Contour lines for the problem of locating a machine shop described in Problem 28, needs to be drawn. Concept Introduction: The line that passes through or drawn through points having the same value is called a contour line.

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Question

Chapter 11.9, Problem 33P

Summary Introduction

Interpretation: Contour lines for the problem of locating a machine shop described in Problem 28, needs to be drawn.

Concept Introduction: The line that passes through or drawn through points having the same value is called a contour line.

Expert Solution & Answer

Answer to Problem 33P

Contour lines have been drawn along the plotted points.

Explanation of Solution

Given information: Contour lines for locating the machine shop is drawn from the values of points derived.

The procedure for drawing contour lines is as follows:

The points need to be plotted on a graph paper, and horizontal and vertical lines need to be drawn along their respective axis.
The horizontal and vertical lines, needs to be clearly numbered from left to right.
It is being assumed, $Di$ to be the sum of weights of points along the horizontal line $i$ and $Ci$ be the sum of weights of points along the vertical line $j$ .
Following is the method for computing numbers:

$M1=M0+2C1 N0=N0+2D1M2=M1+2C2 N1=N1+2D2$

Where $Ci$ represents the total of the weights added up along the vertical line and $Di$ represents the total of the weights added up along the horizontal line.

A region $(i,j)$ can be defined as bounded by the $ith$ vertical line and the $(i+1)th$ vertical line and horizontally by the $jth$ horizontal line and $(j+1)th$ horizontal line. The area below the first horizontal line is represented by $(i,0)$ , and that towards the left of the first vertical line is represented by $(0,j)$ . The contour line’s slope is given by the formula:

$Sij=−MiNj$

After calculating the slopes of the contour lines, it is drawn by starting from any point, moves forward along the predetermined slopes, as calculated from the above mentioned formula and at an angle

The weights and coordinates of the locations of machines are being tabulated as follows:

Location of machines	Weight
(3,3)	$18$
(3,7)	$18$
(8,4)	$14$
(12,3)	1
(14,6)	$16$

Let us assume the following:

$(a1,b1)=(3,3); (a2,b2)=(3,7); (a3,b3)=(8,4); (a4,b4)=(12,3); (a5,b5)=(14,6)$

and $w1=18; w2=18; w3=14; w5=16$ . With these points the horizontal and vertical lines are drawn along the axis and the values of $M0,M1,M2,M3,M4$ and $N0,N1,N2,N3,N4$ are calculated.

$M0=−(18+18+14+1+16)=−56M1=−53+12=−76M2=−76+12=−23M3=−23+2=43M4=43+26=53$

$N0=−(18+18+14+1+16)=−56N1=53+94=712N2=712+12=1312N3=1312+13=1712M4=1712+14=53$

Further, in order to find the slope, the ratios are calculated:

$S0,0=−1S1,0=−(−76)/(−53)=−0.7S2,0=−(−23)/(−53)=−0.4S3,0=−(43)/(−53)=−0.8S4,0=−(53)/(−53)=0.1$ $S0,1=−(−53)/(712)=−2.8S1,1=−(−76)/(712)=12S2,1=−(−23)/(712)=1.1S3,1=−(43)/(712)=−2.2S4,1=−(53)/(712)=−2.8$

$S0,2=−(−53)/(1312)=1.5S1,2=−(−76)/(1312)=1.07S2,2=−(−23)/(1312)=0.61S3,2=−(43)/(1312)=−1.2S4,2=−(53)/(1312)=−1.5$ $S0,3=−(−53)/(1712)=1.17S1,3=−(−76)/(1712)=0.82S2,3=−(−23)/(1712)=0.47S3,3=−(43)/(1712)=−0.94S4,3=−(53)/(1712)=1.17$

$S0,4=−(−53)/(53)=1S1,4=−(−76)/(53)=0.7S2,4=−(−23)/(53)=0.4S3,4=−(43)/(53)=−0.8S4,4=−(53)/(53)=−1$

From the derived points, the contour lines has been plotted as shown in the figure below: EBK PRODUCTION AND OPERATIONS ANALYSIS, Chapter 11.9, Problem 33P