minimize the possible threats to public safety caused by


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accidents etc. These specified goals will determine how well the solutions satisfy the main

objective.

These objective and goals should be measured by some means, in order to compare

different actions. These means could include numerical measurements like total transportation

cost, procurement cost, operating cost; or in short overall total costs, or rates and counts such

as number of cleanings and proportion of the spare time to the all available time. On the other

hand there are some aspects which are difficult to assign numeric values but reveals the

performance of the solutions, like amount of damage given to cleaning workers and amount

of damage caused by tanker accidents

In order to derive a proper solution to the problem, there shall be different actions to

be taken in this problem environment, and those can be listed as the following; determining

number of tankers to buy and sell, number of compartments to clean and finally allocation of

tankers to route. These actions, individually or as a group, will shape the solution provided to

this problem.

Finally, there is an environment in which this problem is emerged and needed to be

solved. Therefore, context of the problem which affects the situation must be presented. There

are some direct relationships that no one has control on them, such as cleaning operations

effects on health problems; also, some legal and operational limitations which cannot be

changed, like the limitation of 16.5 tons for any type of material stated by law and

minimum/maximum level of materials to be delivered to a depot due to the capacities of these

depots. In addition to these, costs of new tankers or salvage, taxes, insurance and

transportation cost per kilometer are taken as given. Since the problem is about planning the

succeeding year, and the manager stated so, demand forecasts of the next year are counted as

completely reliable. Moreover, due to security issues, maintenance operations and their time

requirements, which is 40 % of the total annual time, are also included in the problem

environment.

Secondly, presenting the different roles of individuals which are included in the

problem situation would make the problem statement more clear. The owner of this problem,

who has the full responsibility of the possible consequences of this problem, is the manager ofthe Transportation Department. When the problem is solved, people who are going to execute


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the decisions are the tanker drivers, cleaning and maintenance workers. In addition, there are

the ones who would benefit or would be victim of the consequences of these implemented

decisions. They are other depots which are in Asit Kimyas transportation network, cleaning

and maintenance workers and publicity, due to the possible threats. This problem will be

solved and recommendations will be provided by the analysts in OR Department of the Asit

Kimya.

To conclude, in this part, problem is divided into parts and each part is described in

detail. Following that, different roles of individuals are mentioned in order to provide a full

explanation of the problem statement. In the following part, specific system which this

problem is emerged will be explained.

2.2. Analysis of Specific System

In the previous section, the problem is stated and in this following section, system in

which the problem is emerged is going to be studied and the point of view will be fixed for

the next phases of providing solution.

In this manner, if we consider the Transport Department as a black box then it will

transform existing routes, tankers and workers; with the possible new additions/removals into

a new route scheduling for the following year. In order to accomplish this transformation,

some sub-components of the system; namely, cleaning system, maintenance system, fleet of

tankers, factory and pre-determined routes; are used.

Considering this system as a black box it is convenient to look out for some inputsand outputs. First of all, these inputs can be gathered into two subsets as the ones which are

not in the control of the system and the ones which are going to be decided. The first set

consists of demand forecast, number of on-hand tankers and their types, number of current

workers and wages, costs of procurement, mileage cost, taxes and insurance, limits of other

depots, security limits of carriage, maintenance time limit, current compartment allocation of

tankers, existing routes and capacity of tankers and compartments. And the second set

includes the inputs which could be decided; explicitly, number of new tankers, number of

tankers salvaged, number of cleanings and capacity usage of compartments.


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Besides these controllable and uncontrollable inputs, there should be some outputs of

the system. Although, some of them are difficult to measure; these expected outputs of the

system could be listed as; total cost (transportation cost, procurement cost, operating cost),

damage given to cleaning workers, spare time of tankers, damage caused by tanker accidents

and a new schedule of tankers.

All aspects of the system mentioned could be summarized by the given diagram which

shows the narrow system of our interest specifically:

Diagram 2.2: Narrow system of interest

As summarized in the influence diagram below, controllable inputs are shown with

rectangular and uncontrollable ones are shown with clouds. The different situations of the

system, which are formed by taking these inputs and processing them, are indicated by

circles. Outputs which are shown with ovals should be mentioned according to their

importance in the objective of the system. This system is trying to accomplish its objective by

minimizing total costs, number of cleaning operations, spare time of tankers, damage given to

cleaning workers and damage caused by tanker accidents.


