Operation research finished

8/7/2019 Operation research finished

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Operation research:Jaroslav Knpek

Czech Technical University in Prague

Diogo Vasconcelos

Jimmy Nordstrm

Mats Heyman

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http://www.facebook.com/Diogo.A.M.Vasconceloshttp://www.facebook.com/Diogo.A.M.Vasconcelos


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IntroductionIn these times of snowfall low temperatures and real winter weather we all feel that

Christmas is near, so we (Jimmy, Diogo and Mats) got the idea for this seminar work,

to work out, our Christmas Evening here in Prague.

It seemed interesting for us to optimize our list of to do activities, within the fastest

time, and this in the most efficient way in order to have an idea how long it would

take, to prepare it all, if we do it in the optimal way.

The plan is as followed; we would like to invite 10 friends into our rooms in Strahov

spread over our 3 rooms. We only have 1 small kitchen at our disposal, this kitchen

consist of two hotplates and one oven.

We would like to prepare a real Christmas dinner for our 10 Erasmus friends, on the

evening of December 24. But we are limited in time because we also need to study,

and there are some other constraints, we should try to prepare this diner with a

minimum amount of people and effort.

We use the critical path method (CPM), a mathematically based algorithm for

scheduling a set of project activities. Because its an important tool for effective

project management. And our project is to secure a spectacular Christmas Eve.

On this way we want to achieve an optimal solution. We also uses the simplex

method for optimizing how the different tasks are distributed between us all.

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Description of the taskThe task that is going to be solved consist of 4 different activities cleaning, cooking,

making the table and buying last minute things. So in order to get a better distributionof task to minimize the time needed, we tried to assign the task according to

efficiency on doing it (time needed). This efficiency is a value from 0,5 (taking twice

the normal, estimated time to do it), and 2 (doing it twice as fast as the normal

time).

Also, Diogo cant cook so its efficiency is 0, and Jimmy doesnt have a car license, so

he cant drive and buy the last minute things required for the dinner.

Since cooking the diner is a kind of complex activity which includes several of small

activities it is evaluated separately. The preparations would start in the afternoon of

December 24, we expect 10 people, including a vegetarian, for which we will make a

special vegetarian Christmas menu,Cabbage dish with chopped onions. The othernine eat the original Christmas dinner: turkey with croquettes and warm vegetables.

Overview ingredients and preparation:

Original Christmas dinner: (9 persons) Vegetarian Christmas dinner: (1 person)

Turkey with croquettes and warm

vegetables

Cabbage dish with chopped onions

Ready to cook a 4 kg turkey

2 turkey livers

500 g mixed minced meat (pork)

2 eggs

4 cream

100 grams mushrooms (made

clean)

10 tablespoons breadcrumbs

Salt, pepper and nutmeg

Baking butter

0,250 kg potatoes

2 pcs onions, cut into wedges

2 cups finely chopped vegetarian

meat (a 175 g)

1small table spoon Italian herbs (6

g jar)

1 jars of red cabbage with apple

(a 300 g)

15 g butter, cold

1 tablespoons olive oil

Preparation:

1. Cut the mushrooms and halve that

again. Stew the mushrooms for 10

minutes in some butter.

2. Break the egg into a bowl and mix

with the minced meat. Grind theliver finely in a food processor and

Preparation:

1. Cut the potatoes into quarters and

boil in 300 ml of water 20 minutes

2. Preheat the oven to 180 C. Heat

the oil in a wok and fry the onion

for 3 minutes, stir in the choppedherbs and salt and pepper to

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add to the minced meat. Stir in the

mushrooms, cream and bread

crumbs. So now the filling is

finished.

3. The turkey: Fill the turkey with the

ground beef mixture and sew shut

with needle and thread.

4. Grease a baking tray and place

the turkey. Butter well the turkey

and season to taste with salt,

pepper and nutmeg. Put an extra

piece of butter in the baking tray.

