These constraints translate mathematically to X, 2 .2(x, + x2 + x3 + x4) x2 2 .1(x, + x2 + x3 + x4) x3 + x4 2 25(x + x2 + x3 + x4) The only remaining constraint deals with keeping the demolishition/construction cost within th allowable budget-that is, (Construction and demolition cost) s (A vailable budget) Expressing all the costs in thousands of dollars, we get (50x, + 70x2 + 130x, + 160x,) + 2xs s 15000 The complete model thus becomes Maximize z - 1000.x, + 1900x2 + 2700x + 3400x, subject to .18x, + 28x2 + .4xy + .5x, - 2125xs s 0 Xs s 300 -.8x, + .2x2 + .2x3 + .2x4 .1x, - .9x2 + .1x3 + .1x4 s0 .25.x, + .25x2 - .75x3 - .75x, SOx, + 70x2 + 130.x3 + 160x4 + 2xs s 15000 X1, X2, Xg, Xa, Xg 20
These constraints translate mathematically to X, 2 .2(x, + x2 + x3 + x4) x2 2 .1(x, + x2 + x3 + x4) x3 + x4 2 25(x + x2 + x3 + x4) The only remaining constraint deals with keeping the demolishition/construction cost within th allowable budget-that is, (Construction and demolition cost) s (A vailable budget) Expressing all the costs in thousands of dollars, we get (50x, + 70x2 + 130x, + 160x,) + 2xs s 15000 The complete model thus becomes Maximize z - 1000.x, + 1900x2 + 2700x + 3400x, subject to .18x, + 28x2 + .4xy + .5x, - 2125xs s 0 Xs s 300 -.8x, + .2x2 + .2x3 + .2x4 .1x, - .9x2 + .1x3 + .1x4 s0 .25.x, + .25x2 - .75x3 - .75x, SOx, + 70x2 + 130.x3 + 160x4 + 2xs s 15000 X1, X2, Xg, Xa, Xg 20
Practical Management Science
6th Edition
ISBN:9781337406659
Author:WINSTON, Wayne L.
Publisher:WINSTON, Wayne L.
Chapter2: Introduction To Spreadsheet Modeling
Section: Chapter Questions
Problem 20P: Julie James is opening a lemonade stand. She believes the fixed cost per week of running the stand...
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Transcribed Image Text:These constraints translate mathematically to
X, z .2(x, + x2 + x3 + x4)
X2 2 .1(x, + x2 + x3 + x4)
X3 + x4 2 .25(x, + x2 + x3 + xa)
The only remaining constraint deals with keeping the demolishition/construction cost within the
allowable budget-that is,
(Construction and demolition cost) s (Available budget)
Expressing all the costs in thousands of dollars, we get
(50x, + 70x2 + 130x, + 160x4) + 2xs s 15000
The complete model thus becomes
Maximize z = 1000.x, + 1900x2 + 2700x3 + 3400x4
subject to
.18x, + .28x2 + .4x3 + .5xs – .2125xs = 0
Xg s 300
-.8x, + .2x2 + .2x3 + .2x4
.1x, - .9x2 + .1xz + .1x4
.25.x, + .25x2 - .75x3 – .75x4
SOx, + 70x2 + 130x3 + 160x4 + 2xs s 15000
X1, X2, X3, X4, X5 2 0
30
Chapter 2
Modeling with Linear Programming
Solution:
The optimum solution (using file amplEX2.3-1.txt or solverEx2.3-1.xls) is:
Total tax collection = z = $343,965
Number of single homes = x, = 35.83 - 36 units
Number of double homes = x2 = 98.53 = 99 units
Number of triple homes = x3 = 44.79 - 45 units
Number of quadruple homes = x, = 0 units
Number of homes demolished = xs = 244.49 - 245 units
Transcribed Image Text:Mathematical Model: Besides determining the nuinber of units to be constructed of each type
of housing, we also need to decide how many houses must be demolished to make room for the
new development. Thus, the variables of the problem can be defined as follows:
x1 = Number of units of single-family homes
x2 = Number of units of double-family homes
x3 = Number of units of triple-family homes
X4 = Number of units of quadruple-family homes
xs = Number of old homes to be demolished
The objective is to maximize total tax collection from all four types of homes-that is,
Maximize z = 1000x, + 1900x2 + 2700x3 + 3400.x4
The first constraint of the problem deals with land availability.
Acreage used for new
Net available
home construction
аcreage
From the data of the problem we have
Acreage needed for new homes = .18x, + .28x2 + .4x3 + .5x4
2.3 Selected LP Applications
29
To determine the available acreage, each demolished home occupies a .25-acre lot, thus netting
.25xs acres. Allowing for 15% open space, streets, and casements, the net acreage available is
.85(.25xs) = .2125xs. The resulting constraint is
.18x, + .28.r2 + .4x3 + .5x4
s.2125xs
or
.18x, + .28x2 + 4x3 + .5x4 – .2125xs s 0
The number of demolished homes cannot exceed 300, which translates to
Xs s 300
Next we add the constraints limiting the number of units of each home type.
(Number of single units) 2 (20% of all units)
(Number of double units) > (10% of all units)
(Number of triple and quadruple units) > (25% of all units)
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