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Diagram 2.3: Influence diagram


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2.3. Scale Decisions and Critiques

Having settled the system to be studied in the last section, assumptions and critiques

will be presented with their justifications before going any further in analyzing the system, in

the following part of this report.

First of all, as mentioned before, demand forecasts are taken as given for the planning

of next years operations without any concern. As known, Transportation Department

Manager implied that there are negligible deviations in the numbers when previous years

forecasts considered. Therefore, it is reasonable to assume that forecasts are totally true for

next year.

Secondly, being a decision maker, Transportation Manager is presumed that has some

authority limitations and relaxations. Although there are any clear evidence; Transportation

Manager is counted as having the power of buying and selling tankers without any restrictions

like smooth level of resources or monetary issues. On the other hand, manager has some

restrictions, for instance; it is regarded as he has no chance to change the other depots

load/unload demands and capacities.

Thirdly, there are some assumptions about operations of the factory and its sub-

departments. All considerations are made in an environment such that the workforce is fixed.

That is because, any cost of hiring new drivers or workers for maintenance when new tankers

are bought is given; in addition to these, also any cost for firing employees when the size of

fleet is decreased is mentioned. Considering all of these, workforce is taken as constant for

the following year or there would be no effect of changing the level of workforce for our

concern. The second issue about operations is the currently used routes. Routes which are

currently using will be used without any change in the following year. This assumption would

not only will ease the analysis of the recommendations, but also, fix the focus of the study to

more important aspects of the situation. Finally, since it is mentioned that, due to security

issues, any change is considered on the maintenance checks, therefore, total available time of

one tanker is taken as 5240 hours annually.

Fourthly, there are some assumptions about types and assignments of tankers. It is

assumed that when the manager wants to buy new tankers, the only available tankers are the

ones which are mentioned in the question text, i.e. A, B, C, D; and there is no other option. In


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addition, it is assumed that any type of tanker could be assigned to any of the routes without

any restriction like road conditions, insurance restrictions etc.

Fifthly, there are some immeasurable aspects which cannot be directly included in the

cost calculations. For instance, there is no assigned cost of cleaning one tanker and more

importantly there is no fixed cost of the damage given to the cleaning workers. Therefore, it is

assumed that minimizing the number of cleanings could minimize both of these and it will

satisfy the concerns of the manager.

Finally, in this project it is tried to give a scheduling output which only includes

assignment of one type of tanker to one route and this approach is undertaken in order to

avoid combining or dividing routes throughout the next planning horizon. Since this approach

of assignment seems to ignore idle time of the tankers by not assigning them to other routes in

their idle times, and since spare time of the tankers are tried to be minimized, at all steps of

analysis, idle time percentages are checked prior to making any recommendations.

Considering these assumptions and decisions made to fix the scale of our view, our

solution approach to the problem will be presented in the following part.

2.4. Major Steps of Analysis and Findings

First of all, the main thing that should be focused on, while constructing the model of

the system, is the trade-off between the number of cleanings and the number of new tankers to

be bought. If a tanker is cleaned, it will not affect the total cost considerably but the harm that

will be given to the cleaning workers must be taken into consideration. On the other hand, if a

new tanker is bought, only the high level of cost will be considered; however, the

responsibility on the health of the cleaning workers is not so easy to consider. So the manager

is willing to undertake the cleaning if he can see that it will save him from purchasing another

vehicle to transport the forecast of the coming year.

Due to the trade-off explained above, it is thought that solving the problem using one

model will not be effective on finding a solution. Therefore, to solve the problem different

models; in fact modified models, are used in each step. Note that the number of tankers

allocated to routes, number of tankers cleaned and the number of new tankers bought are


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defined as integer numbers in the models in order to find a logically valid solution. However,

number of times a tanker completes its route is assumed to be a real number to make the

solver of the problem, GAMS, solve efficiently. Moreover, salvaging the unnecessary tankers

is also considered in each step.

The major steps followed in order to reach the best solution to the problem are

summarized below:

a) First Step:

In this step, the aim of the model is to be acquainted with the system and to see

whether the tankers on hand will be sufficient to meet the demand of the next year without

cleaning the existing ones. The model is constructed using the idea that depots are covered byroutes. It means that if the demand of the depot is going to be satisfied then there must be at

least one tanker which is assigned to the routes covering that depot.

