PROGRAMMING
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Pure INTEGER Integer Program. max z = 3x1 + 2x2 x1 + x2 6 x1 int x2 int x1 0 x2 0
Mixed Integer Program. max z = 3x1 + 2x2 x1 + x2 6 x1 int x2 real x1 0 x2 0
Saigal (U of M)
IOE 310
1/1
INTEGER PROGRAMMING
0-1 IP. max z =
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3x1 + 2x2 x1 + 2x2 2 2x1 - x2 1 x1 = 0 or 1 x2 = 0 or 1
LP Relaxation. max z = 3x1 + 2x2 x1 + x2 6 x1 0 x2 0
Saigal (U of M)
IOE 310
2/1
FORMULATION Fixed Charge Case
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A company produces shirts, shorts and pants. It must rent machines for the production. The data follows: Item M/c Rental Fee Labor/unit Cloth/unit Sales Price Variable cost Shirt $200 3 hrs 4 yrds $ 12 $6 Short $150 2 hrs 3 yrds $8 $4 Pant $100 6 hrs 4 yrds $ 15 $8 Max Avail 150 hrs 160 yrds
Formulate an IP to maximize profit.
Saigal (U of M)
IOE 310
3/1
FORMULATION Continued
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Decision Variables: x1 = number of shirts produced x2 = number of shorts produced x3 = number of pants produced y1 = 1 0 if any shirts produced if no shirts produced , y2 = 1 0 if any shorts produced if no shorts produced
, y3
Saigal (U of M)
IOE 310
4/1
FORMULATION Continued
Rental Cost:
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200y1 + 150y2 + 100y3 . Profit: (12 - 6)x1 + (8 - 4)x2 + (15 - 8)x3 . THE MODEL: max 6x1 + 4x2 + 7x3 - 200y1 - 150y2 - 100y3 labor: 3x1 + 2x2 + 6x3 cloth: 150 4x1 + 3x2 + 4x3 160 In addition we need: x1 M1 y1 x2 M2 y2 x3 M3 y3
Saigal (U of M) IOE 310 5/1
FORMULATION Continued
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Mi are large numbers. OPTIMUM SOLUTION: z = 75, x3 = 25, y3 = 1.
Saigal (U of M)
IOE 310
6/1
EITHER OR CONSTRAINTS
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Given two constraints expressed by functions f and g : f (x1 , x2 , , xn ) 0 g (x1 , x2 , , xn ) 0 how do we express that at least one of these constraints is satisfied? Answer: Define y= 0 1 if constraint 1 must be satisfied if constraint 2 must be satisfied
and add the constraints below to the formulation: f (x1 , x2 , , xn ) My g (x1 , x2 , , xn ) M(1 - y )
Saigal (U of M) IOE 310 7/1
EITHER OR CONSTRAINTS
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where M is a very large number that renders the constraints f (x1 , x2 , , xn ) M g (x1 , x2 , , xn ) M redundant.
Saigal (U of M)
IOE 310
8/1
IF THEN CONSTRAINTS
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In many cases if some constraint f (x1 , x2 , , xn ) > 0 is satisfied, then the constraint g (x1 , x2 , , xn ) 0 must be satisfied, i.e., f (x1 , x2 , , xn ) > 0 implies g (x1 , x2 , , xn ) 0 This is satisfied by including the following in the formulation: -g (x1 , x2 , , xn ) My f (x1 , x2 , , xn ) M(1 - y ) y = 0 or 1
Saigal (U of M)
IOE 310
9/1