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ENGM 732 Network Flow Programming

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ENGM 732Network Flow Programming

Network Flow Models

Hirem-Firem supplies clerical employees upon demand to various business concerns in the area. The best available estimates are

Month1234

Demand20253218

Available25202520

Costs to hire and / layoff each month are

Month1234

Hire170200200150

Fire220240200200

Pay equals $250 per month.

- Build a network flow model to capture this problem

[0,25,170]

1

(lower bound, capacity, cost()

[Fixed, Max Slack, Cost]

(20,M,250)

[0,20,200]

2

(0,M,220)

(25,M,250)

(0,M,240

[0,25,200]

[0,-M,0]

3

6

(0,M,200)

(22,M,250)

[0,20,50]

4

(0.M,200)

(18,M,250)

5

[0,0,170]

AGE is based in RC, SD with branch offices in every major grain producing country in the world. Recent crop failures in ADXP have given rise for humanitarian service (and profit). Since it is probable the next wheat harvest will be successful, AGE believes it has a five month window to distribute grain to ADXP. Costs and revenues follow.

Purchase Sale Holding TonsWarehouse

Month Cost Price Cost Available CapacityDemand

1 85 95 2 6000 5000 3500

2709047000 40002500

3 10510065000 40004000

49510043000 60006000

5103 11043000 70004500

[0,60,85]

1

[Fixed, Slack Flow, Cost]

Purchase the wheat each month

[0,70,70]

2

[0,50,105]

3

[0,30,95]

4

[0,30,103]

5

[0,60,85]

1

[Fixed, Slack Flow, Cost]

(Capacity, Cost)

(50,2)

Purchase the wheat each month

Include a model for holding some

wheat from month to month

[0,70,70]

2

(40,4)

[0,50,105]

3

(40,6)

[0,30,95]

4

(60,4)

[0,30,103]

5

(M,-95)

[0,60,85]

1

6

[Fixed, Slack Flow, Cost]

(Capacity, Cost)

(50,2)

(M,-90)

Purchase the wheat each month

Include a model for holding some

wheat from month to month

Sell the wheat

[0,70,70]

2

7

(40,4)

(M,-100)

[0,50,105]

3

8

(40,6)

(M,-100)

[0,30,95]

4

9

(60,4)

(M,-110)

[0,30,103]

5

10

(M,-95)

[0,60,85]

[-35]

1

6

[Fixed, Slack Flow, Cost]

(Capacity, Cost)

(50,2)

(M,-90)

Purchase the wheat each month

Include a model for holding some

wheat from month to month

Sell the wheat

Meet the demand

[0,70,70]

2

7

[-25]

(40,4)

(M,-100)

[0,50,105]

3

8

[-40]

(40,6)

(M,-100)

[0,30,95]

4

9

[-60]

(60,4)

(M,-110)

[0,30,103]

5

10

[-45]

Generous owns primary warehouses in Minneapolis, Lincoln, and Denver. We want to resupply secondary warehouses from these three at lowest cost while meeting the demand. Transport costs are

MinneapolisLincoln Denver Demand

Aberdeen4 5 830

Vermillion74925

Chamberlain66835

Rapid City76550

Available80 70 90

[Fixed]

M

Push available supply at

primary warehouses

[80]

[70]

L

D

[90]

[Fixed]

(cost)

A

(4)

(7)

M

Push available supply at

primary warehouses

Ship to secondary

[80]

V

(6)

(7)

[70]

L

C

D

[90]

R

[Fixed]

(cost)

A

(4)

(7)

M

Push available supply at

primary warehouses

Ship to secondary

[80]

V

(6)

(7)

[70]

L

C

D

[90]

R

[Fixed]

(cost)

A

[-30]

(4)

(7)

M

Push available supply at

primary warehouses

Ship to secondary

Meet demand at secondary

[80]

V

[-25]

(6)

(7)

[70]

L

C

[-35]

D

[90]

R

[-50]

[Fixed]

(cost)

A

[-30]

(4)

(7)

M

Push available supply at

primary warehouses

Ship to secondary

Meet demand at secondary

We have 80+70+90=240 flow in

And

30+25+35+50=140 out.

Flow in must = flow out

How do we resolve?

[80]

V

[-25]

(6)

(7)

[70]

L

C

[-35]

D

[90]

R

[-50]

[Fixed]

(cost)

A

[-30]

(4)

(7)

M

Push available supply at

primary warehouses

Ship to secondary

Meet demand at secondary

Add a dummy node

[80]

V

[-25]

(6)

(7)

[70]

L

C

[-35]

D

[90]

R

[-50]

(0)

(0)

(0)

D

[-100]