The idea behind this model is to assign types of tankers (A,B,C,D) to the routes. So an

assumption made here is that this assignment is done at the beginning of a year and it does not

change during the year. Decision variables are defined for each route and each tanker type

indicating the number of tankers of given type assigned to the given route. In addition to

these, there are also decision variables for each tanker type indicating the number of salvaged

tankers and one more decision variable for number of times a tanker completes its given

route. The model includes constraints mentioned by the manager of the company. First of all,

there are demand satisfaction constraints to satisfy the demands of the next year. Moreover,

maintenance constraints saying that 40% of a year a tanker should be in maintenance, legal

restrictions obligating a delivery amount no more than 16,5 tons and the restrictions of the

depots on the delivery amounts are also included.

The details of the model and brief explanations of the constraints can be found in

Appendix A.

Findings: In this step, it is expected to find out whether or not the current situation is

sufficient for the next periods. Solving the model explained in the first step of analysis, it is

found that it is not possible to meet the demands of the next year without changing the

number of on hand tankers. Therefore, the manager should take any actions like either

existing tankers should be cleaned or the manager should buy new tankers. Details of thesolution are given in Appendix B showing that the solution is infeasible.


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b) Second Step:

In the last part, it is shown that it is impossible to meet the demand of the next

planning horizon with the current number of tankers. Since manager is willing to undertake

cleaning only if it saves him from buying a new tanker, and current number of tankers is not

enough, it is thought that, what would happen if the model gives full permission to cleaning

operation?

With this reasoning, firstly, new decision variables are added in order to determine

whether or not a cleaning operation is undertaken, how many of tankers are cleaned and

combining these with the current level of the tankers, number of tankers is updated. Secondly,

some constraints related to change in the number of tankers due to cleaning operation are

added and one more constraint related to assigning number of cleaned tankers is added.Thirdly, since there is no determined cost of the cleaning, and this step only checks whether

cleaning is a solution to infeasibility of the first step, no cleaning related cost is added to

objective function. Finally, it is thought that only cleaning between A-B types and C-D types

considered since, for instance, A-D cleaning exceeds the 16,5 tons of legal carriage amount.

Changes described above are made to the model of the first step and it is provided in

the Appendix C.

Findings: By solving the model of the second step, it is found that it is impossible to

meet the demand of the next period even with the restrictions on cleanings are removed. This

result means that there should be an increase in the total numbers of tankers because in this

step all operationally feasible compartment combinations of the currently available trucks are

tested. As given in the Appendix D, relaxing the considerations on cleaning will also yield an

infeasible solution.

c) Third Step:

In the last part, it is shown that it is impossible to meet the demand of the next

planning horizon with the current number of tankers even if the cleaning operations are

undertaken. Therefore, this step is focused on what would happen if we let manager buy any

number of tankers which yields the minimum cost. This step will going to be a base step for

the following steps, because this will provide a direction on the type and number of tankers tobuy and trade-offs between new tankers and cleanings will be made on this direction.


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With this reasoning, some changes are made to the model in the first step. First of all,

number of tankers and their types are added to decision variables and this will determine the

total number of tankers with their types. Secondly, a constraint added to ensure that the new

bought tankers are added to the number of the related tankers. Thirdly, objective function is

updated to include new purchase cost.

Changes described above, which are made to the model in the first step, are given in

the Appendix E in detail.

Findings: In this step, it is expected to find the optimal number of trucks with the

minimum total cost. Because, since in the last two steps yield infeasible solutions, in this step

purchase is allowed with no number limitation. As given in the Appendix F, this step has a

minimum cost level of $ 110.846,73 and route-tanker assignments with related numbers can

be seen from the Table 2.5 below:

As tabulated above, result of this step shows that the minimum cost is attained when 4

new D type tankers are purchased and shown assignments are made. As mentioned in the last

parts related step above, since this step creates a basis for the direction ofthe following steps,

in the next steps trade-offs will be made considering this 4 new D type tankers.