5. Put the turkey for three hours in a

preheated oven at 180

degrees. Put in the oven, a dish

with water to combat

dehydration. When cooked, add

any remaining butter and baste

the turkey regularly with the

juices.

taste. Spoon the minced mixture

into the baking dish. Divide the

cabbage with apple over it.

3. Mash potatoes with

masher. Spread the mixture on

the cabbage. Cut the butter into

pieces and arrange over the

potatoes.

4. Put the dish in the oven for 30

minutes until the mash is golden

brown.

We have summarized and filled in proceedings; this into the following list of activities

se table 1 below:

DiscriptionsProceding

Pessimistic,b

Reference,m Optimistic,a

A Cutting Potatoes 12 10 8

B Boil Potatoes A 25 20 18

C Smash potatoes B 11 10 9

D Make the cover 9 5 4

E Roll in the cover D,C 48 40 35

F 1# fry E 11 7 6

H Prepare the minced meat 17 15 13

I Cook the minced meat H 26 20 17

J Cut mushrooms 12 7 5

K Put all inside I,J 10 6 5

L Start the oven J 16 15 14

M Turkey in the oven K,L 250 200 180

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N Everything in Oven (Vegetarian) M,F,Q 40 30 26

P Prepare the vegetables 20 15 10

Q Put potatoes and vegetables together P,C 6 5 3

Table 1: Description of the activities with proceeding activities and time estimations.

We solved this problem with the critical path method, the PERT method because our

problem has a stochastic behavior, in each activity things can go wrong, for example

while cutting the mushrooms you can cut your finger or the mushroom can fall on the

ground, then you lose time because you need to clean the mushroom again an clean

the ground. Thats why the PERT method is useful, because you have the time

reference, optimistic and pessimistic time.

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SolvingCooking task

The strategy when solving the task is first to make a scheme over the whole project

to be able to calculate the critical path and see approximately how long the project

will take. This can be seen in figure XX below where also the earliest possible and

latest acceptable starting times for each activity are written. By looking at the first

figure we can see that the project is going to take 5 people to solve since there are at

least in one place 5 parallel activities that requires a person. This is not so optimal

though most of these persons have nothing to do after they have finished their

activities. By making another scheme (figure XX) and consider the fact that we want

to minimize the number of persons that is used. In the second figure the same project

is done but instead of having 5 people it only requires 2 people, despite that the end

time and critical path is the same. Preparing it with only one person would also bepossible but that would affect the end time negatively which is not desirable.

The critical path for the project in both of the cases is the activities H-I-K-M-N which

also can be seen in the figures.The critical path is the path of activities where the

total time reserve is zero, the total time reserve can be calculated with the formula

rc=t'j-ti-tijwhich is (Total time reserve= latest acceptable-earliest possible-time for

activity)

Figure XX: Time schedule for the activities with earliest possible and latest

acceptable starting time for each activity.

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Figure XX: Time schedule for the activities designed to be performed by two persons

with earliest possible and latest acceptable starting time for each activity.

By using the Perth method the meantime and variance can be calculated (see table

2). With these the probability of finishing within certain time can be calculated, the

result is shown in figure XX. The calculations is as follows

MeantimeVariances

A 10 0,44

B 20,5 1,36

C 10 0,11

D 5,5 0,69

E 40,5 4,69

F 7,5 0,69

H 15 0,44

I 20,5 2,25

J 7,5 1,36

K 6,5 0,69

L 15 0,11

M 205 136,11

N 31 5,44

P 15 2,77

Q 4,83 0,25

Table 2: Calculation of meantime

and variances for all of the activities

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Meantime: t=a+4m+b6 (see table 1)Variances: D=b-a236Specify the goal time as TG

The value for the Gaussian distribution function can

be calculated as Tg-TmDtotal, where Tmis the

meantime for the project (278 minutes) and Dtotal is

the sum of the variances from the critical path. From

the Gaussian distribution function the probability of

finishing within the goal time is obtained.

The result is shown in figure XX below.


