Notation and Case Definitions
Definitions
| cj | º | external price or variable cost (if not sold externally) of product j |
| bi | º | capacity limit of resource input i |
| aji | º | technological coefficient depicting units of resource i required per unit of product j |
| xj | º | quantity of output (or input) of product j |
| si | º | unused (slack) capacity of resource input i |
| xo | º | aggregate contribution margin (revenue less variable cost) of all product output |
| thk | º | artificial variate introduced as a slack variable when two inequality constraints are substituted for one equality constraint for purposes of simplex tableau calculation |
| yi | º | opportunity (shadow price) value (loss) per unit of resource i at its marginal limit of capacity |
| pj | º | reciprocal cost per unit of product j output after adjusting for internal and external losses |
Base Illustration
| b1 | = | gallons of Water Plant capacity |
| b2 | = | cubic feet of steam capacity of Steam Room |
| b3 | = | k.w.h. capacity of Electric Plant |
| b4 | = | units of paper demand capacity of Paper Plant |
| x1 | = | output of Water Plant at variable cost c1 = -13.33 per gal. |
| x2 | = | output of Steam Room at variable cost c2 = -100 per cu. ft. |
| x3 | = | output of Electric Plant at variable cost c3 = -10 per k.w.h. |
| x4 | = | output of Paper Plant at external price c4 = 700 per unit |
| x5 | = | external purchase of steam at variable cost c5 = 200 per cu. ft. |
| 0 | 0 | .8 | a13 | = | .8 k.w.h. of electricity internal requirement per gallon of x1 | |||
| A | = | .5 | 0 | 0 | , where | a21 | = | .5 gallons of water internal requirement per cu. ft. of x2 |
| 0 | .15 | 0 | a22 | = | .15 cu. ft. of steam internal requirement per k.w.h. of x3 |
Alternate Case Situations
| Problem 1 | (Demand
Constraint Active; No Outside Purchase of Steam) xo > 0; s1 > 0; s2 > 0; s3 > 0; s4 = 0; x5 = 0 |
| Problem 2-A | (Water
Plant Constraint Active; No Ourside Purchase of Steam) xo > 0; s1 = 0; s2 > 0; s3 > 0; s4 > 0; x5 = 0 |
| Problem 2-B | (Steam
Room Constraint Active; No Outside Purchase of Steam) xo > 0; s1 > 0; s2 = 0; s3 > 0; s4 > 0; x5 = 0 |
| Problem 2-C | (Electric
Plant Constraint Active; No Outside Purchase of Steam) xo > 0; s1 > 0; s2 > 0; s3 = 0; s4 > 0; x5 = 0 |
| Problem 3-A | (Water
Plant Constraint Active; External Purchase of Steam
Possible) xo > 0; s1 = 0; s2 > 0; s3 > 0; s4 > 0; x5 > 0 |
| Problem 3-B | (Steam
Room Constraint Active; External Purchase of Steam
Possible) xo > 0; s1 > 0; s2 = 0; s3 > 0; s4 > 0; x5 > 0 |
| Problem 3-C | (Electric
Plant Constraint Active; External Purchase of Steam
Possible) xo > 0; s1 > 0; s2 > 0; s3 = 0; s4 > 0; x5 > 0 |