It is shown that it is impossible to meet the demand of the next planning horizon

without any purchase. In addition, it is found that if all new purchases made on type D and no

cleaning is made then the minimum cost will be attained. Therefore, in the next steps,

purchasing one more D type tanker and cleaning trade-off will be compared. With this

reasoning, in the next steps, 8000 of new purchase cost will be added to total cost function

and on hand D type tanker will be increased by one in each step. Then it is checked whether

or not the model is feasible without buying any additional tanker and cleaning the on hand

tankers. This iterative controls are made for considering that we have one, two and three D

types of tankers on hand, because if were to buy 4 D type tankers, considering any cleaning

R O U T E S TOTAL

TANKERS

1 2 3 4 5 6 7 8 9 10 11 12

TANKERS A 1 1

B 3 3

C 1 1 1 1 1 1 6

D 1 1 2 4

Table 2.5.1: Information gathered from Appendix F


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will be unnecessary since the manager wants to undertake cleaning only if it saves him from

buying any additional tanker.

d) Fourth Step:

In this step, it is checked whether or not it is possible to purchase one D type tanker

and undertake cleaning operations to meet demand. With this approach, changes made to

model of the Second Step are provided in Appendix G.

Findings: By solving this model, it is found impossible to meet demand with one

additional D type tanker and letting cleaning operation if there is any need.

e) Fifth Step:

In this step, it is checked whether or not it is possible to purchase two D type tankers


model of the Second Step are provided in Appendix H.

Findings: By solving this model, it is found impossible to meet demand with two

additional D type tankers and letting cleaning operation if there is any need.

F) Sixth Step:

In this step, it is checked whether or not it is possible to purchase three D type tankers


model of the Second Step are provided in Appendix J.

Findings: By solving this model, it is found impossible to meet demand with three

additional D type tankers and letting cleaning operation if there is any need.

Considering the fourth, fifth and sixth steps, it is found that the manager cannot use cleaning

to save himself from buying additional D type tankers.


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2.5. Alternative Actions

As explained above, the first feasible solution to this problem is found in the third step

in which the cleaning option is not considered and buying new tankers is taken into

consideration only. The decision was to buy 4 new type D tankers. Then in order to reflect the

trade-off between cleaning decision and buying decision, the model including cleaning in

second step is modified to reach a feasible solution by increasing number of type D tankers on

hand by one in each step up to three type D tankers. In these steps, it is found that cleaning

cannot overcome the need for type D tankers. So it is concluded that the best solution to that

problem is to buy 4 new type D tankers. However, some alternative scenarios are also found.

While deciding on each scenario, the idea was to see whether other tanker types can be

bought instead of type D tankers when the cleaning is also considered. To be able to compare

each scenario easily all combinations of options are shown by using tables. Since buying a

type B tanker will result in cleaning them and increasing the number of type A tankers due to

larger demand for acidic than for caustic and larger acidic capacity of type A tankers than of

type B tankers; while tabulating only purchase of type A tankers are combined with purchase

of type C tankers. Each feasible scenario has the cost value and the number of cleaning donein that scenario. Moreover, some of the scenarios are cancelled since they include buying

more than four tankers, which will directly be more costly than buying four new type D

tankers.

Number of type A tankers bought

Number of

type C

tankers

bought

0 1 2 3

0Fourth Step

Infeasible Infeasible Infeasible

Cost:

116.204,48

# of Cleanings:

2 (C type)

1 Infeasible Infeasible

Cost: 111.646,73

# of Cleanings:

3 (C type)

-

2 Infeasible

Cost: 111.646,73

# of Cleanings:3 (C type) 1 (B Type)

- -

3

Cost: 111.646,73

# of Cleanings:

3 (C type)

- - -

Table 2.5.2 : Number of A and C type tankers ara compared when number of on hand D type tankers is 1


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Number of

type C

tankers

bought

0 1 2 3

0Fifth Step

InfeasibleInfeasible

Cost: 111.646,73

# of Cleanings:

2 (C type)

-

1 Infeasible

Cost: 111.646,73

# of Cleanings:2 (C type) 1 (B Type)

- -

2

Cost: 111.646,73

# of Cleanings:

3 (C type)

- - -

3 - - - -



Number of

type C

tankers

bought

0 1 2 3

0Sixth Step

Infeasible

Cost: 111.646,73

# of Cleanings:

2 (C type)

- -

1

Cost: 111.646,73

# of Cleanings:

2 (C type)

- - -

2 - - - -

3 - - - -


Above tables show the options when one, two and three type D tankers are bought

respectively. The options are elected by comparing the costs first, then number of cleanings

done is considered and the ones having larger number of cleanings are omitted. Thereafter,

options having larger caustic capacity are ignored since caustic demand is not high. Finally,

two alternative actions are found which are shown in Table 2.5.5 below. GAMS outputs of the

models for Option 1 and Option 2 are in Appendix K and L respectively.


