Ace construction currently has three projects under way in various counties in I

Question

Ace construction currently has three projects under way in various counties in Iowa. Each requires a specific supply of gravel. Three gravel pits are available in Iowa to provide for Ace’s needs, but shipping costs per ton differ from location to location. The following lists the shipping costs per ton from each pit to each job, the tonnage allowance (capacity) of each pit, and the tons required for each job. Determine the optimal shipping schedule so as to minimize total costs:

TO

FROM

JOB 1

JOB 2

JOB 3

TONNAGE
ALLOWANCE

Dubuque pit

$6

$8

$10

150

Davenport pit

$7

$11

$11

175

Des Moines pit

$4

$5

$12

275

JOB REQUIREMENTS (TONS)

200

100

300

600

TO

FROM

JOB 1

JOB 2

JOB 3

TONNAGE
ALLOWANCE

Dubuque pit

$6

$8

$10

150

Davenport pit

$7

$11

$11

175

Des Moines pit

$4

$5

$12

275

JOB REQUIREMENTS (TONS)

200

100

300

600

Explanation / Answer

Let X1 be the tonnage from Dubuque pit to Job 1,X2 be the tonnage from Davenport pit to Job 2 , so on till x9

Now Objective condition is to minimise the cost hence

MIN z = 6x1 + 8x2 + 10x3 + 7x4 + 11x5 + 11x6 + 4x7 + 5x8 + 12x9
Subject to,

x1 + x2 + x3 <= 150
x4 + x5 + x6 <= 175
x7 + x8 + x9 <= 275
x1 + x4 + x7 >= 200
x2 + x5 + x8 >= 100
x3 + x6 + x9 >= 300
and x1,x2,x3,x4,x5,x6,x7,x8,x9 >= 0

Hence, optimal solution using online calculator is arrived with value of variables as :
x1=25,x2=0,x3=125,x4=0,x5=0,x6=175,x7=175,x8=100,x9=0

Min z=4525

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