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Figure XX: Graph showing the probability of finishing related to different goal times.

So the result for the cooking task is shown in the figure XX above. As seen the

probability of finishing within a specified time increase as the time increases. 50%

chance is obtained with the meantime that was calculated before, and with shorter

time the change of doing it within the time decreasing.

Distribution of people task

For the Christmas dinner there are some tasks that need to be done. So in order to

get a better distribution of task to minimize the time needed, we tried to assign the

task according to efficiency on doing it (time needed). This efficiency is a value from

0,5 (taking twice the normal, estimated time to do it), and 2 (doing it twice as fast as

the normal time).

Also, Diogo cant cook so its efficiency is 0, and Jimmy doesnt have a car license, so

he cant drive and buy the last minute things required for the dinner.

The time that one single person can work will be 300minutes (because its the time

needed to finish the cooking, with 96,6% of probability to finish on time, according to

previous problem). But with efficiency = 1, it would take 830min to finish all the tasks,

and because Diogo will get there 2 hours later, he can only work 180min, and

therefore the total working time is 780 min, 50min less than the normal time needed

to finish. So we need to distribute the tasks in an effective way.

In order to solve to optimize this dinner preparation, we used to simplex method, to

solve this problem of linear programming.

Simplex method Objective: get the best task distribution in order to get minimal time needed.

Tasks: Cleaning, Cooking, preparing table, Buying last minute things

People: Diogo, Jimmy, Mats

Efficiency Cleaning (1) Cooking (2) Table (3) Buying (4)

Diogo (D) 1,2 0 0,6 0,9

Jimmy (J) 0,8 1,15 0,9 0

Mats (M) 1,3 1,2 0,9 0,6

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Goal time 240 255 270 278 285 300 315

Probalility of

finishing

0,08

% 2,80% 25,32% 50,00%

71,95

%

96,62

% 99,89%


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Criteria Function :

D1+D2+D3+D4+M1+M2+M3+M4+J1+J2+J3+J4= min Tasks requirements :

1,2 D1+0,8 J1+ 1,3 M1 = 180 (cleaning)

0 D2 + 1,15 J2 + 1,2 M2 = 300 (cooking)

0,6 D3+0,9 J3 + 0,9 M3 = 150 (table)

0,9 D4 + 0 J4 + 0,6 M4 = 100 (buying)

Time limitation (source limitation):

D1 + D2 + D3 + D4 180 (Diogo)

J1 + J2 + J3 + J4 300 (jimmy)

M1 + M2 + M3 + M4 300 (Mats)

i)The first step in Simplex method is to put this problem in Standard Form

Criteria Function : (needs to be minimization)

D1+D2+D3+D4+M1+M2+M3+M4+J1+J2+J3+J4= min

Tasks requirements: (Adding artificial variable S, to ensure initial basic

feasible solution).

1,2 D1+0,8 J1+ 1,3 M1 +S1= 180 (cleaning)

0 D2 + 1,15 J2 + 1,2 M2 +S2= 300 (cooking)

0,6 D3+0,9 J3 + 0,9 M3 +S3= 150 (table)

0,9 D4 + 0 J4 + 0,6 M4 +S4= 100 (buying)

Time limitation (source limitation):(For inequalities, turn into equalities,

adding slack variables )

D1 + D2 + D3 + D4 +S5 = 180 (Diogo)

J1 + J2 + J3 + J4 +S6 = 300 (Jimmy)

M1 + M2 + M3 + M4 +S7 = 300 (Mats)

Assuming non-negativity in all the variables.

ii) Start with an elementary base:

Simplex method is based in the principle that the solution is in a corner point, and

that each corner point has a related feasible base.

So we start with an elementary base (using the slack variables).

iii) Test to see if solution if optimal

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Than we test to see if the solution can be improved, and if so, we substitute one of

the vectors in base, and so by changing the base we change the corner point were

testing.

iv) Change to another corner point

Until we finally reach a corner point, where base cannot be improved, and this is our

optimal solution.

