# Equation Descriptions

## Description of Equation 1

Uppercase P equals lowercase a plus the product of lowercase b and uppercase Q. Uppercase P, therefore, equals the sum of two expressions: uppercase P subscript zero minus the quotient of uppercase P subscript zero over uppercase N, and uppercase Q times the quotient of uppercase P subscript zero over the product of uppercase N and uppercase Q subscript zero.

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## Description of Equation 2

Uppercase C equals lowercase a prime minus the product of lowercase b prime and uppercase V M.

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## Description of Equation 3

Lowercase a prime equals the quotient of two expressions: uppercase C subscript zero plus the quotient of uppercase C subscript zero over uppercase N, and 1 plus uppercase D V M. Lowercase b prime equals the quotient of two expressions: uppercase C subscript zero divided by uppercase N times uppercase V M subscript zero, and the square of 1 plus uppercase D V M.

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## Description of Equation 4

Uppercase B E N subscript Net equals uppercase S PLUS subscript 1 minus uppercase S PLUS subscript zero.

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## Description of Equation 5

Uppercase S PLUS subscript zero equals the product of uppercase V M subscript zero and the expression uppercase C subscript zero plus the quotient of uppercase C subscript zero over uppercase N, minus zero point five times the square of uppercase V M subscript zero times the quotient of uppercase C subscript zero over the product of uppercase N and uppercase V M subscript zero, minus the product of uppercase C subscript zero and uppercase V M subscript zero.

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## Description of Equation 6

Uppercase S PLUS subscript 1 equals the product of the expression uppercase C subscript zero plus the quotient of uppercase C subscript zero over uppercase N and the expression uppercase V M subscript 1 divided by 1 plus uppercase D V M, minus the expression zero point five times the square of uppercase V M subscript 1 times the product of the quotient of uppercase C subscript zero over uppercase N times uppercase V M subscript zero and the square of 1 plus uppercase D V M, minus the product of uppercase C subscript zero, 1 minus uppercase D C, and uppercase V M subscript 1.

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## Description of Equation 7

Uppercase C S subscript 1 equals the product of uppercase C subscript zero, 1 minus uppercase D C, and 1 plus uppercase D V M.

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## Description of Equation 8

Uppercase Y equals the output of the function lowercase f with input uppercase S subscript uppercase I, uppercase I subscript lowercase j, uppercase B subscript lowercase k, and uppercase I T subscript lowercase l.

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## Description of Equation 9

Uppercase W T P equals the product of uppercase T S, alpha subscript Log, and the output of the function uppercase F with input uppercase T, eta subscript uppercase T, sigma subscript uppercase T, and eta subscript the variable sigma subscript uppercase T.

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## Description of Equation 10

Uppercase W T P equals the product of uppercase T S, alpha subscript Log, and the expression the product of eta subscript uppercase T and the quotient of delta uppercase T over uppercase T plus the product of eta subscript sigma and the quotient of delta sigma over sigma.

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## Description of Equation 11

Productivity equals the quotient of Output over Input.

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## Description of Equation 12

Delta uppercase P equals the quotient of uppercase O over uppercase I subscript 2 minus the quotient of uppercase O over uppercase I subscript 1. Delta uppercase P therefore equals uppercase O times the quotient of uppercase I subscript 1 minus uppercase I subscript 2 over the product of uppercase I subscript 1 and uppercase I subscript 2. Delta uppercase P is therefore approximately equal to uppercase O times the quotient of delta uppercase C subscript Log over the product of uppercase I subscript 1 and uppercase I subscript 2, which is exactly equal to uppercase P subscript 1 times the quotient of delta uppercase C subscript Log over uppercase I subscript 2.

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## Description of Equation 13

Delta uppercase P prime equals the quotient of uppercase O subscript 2 over uppercase I subscript 2 minus the quotient of uppercase O subscript 1 over uppercase I subscript 1. Delta uppercase P prime therefore equals the quotient of the expression uppercase O subscript 2 times uppercase I subscript 1 minus uppercase O subscript 1 times uppercase I subscript 2 over the product of uppercase I subscript 1 and uppercase I subscript 2.

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## Description of Equation 14

Eta superscript uppercase V M subscript uppercase C equals the quotient of delta uppercase V M divided by uppercase V M over delta uppercase C divided by uppercase C. Eta superscript uppercase V M subscript uppercase C therefore equals the quotient of two expressions: the quotient of delta lowercase V M divided by uppercase V M over delta uppercase T divided by uppercase T, and the quotient of delta uppercase C divided by uppercase C over delta uppercase T divided by uppercase T. Eta superscript uppercase V M subscript uppercase C therefore equals eta superscript uppercase V M subscript uppercase T divided by eta superscript uppercase C subscript uppercase T.

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## Description of Equation 15

Uppercase B lowercase e lowercase n subscript Net equals delta uppercase C S. Uppercase B lowercase e lowercase n subscript Net therefore equals uppercase C S subscript 1 minus uppercase C S subscript zero. Uppercase B lowercase e lowercase n subscript Net therefore equals the difference of two expressions: the integral from zero to lowercase q subscript 1 of the function lowercase p lowercase q lowercase d lowercase q minus the product of lowercase p subscript 1 and lowercase q subscript 1, and the integral from zero to lowercase q subscript zero of lowercase p lowercase q lowercase d lowercase q minus the product of lowercase p subscript zero and lowercase q subscript zero. Uppercase B lowercase e lowercase n subscript Net therefore equals the integral from lowercase q subscript zero to lowercase q subscript 1 of lowercase p lowercase q lowercase d lowercase q minus the product of lowercase p subscript 1 and delta lowercase q plus the product of lowercase q subscript zero and delta lowercase p.

