Quantitative Analysis
Linear Programming
D.V. – Decision Variables
O.F. – Objective Funtion
S.T. or CONST - Constraints
Constrained Mathematical Model – a model with an objective and one or more constraints EXAMPLE: 50D + 30C + 6M is the total profit for a production run($50 profit for Desk, $30 profit for Chair and $6 per pound for steel)
Functional Constraints - ≤ ≥ or = --Restrictions that involve expressions with 1 or more variables EXAMPLE: 7d+3c+1.5M <= 2000 (constraint on raw steel)
Variable Constraints – Involve only 1 variable –
Nonnegativity Constraint - X≥0
Lower Bound Constraint - X≥L(a number other than 0)
Upper Bound Constraint - X≤U
Interger Constraint - X=integer
Binary Constraint - X=0 or 1
EXAMPLE FROM HOMEWORK 1 of Constraints
CONSTRAINTS
FORMULA
WOOD
9X1 + 3x2 ≤ 2250
CUSHIONS
X2 ≤ 500
CHAIRS TO TABLE MIN
4x2:x1 -4x1+x2≤0
CHAIRS TO TABLE MAX
6x2:x1 -6x1+x2≥ 0
PRODUCTION HOURS
X1 + .6x2 ≤375
Given
X1 , x2 ≥ 0
See SELF QUIZ 1 for more info
Step 1 – Define Decision Variables
Step 2 – Overall objective Maximized or Minimized
Step 3 – Set of Constraints
Figure 1 - SAMPLE
Another Example from HW/Quiz
Golden Electronics manufactures several products, including 45-inch GE45 and 60-inch GE60 television. It makes a profit of $50 on each GE45 and $75 on each GE60 television produced. During each shift, Golden allocates up to 300 man-hours in its production area and 240 man-hours in its assembly area to manufacture the televisions. Each GE45 requires two man-hours in the production area and one man hour in the assembly area, whereas each GE60 requires two man-hours in the production area and three man-hours in the assembly area. Formulate the linear program for television production for Golden Electronics.
Answer
X1 = Number of GE45 televisions produced per shift
X2 = Number of GE60 televisions produced per shift
MAX 50X1 + 75X2
S.T. 2X1 + 2X2 300 (Production hours) X1 + 3X2 240 (Assembly hours) X1, X2 0
Recap of Analytic Geometry
D.V. – Decision Variables
O.F. – Objective Funtion
S.T. or CONST - Constraints
Constrained Mathematical Model – a model with an objective and one or more constraints EXAMPLE: 50D + 30C + 6M is the total profit for a production run($50 profit for Desk, $30 profit for Chair and $6 per pound for steel)
Functional Constraints - ≤ ≥ or = --Restrictions that involve expressions with 1 or more variables EXAMPLE: 7d+3c+1.5M <= 2000 (constraint on raw steel)
Variable Constraints – Involve only 1 variable –
Nonnegativity Constraint - X≥0
Lower Bound Constraint - X≥L(a number other than 0)
Upper Bound Constraint - X≤U
Interger Constraint - X=integer
Binary Constraint - X=0 or 1
EXAMPLE FROM HOMEWORK 1 of Constraints
CONSTRAINTS
FORMULA
WOOD
9X1 + 3x2 ≤ 2250
CUSHIONS
X2 ≤ 500
CHAIRS TO TABLE MIN
4x2:x1 -4x1+x2≤0
CHAIRS TO TABLE MAX
6x2:x1 -6x1+x2≥ 0
PRODUCTION HOURS
X1 + .6x2 ≤375
Given
X1 , x2 ≥ 0
See SELF QUIZ 1 for more info
Step 1 – Define Decision Variables
Step 2 – Overall objective Maximized or Minimized
Step 3 – Set of Constraints
Figure 1 - SAMPLE
Another Example from HW/Quiz
Golden Electronics manufactures several products, including 45-inch GE45 and 60-inch GE60 television. It makes a profit of $50 on each GE45 and $75 on each GE60 television produced. During each shift, Golden allocates up to 300 man-hours in its production area and 240 man-hours in its assembly area to manufacture the televisions. Each GE45 requires two man-hours in the production area and one man hour in the assembly area, whereas each GE60 requires two man-hours in the production area and three man-hours in the assembly area. Formulate the linear program for television production for Golden Electronics.
Answer
X1 = Number of GE45 televisions produced per shift
X2 = Number of GE60 televisions produced per shift
MAX 50X1 + 75X2
S.T. 2X1 + 2X2 300 (Production hours) X1 + 3X2 240 (Assembly hours) X1, X2 0
Recap of Analytic Geometry