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Third Step Option 1 Option 2

Total Cost 110.846,73 111.646,73 111.646,73

# of cleanings - 2 ( C type ) 2 ( C type )

# of type A tankers 1 3 2

# of type B tankers 3 3 3

# of type C tankers 6 4 4

# of type D tankers 4 4 5

Table 2.5.5: Alternative Actions and Third Step Compared

2.6. Recommendations

As seen in Table 2.5.5 the cost value of buying four type D tankers is the optimal one.

Therefore, it will be better to choose this option. Moreover, in these two alternative options,

number of type A tankers are increased by buying new ones. However, buying type D tankers

is always more meaningful than buying type A tankers. The reason for that is that type D

tankers have extra caustic capacity compared to type A tankers. In addition to that, they

have the same price, 8000TL.Therefore, buying four new type D tankers is again the optimal

policy when looked from that point of view.

As mentioned before, outputs of the system are schedule of the tankers which is given

in Appendix F for the optimal case, number of cleanings done which is zero, total cost

including procurement, operating and transportation cost minus salvages, which is found to be

110,846.73TL. In addition to these, Table 2.6.1 below shows the spare time percentages of a

tanker assigned to each route for the optimal policy.

Route 1 Route 2 Route 3 Route 4 Route 5 Route 6

% spare time of tankers 0 0 45 36 66 0

Route 7 Route 8 Route 9 Route 10 Route 11 Route 12

% spare time of tankers18 31 0 69 90 96

Table 2.6.1: Spare time of tankers


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As it can be seen from the above table some tankers assigned to specific routes have a

large amount of spare time. The reason for that is the assumption made which says that if a

tanker is assigned to a route at the beginning of the year, then it will not be assigned to

another route. If this assumption is considered again, then the number of tankers needed to

meet the requirements of the following year could be less than 14 since the tankers would be

assigned to different routes at their spare times.

3. CONCLUSION

In this project, scheduling the fleet of Transport Department of Asit Kimya is

analyzed. The problem is stated with all its sub-components and then the system which this

problem is emerged is described in this report. Making related assumptions, a step-by-step

solving approach is undertaken with considering the managers approach to cleaning and

purchasing. At each step, cleaning or buying one more additional tanker options are

compared. In addition, considering business environment and its future, alternative actions are

introduced. Considering all alternative actions, it could be stated that buying four D type

tankers and increasing fleet size without any cleaning is recommended and this results with

the total cost of $ 110.846,73. This is found to be the best alternative since it considers the

effect of cleanings on workers health without comparing it with any financial gain. In

addition considering the compartment sizes of D type tankers and currently having no D type

tankers on hand, it is the recommended action for manager.


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4. GLOSSARY

In this part, technical expressions which are used in the report will be explained.

Mathematical model: Mathematical model is a description of a system using

mathematical concepts and language.

Decision variable: Decision variables are the variables within a mathematical model

that one can control. They are not random variables and they are related to the managers

decisions in this content.

Constraint: Constraints are the conditions in an optimization environment which are

needed to be satisfied.

Infeasible: A mathematical model is said to be infeasible if it cannot satisfy all or

some of the constraints it contains.


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Appendix - I

5. APPENDIX

PAGE

Appendix A - Mathematical Model of the First Step Analysis ............................................ II

Appendix B - GAMS Output of the First Step ..................................................................... IV

Appendix C - Mathematical Model Changes for the Second Step Analysis V

Appendix D - GAMS Output of the Second Step ................................................................ VI

Appendix E - Mathematical Model Changes for the Third Step Analysis ........................... VII

Appendix F - GAMS Output of the Third Step .................................................................... VIII

Appendix G - Mathematical Model Changes for the Fourth Step Analysis ........................ IX

Appendix H - Mathematical Model Changes for the Fifth Step Analysis ........................... IX

Appendix J - Mathematical Model Changes for the Six Step Analysis ................................ X

Appendix K - GAMS Output of the Option 1 ...................................................................... XI

