Powerhouse produces capacitors at three locations: Los Angles, Chicago and New Y
ID: 3065619 • Letter: P
Question
Powerhouse produces capacitors at three locations: Los Angles, Chicago and New York. Capacitors are shipped from these locations to public utilities in five regions of the country: northeast (NE), northwest (NW), Midwest (MW), southeast (SE) and Southwest (SW). The cost of producing and shipping a capacitor from each plant to each region of the country is given in the table below:
To
From
NE
NW
MW
SE
SW
Los Angeles
$27.86
$4.00
$20.54
$21.52
$13.87
Chicago
$8.02
$20.54
$2.00
$6.74
$10.67
New York
$2.00
$27.86
$8.02
$8.41
$15.20
Each plant has an annual production capacity of 75,000 capacitors. Each year, each region of the country must receive the following number of capacitors: NE, 55,000; NW, 50,000, MW, 60,000; SE, 60,000 and SW 45,000. Powerhouse feels that shipping cost are too high and the company is therefore considering building one or two more production plants. Possible sites are Atlanta and Houston. Each of these plants will have production capacity of 150,000 capacitors. The cost of producing a capacitor and shipping it to each region of the country for these new possible plants are shown below:
From
NE
NW
MW
SE
SW
Atlanta
$8.41
$21.52
$6.74
$3.00
$3.20
Houston
$15.20
$13.87
$10.67
$7.89
$3.50
Answering the following questions(use transportation model with excel solver):
What is the optimal cost and shipping allocation from the original three plants (Los Angeles, Chicago and New York) to the final destinations?
Are there any shortage problems that occur with this original solution? If so, how much of a shortage in number of total capacitors and what regions are affected by the shortage?
What is the total optimal cost of operation with addition of the new plants?
Should both plants be added to the operation or just one plant? If one new plant, which plant?
From
NE
NW
MW
SE
SW
Los Angeles
$27.86
$4.00
$20.54
$21.52
$13.87
Chicago
$8.02
$20.54
$2.00
$6.74
$10.67
New York
$2.00
$27.86
$8.02
$8.41
$15.20
Explanation / Answer
We may either convert the construction costs of the proposed plants into equivalent annual costs, or convert the annual costs over an infinite time period into present values. I have arbitrarily selected the latter. With the given discount rate 0.1111111, an infinite sequence of annual costs of $1/year is equivalent to a present value of ($1/0.111111) = $9. There are four options to consider. For each option, we solve a transportation problem to compute the annual production & shipping cost (exclusive of the fixed operating cost and construction costs). I. No added plants: Shipments f| to r| ¯¯ o| NE NW MW SE SW dummy m|¯¯¯¯¯ ¯¯¯¯¯ ¯¯¯¯¯ ¯¯¯¯¯ ¯¯¯¯¯ ¯¯¯¯¯ 1| 0 50000 0 0 20000 30000 2| 0 0 60000 15000 25000 0 3|55000 0 0 45000 0 0 Cost = 1453700 ($ per year) Present value production & shipping costs : 9×1453700 = $13,083,300 operating costs of plants: 9×3×$50,000 = $1,350,000 construction costs: 0 Total present value: $14,433,300 II. Add plant at Atlanta only Shipments f| to r| ¯¯ o| NE NW MW SE SW dummy m|¯¯¯¯¯ ¯¯¯¯¯ ¯¯¯¯¯ ¯¯¯¯¯ ¯¯¯¯¯ ¯¯¯¯¯ 1| 0 50000 0 0 0 50000 2| 0 0 60000 0 5000 35000 3|55000 0 0 0 0 45000 4| 0 0 0 60000 40000 0 Cost = 978950 ($ per year) Present value production & shipping costs : 9×978950= $8,810,550 operating costs of plants: 9×4×$50,000 = $1,800,000 construction cost of Atlanta plant: $3,000,000 Total present value: $13,610,550 III. Add plant at Houston only Shipments 56:171 O.R. HW#5 Solutions Fall 2001 page 4 of 5 f| to r| ¯¯ o| NE NW MW SE SW dummy m|¯¯¯¯¯ ¯¯¯¯¯ ¯¯¯¯¯ ¯¯¯¯¯ ¯¯¯¯¯ ¯¯¯¯¯ 1| 0 50000 0 0 0 50000 2| 0 0 60000 40000 0 0 3|55000 0 0 0 0 45000 4| 0 0 0 20000 45000 35000 Cost = 992400 Present value production & shipping costs : 9×992400= $8,931,600 operating costs of plants: 9×4×$50,000 = $1,800,000 construction cost of Houston plant: $3,000,000 Total present value: $13,731,600 IV. Add plants at both Atlanta & Houston Shipments f| to r| ¯¯ o| NE NW MW SE SW dummy m|¯¯¯¯¯ ¯¯¯¯¯ ¯¯¯¯¯ ¯¯¯¯¯ ¯¯¯¯¯ ¯¯¯¯¯ 1| 0 50000 0 0 0 50000 2| 0 0 60000 0 0 40000 3|55000 0 0 0 0 45000 4| 0 0 0 60000 0 40000 5| 0 0 0 0 45000 55000 Cost = 745000 Present value production & shipping costs : 9×745,000= $6,705,000 operating costs of plants: 9×5×$50,000 = $2,250,000 construction cost of Atlanta & Houston plants: $6,000,000 Total present value: $14,955,000 The minimum-cost decision is to build the plant at Atlanta. The L.A. plant will then ship 50,000 annually to the NW region. The Chicago plant will ship 60,000 annually the the MW region and 5000 to the SW region. The NY plant will ship 55,000 to the NE region. The Atlanta plant will ship 60,000 to the SE region and 40,000
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