Goal Programming Answers:

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1)  Note that since the demand goal stated "meet the demand as much as possible" production deviation either above or below demand is considered undesirable for both types of televisions. Also note that all apparent "hard" constraints are superceded by goal constraints.
 

Let:	C = # of color televisions produced
	B = # of black and white televisions produced
	ds+ = deviation above skilled labor utilization target
	ds- = deviation below skilled labor utilization target
	du+ = deviation above unskilled labor utilization target
	du- = deviation below unskilled labor utilization target
	dc+ = deviation above color television demand target
	dc- = deviation below color television demand target
	db+ = deviation above black and white television demand target
	db- = deviation below black and white television demand target
	do+ = deviation above unskilled labor overutilization target
	do- = deviation below unskilled labor overutilization target

Minimize Z =

ds+ + du- + dc+ + dc- + db+ + db- + do+

s.t.	10 C + 5 B - ds+ + ds-	=	100	(Skilled use goal)
	100 C + 150 B - du+ + du-	=	1,500	(Unskilled use goal)
	C - dc+ + dc-	=	70	(Color demand goal)
	B - db+ + db-	=	45	(B&W demand goal)
	du+ - do+ + do-	=	100	(Unskilled overuse goal)

where:

C, B, ds+,  ds-, du+, du-, dc+, dc-, db+, db-, do+, do- > 0 and C, B are integer

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2)  Note that an arbitrarily high number has been chosen for the RHS of the final goal constraint to "maximize the audience points." Also note that all apparent "hard" constraints are superceded by goal constraints.
 

Let:	R = # of local radio spots purchased
	T = # of local television spots purchased
	N = # of newspaper ads purchased
	db+ = deviation above budget target
	db- = deviation below budget target
	dc+ = deviation above television contract target
	dc- = deviation below television contract target
	dp+ = deviation above newspaper policy target
	dp- = deviation below newspaper policy target
	da+ = deviation above audience points target
	da- = deviation below audience points target

Minimize Z =

10 db+ + 6 dc- + 3 dp+ + da-

s.t.	100 R + 500 T + 200 N - db+ + db-	=	200,000	(Budget goal)
	500 T - dc+ + dc-	=	30,000	(TV contract goal)
	200 N - dp+ + dp-	=	50,000	(Newspaper goal)
	30 R + 150 T + 150 N - da+ + da-	=	1,000,000	(Audience goal)

where:

R, T, N, db+,  db-, dc+, dc-, dp+, dp-, da+, da- > 0 and  R, T, N are integer

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3)  Note that a heuristic is employed in assigning objective coefficients in order to prioritize goal compliance. The variable representing the deviation with the lowest priority is assigned a weight of "1" and the weights are increased by a factor of 10 for each successively more important goal. The resulting solution, however, does not allow compliance with goal 3 before that for goal 4. There is no feasible solution that allows compliance with goal 3 if goals 1 and 2 are both satisfied. Also note that only the total acres available for building remains as a "hard" constraint.
 

Let:	S = # of single-familiy homes built
	C = # of deluxe condominiums built
	A = # of apartments built
	D = # of acres of land not used for building
	df+ = deviation above families housed target
	df- = deviation below families housed target
	dt+ = deviation above tax base target
	dt- = deviation below tax base target
	ds+ = deviation above city services tax target
	ds- = deviation below city services tax target
	dp+ = deviation above park set-aside target
	dp- = deviation below park set-aside target

Minimize Z =

1,000 df- + 100 dt- + 10 ds+ + dp-

s.t.	.25 S + .2 C + .125 A + D	=	50	(Acres available)
	S + 4 C + 6 A - df+ + df-	=	500	(Families goal)
	5 S + 10 C + 15 A - dt+ + dt-	=	500	(Tax base goal)
	4 S + 8 C + 10 A - ds+ + ds-	=	250	(City service tax goal)
	D - dp+ + dp-	=	5	(Park goal)

where:

S, C, A, D, df+,  df-, dt+, dt-, ds+, ds-, dp+, dp- > 0 and  S, C, A are integer

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4)

Let:	X1 =	# of tons of plywood produced
	X2 =	# of tons of chipboard produced
	X3 =	# of tons of pulp produced
	X4 = {	1 if plywood is produced 0 if otherwise
	X5 = {	1 if chipboard is produced 0 if otherwise
	X6 = {	1 if pulp is produced 0 if otherwise
	dp+ =	deviation above profit target
	dp- =	deviation below profit target
	dg+ =	deviation above glue requirement target
	dg- =	deviation below glue requirement target
	da+ =	deviation above additive requirement target
	da- =	deviation below additive requirement target

Minimize Z =

5 dp- + dg+ + da+

s.t.	.95 X1 + .91 X2 + .86 X3	<	15,000	(Wood available)
	X1 - 10,000 X4	<	0	(Ply demand)
	X2 - 5,000 X5	<	0	(Chip demand)
	X3 - 5,000 X6	<	0	(Pulp demand)
	10 X1 + 9 X2 + 6 X3 - 50,000 X4 - 25,000 X5 - 2,000 X6 - dp+ + dp-	=	70,000	(Profit goal)
	.05 X1 + .05 X2 - dg+ + dg-	=	500	(Glue goal)
	.04 X2 + .14 X3 - da+ + da-	=	500	(Additives goal)

where:

X1, X2, X3, X4, X5, X6, dp+, dp-, dg+, dg-, da+, da- > 0,  X1, X2 are integer, and X4, X5, X6 are binary

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