Appendix L - GAMS Output of the Option 2 ....................................................................... XII


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Appendix - II

Appendix A - Mathematical Model of the First Step Analysis:

Sets:

i : Type of tankers {A, B, C, D}

r: Routes defined in the question {1, 2, , 12}

j: Depots {1, 2, , 11}

Parameters:

Ni : Number of on hand tankers of type i

CAi : Capacity of type i tankers for acidic

CCi : Capacity of type i tankers for caustic

Dr : Duration of route r

DTr : Distance of route r

Tj : Total demand of depot j

Decision Variables:

Xir :

{

Yir : Number of type i tankers assigned to route r

Lr : Number of times a tanker assigned to route r completes its assigned route

Si : Number of type i tankers salvaged

Objective function:

min Z = DTr * Lr *0.1 + Yir * 200 - Si * 3000Constraints:

Yir Ni * Xir ( If type i is not assigned to route r, number of total tankers assigned to

route r should be zero )

Yir Ni for all i (Sum of all type i tankers assigned to a route is smaller than its onhand amount)


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Appendix - III

Yir * CCi * 5240 / Dr 53000 (Caustic demand of Tepe) Yir * CCi * 5240 / Dr Tj for all j ( Acidic demand of all depots )Lr ( 5240 / Dr )

*

Yir for all r ( Maintenance restriction on total number of

times a tanker assigned to route r completesthe route)

Si Ni - Yir for all i ( Number of unnecessary tankers of each typeis salvaged )

16,5 for all j

( Legal restriction on acidic )

16,5 for all j

( Legal restriction on caustic )

5,5 for all j

( Delivery amount of at most 5,5 for Hanya )

5,5 for all j

( Delivery amount of at most 5,5 for Maras3 )

15for all j except Hanya and

Mara3

( Delivery amount of at least 15 tons for

depots )

Xir , Yir , Lr , Si 0 Xir , Yir integer

THE ROUTES

1: {Tepe, Hanya}

i.e: Tepe-Hanya-Tepe

2: {Tepe, Bor}

3: {Tepe, Bor, Hanya}

i.e: Tepe-Bor-Hanya-Tepe

4: {Tepe, Hanya, Mara1}

i.e: Tepe-Hanya-Mara1-Tepe OR

Tepe-Mara1-Hanya-Tepe OR

Tepe-Mara1-Tepe



7: {Tepe, Hanya, eme}

8: {Tepe, Hanya, Koru}

9: {Tepe,Lara}

10: {Tepe, Lara, Hanya}

11: {Tepe, Geyve}

12: {Tepe, Hora}


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Appendix - IV

Appendix B - GAMS Output of the First Step

GAMS Rev 227 x86/MS Windows 05/19/1120:49:34 Page 5G e n e r a l A l g e b r a i c M o d e l i n g S y s t e mSolution Report SOLVE system_1 Using MIP From line 162

S O L V E S U M M A R Y

MODEL system_1 OBJECTIVE zTYPE MIP DIRECTION MINIMIZESOLVER CPLEX FROM LINE 162

**** SOLVER STATUS 1 NORMAL COMPLETION**** MODEL STATUS 10 INTEGER INFEASIBLE**** OBJECTIVE VALUE 0.0000

RESOURCE USAGE, LIMIT 0.140 1000.000ITERATION COUNT, LIMIT 0 10000

ILOG CPLEX May 1, 2008 22.7.2 WIN 4792.4799 VIS x86/MSWindowsCplex 11.0.1, GAMS Link 34Cplex licensed for 1 use of lp and barrier.

Problem is integer infeasible.

No solution returned


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Appendix - V

Appendix C - Mathematical Model Changes for the Second Step Analysis:

Additions

Decision Variable:

Mi : New number of tankers of type i

Ti : {

Ci : Number of type i tankers cleaned

Costraints:

Ci 10* Ti ( If there is not any cleaning on type i, the number of cleaned tankers

should be zero)MA = NA - CA + CB ( Change in tanker numbers due to cleaning )

MB = NB + CA + CB ( Change in tanker numbers due to cleaning )

MC = NC - CC + CD ( Change in tanker numbers due to cleaning )

MD = ND - CD + CC ( Change in tanker numbers due to cleaning )

Ci , Mi 0 Ti, Ci integer

Changes Made

Old:

Si Ni - Yir for all i ( Number of unnecessary tankers of each type is salvaged )New:

Si Mi - Yir for all i ( Number of unnecessary tankers of each type is salvaged )


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Appendix - VI

Appendix D - GAMS Output of the Second Step

GAMS Rev 230 WIN-VIS 23.0.2 x86/MS Windows 05/21/1117:41:38 Page 5G e n e r a l A l g e b r a i c M o d e l i n g S y s t e mSolution Report SOLVE system_1 Using MIP From line 188



**** SOLVER STATUS 1 NORMAL COMPLETION**** MODEL STATUS 10 INTEGER INFEASIBLE**** OBJECTIVE VALUE 0.0000


...

Problem is integer infeasible.

No solution returned


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Appendix - VII

Appendix E - Mathematical Model Changes for the Third Step Analysis:

Additions

Decision Variable:

Ji : Number of bought tankers of type i

Mi : New number of tankers of type i

Costraints:

Mi = Ni + Ji for all i ( Updating the total number of tankers )

Ji , Mi 0 Ji integer

Changes Made

Old:

Objective function:

min Z = DTr * Lr *0.1 + Yir * 200 - Si * 3000

New:

Objective function:

min Z = DTr * Lr *0.1 +( Yir ]- Ji ) ) * 200 - Si * 3000+ Ji * 8000


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Appendix - VIII

Appendix F - GAMS Output of the Third Step



**** SOLVER STATUS 1 NORMAL COMPLETION**** MODEL STATUS 8 INTEGER SOLUTION**** OBJECTIVE VALUE 110846.7273


MIP status(102): integer optimal, toleranceFixed MIP status(1): optimalSolution satisfies tolerances.

MIP Solution: 110846.727273 (31 iterations, 0 nodes)Final Solve: 110846.727273 (2 iterations)

Best possible: 106318.938883Absolute gap: 4527.788390Relative gap: 0.040847

---- VAR Y Number of type i tankers assigned to route r

LOWER LEVEL UPPER MARGINAL...A.route8 . 1.000 100.000 3200.000...B.route1 . 3.000 100.000 -3088.000...C.route3 . 1.000 100.000 3200.000...

C.route5 . 1.000 100.000 3200.000C.route6 . 1.000 100.000 143.333...C.route10 . 1.000 100.000 3200.000C.route11 . 1.000 100.000 3200.000C.route12 . 1.000 100.000 3200.000...D.route3 . 1.000 100.000 3200.000D.route4 . 1.000 100.000 3200.000...D.route7 . 2.000 100.000 3200.000...


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Appendix - IX

Appendix G - Mathematical Model Changes for the Fourth Step Analysis:

Old:

Objective function:

min Z = DTr * Lr *0.1 + Yir * 200 - Si * 3000New:

Objective function:

min Z = DTr * Lr *0.1 + Yir * 200 - Si * 3000 + 8000

On hand number of type D tankers is increased to one from zero.

Appendix H - Mathematical Model Changes for the Fifth Step Analysis:

Old:

Objective function:


Objective function:

min Z = DTr * Lr *0.1 + Yir * 200 - Si * 3000 + 16000

On hand number of type D tankers is increased to two from one.


29/31

Appendix - X

Appendix J - Mathematical Model Changes for the Six Step Analysis:

Old:

Objective function:


Objective function:

min Z = DTr * Lr *0.1 + Yir * 200 - Si * 3000 + 24000

On hand number of type D tankers is increased to three from two.


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Appendix - XI

Appendix K - GAMS Output of the Option 1







R O U T E S TOTAL

TANKERS

1 2 3 4 5 6 7 8 9 10 11 12

TANKERS A 1 1 1 3

B 3 3

C 1 1 1 1 4

D 1 1 2 4

Table Appendix K-1: Information gathered from the related GAMS Output


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Appendix L - GAMS Output of the Option 2







Best possible: 110681.488289

Absolute gap: 965.238984Relative gap: 0.008645

R O U T E S TOTAL

TANKERS

1 2 3 4 5 6 7 8 9 10 11 12

TAN

KERS A 1 1 2

B 3 3

C 1 1 1 1 4D 1 1 2 1 5

Table Appendix L-1: Information gathered from the related GAMS Output

Documents

Systems Thinking Term Project: Scheduling a Fleet of Road Tankers