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Base Cb P0 P1 P2 P3 P4 P5 P6 P7 P8

P 0 1 1 1 1 1 1 1 1

S1 0 180 1,2 0 0 0 0,8 0 0 0

S2 0 400 0 0 0 0 0 1,15 0 0

S3 0 150 0 0 0,8 0 0 0 0,9 0

S4 0 100 0 0 0 0,9 0 0 0 0

S5 0 180 1 1 1 1 0 0 0 0

S6 0 300 0 0 0 0 1 1 1 1

S7 0 300 0 0 0 0 0 0 0 0

Zj-Cj

P9 P10 P11 P12 P13 P14 P15 P16 P17 P18 P19 h-ratio

1 1 1 1 0 0 0 0 0 0 0

1,3 0 0 0 1 0 0 0 0 0 0

0 1,2 0 0 0 1 0 0 0 0 0

0 0 0,9 0 0 0 1 0 0 0 0

0 0 0 0,6 0 0 0 1 0 0 0

0 0 0 0 0 0 0 0 1 0 0

0 0 0 0 0 0 0 0 0 1 0

1 1 1 1 0 0 0 0 0 0 1


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Because manual solution of this problem would take a lot of work, we used

Excel to solve it.

Solution:

Total time, Criteria function = 758,80 min = 12 hours and 39 minutes

D1 = 0 min J1=0 min M1=138,46 min

D2 = 0 min J2=92,31 min M2= 161,53 min

D3 = 0 min J3=166,67 min M3=0 min

D4 = 111,11 min J4 = 0 min M4 = 0 min

Total time each person worked:

Diogo = 158,79 min = 2 hours and 39 minutes

Jimmy = 300 min = 5 hours

Mats = 300 min = 5 hours

With efficiency = 1, (normal estimated time), it would take 830 minutes to finish all

the tasks. But by distributing the tasks to the most effective people , we got 759

minutes, which is less on hour and 10 minutes of work (8,58% reduction).

Mats and Jimmy work the full 5 hours, exhausting all their time. Its a bit unfair that

Diogo only work 2 hours and 39 minutes, when the two others work a lot more, and

dont even exhaust all his time, but this is the most effective way to prepare thedinner, so just taking into account doing this task with the least consumption of

working time, this would be the best solution.

In reality, as there are 3 people working, the quickest/most effective way to get the

dinner preparation (because people can work simultaneously) would be to get the

lower (minimize) the time that the person that worked the most works and not the

total time. But as said before, we also need to take into account other factors, like for

instance, that to cook we need 278 minutes to finish the task, (300 minutes to have

96,6% probability of finishing on time) so we know that people will be working for at

least 300 minutes. So with this model, we go it, the minimal possible time (300minutes).

Also with this model we got 2 people working on the cooking, like we get in the

previous problem.

Results:

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So now we have an idea, how much it time it would take to have the perfect

Christmas evening, using the CPM, PERT method we found that we need at least

315 min to be sure that the dinner is finished on time.

From the distribution task where we took also into account, activities as cleaning,

covering the table and buying the ingredients. We found that Diogo needs to work 2

hours and 39 minutes, Jimmy and Mats need to work 5 hours. More precise, Diogo

buys all the ingredients (111,11 min), Jimmy covers the table (166,67min) and

helps Mats with the cooking (92,31min) and Mats does the cleaning (138,46 min)

and the cooking (161,53 min).

References:

Cooking Recipes:

http://www.receptjes.be/recept_gevulde_kalkoen.html

http://www.ah.nl/recepten/recept?id=678698

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http://www.receptjes.be/recept_gevulde_kalkoen.htmlhttp://www.receptjes.be/recept_gevulde_kalkoen.htmlhttp://www.ah.nl/recepten/recept?id=678698http://www.ah.nl/recepten/recept?id=678698http://www.receptjes.be/recept_gevulde_kalkoen.htmlhttp://www.ah.nl/recepten/recept?id=678698

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