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## Description of Equation 16

Uppercase B lowercase e lowercase n subscript Net equals delta uppercase C S. Uppercase B lowercase e lowercase n subscript Net therefore equals the product of two expressions: the quotient of lowercase a over the product of lowercase q subscript zero and lowercase q subscript 1, all raised to the power of the quotient of 1 over lowercase b, and the sum of 2 times lowercase q subscript zero to the power of the sum of 1 and the quotient of 1 over lowercase b times the expression lowercase b minus 1, plus the product of lowercase q subscript 1 and lowercase q subscript zero to the power of the quotient of 1 over lowercase b, minus the expression lowercase q subscript 1 to the power of the quotient of 1 over lowercase b times lowercase q subscript zero times the expression 2 times lowercase b minus 1, all divided by the expression lowercase b minus 1.

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## Description of Equation 17

Uppercase B lowercase e lowercase n subscript Net equals the product of lowercase a and uppercase V M subscript 2 minus uppercase V M subscript zero, minus the product of the quotient of lowercase b over 2 and uppercase V M subscript 2 squared minus uppercase V M subscript zero squared, minus the product of uppercase C subscript 1 and delta uppercase V M, plus the product of uppercase V M subscript zero and delta uppercase C. In this equation, lowercase a equals uppercase C subscript zero plus the quotient of uppercase C subscript zero over eta superscript uppercase V M subscript uppercase C, and lowercase b equals the product of the quotient of uppercase C subscript zero over eta superscript uppercase V M subscript uppercase C and uppercase V M subscript zero.

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## Description of Equation 18

Delta lowercase p equals delta uppercase C. Delta lowercase p therefore equals the product of uppercase C subscript zero, eta superscript uppercase C subscript uppercase T, and the quotient of delta uppercase T over uppercase T.

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## Description of Equation 19

Lowercase p subscript 1 equals lowercase p subscript zero minus the product of delta uppercase C and eta.

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## Description of Equation 20

Delta lowercase q equals delta uppercase V M. Delta lowercase q therefore equals the product of uppercase V M subscript zero, eta superscript uppercase V M subscript uppercase T, and the quotient of delta uppercase T over uppercase T.

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## Description of Equation 21

Uppercase B lowercase e lowercase n subscript Net equals delta uppercase C S plus delta uppercase C S prime.

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## Description of Equation 22

Uppercase B lowercase e lowercase n subscript Net equals delta uppercase C S plus delta uppercase C S prime. Uppercase B lowercase e lowercase n subscript Net therefore equals the product of three expressions: the quotient of 2 times lowercase b minus 1 over lowercase b, the quotient of lowercase a over the product of lowercase q subscript zero and lowercase q subscript 1, raised to the power of 1 divided by lowercase b, and the quotient of the expression the product of 2, lowercase q subscript zero to the power of the sum of 1 and the quotient of 1 over lowercase b, and lowercase b minus 1 plus the product of lowercase q subscript 1 times lowercase q subscript zero to the power of the quotient of 1 over lowercase b minus lowercase q subscript 1 to the power of the quotient of 1 over lowercase b times lowercase q subscript zero times the expression the product of 2 and lowercase b minus 1, over lowercase b minus 1.

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## Description of Equation 23

Uppercase T C equals the product of uppercase N and the output of the function uppercase C with input uppercase N and uppercase K, plus the output of the function lowercase f with input uppercase K.

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## Description of Equation 24

Delta uppercase T C divided by delta uppercase N equals the output of the function uppercase C with input uppercase N and uppercase K plus the quotient of the product of uppercase N and delta uppercase C over delta uppercase N.

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## Description of Equation 25

Uppercase C equals lowercase t times the sum of lowercase c subscript lowercase h lowercase r times uppercase D and mu times uppercase D, plus the quotient of the product of lowercase a and uppercase D over uppercase Q, plus the quotient of the product of lowercase w and uppercase Q over 2, plus the expression the sum of uppercase Q, lowercase t times uppercase D, and lowercase k prime times sigma times uppercase D, all raised to the power of one-half, times lowercase w times lowercase k, plus the output of the function Backorder Cost with input lowercase t and sigma.

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## Description of Equation 26

Uppercase C superscript uppercase T subscript zero equals uppercase C superscript uppercase T P T subscript zero, plus uppercase C superscript uppercase I N V subscript zero, plus uppercase C superscript uppercase P R O C subscript zero.

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## Description of Equation 27

Uppercase C subscript zero equals uppercase C superscript uppercase T subscript zero divided by uppercase V M subscript zero. Uppercase C subscript zero therefore equals the quotient of uppercase C superscript uppercase T P T subscript zero over uppercase V M subscript zero plus the quotient of uppercase C superscript uppercase OTHER subscript zero over uppercase V M subscript zero.

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## Description of Equation 28

Uppercase D C equals the quotient of uppercase C subscript zero minus uppercase C subscript 1 over uppercase C subscript zero. Uppercase D C therefore equals the sum of two expressions: the quotient of uppercase C superscript uppercase T P T subscript zero minus uppercase C superscript uppercase T P T subscript 1 over uppercase C subscript zero, and the quotient of uppercase C superscript uppercase OTHER subscript zero divided by uppercase V subscript zero minus uppercase C superscript uppercase OTHER subscript 1 divided by uppercase V subscript 1 over uppercase C subscript zero. Uppercase D C therefore also equals the sum of two expressions: the quotient of uppercase C superscript uppercase T P T subscript zero minus uppercase C superscript uppercase T P T subscript 1 over uppercase C subscript zero, and the quotient of the product of uppercase C superscript uppercase OTHER subscript zero and uppercase V subscript 1 divided by uppercase V subscript zero, minus uppercase C superscript uppercase OTHER subscript 1, over the product of uppercase C subscript zero and uppercase V subscript 1.

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