The Beaver Creek Pottery Company produces bowls and mugs. The profits of bowls and mugs are $40 and $50 respectively. A bowl requires 1 hour of labor; whereas, a mug requires 2 hours of labor. 40 hours of labor are available daily. Each bowl requires 4 pounds of clay and a mug requires 3 pounds. The daily limit of clay is 120 pounds.

The pottery company has several goals:

achieving a profit level of at least $1600 per day;

maintaining the current level of labor hours per day (i.e. 40 hours, no overtime);

preferring not to keep more than 120 pounds on hand each day (Because the clay must be stored in a special place so that it does not dry out).

Let

x₁ = number of bowls produced; x₂ = number of mugs produced

Further, given the penalty weights in the rightmost column of Table 1, let W be the number of penalty points incurred by missing these goals. The overall objective is to choose the values of x1 and x2 so as to

Minimize W = 5(amount under the profit goal)

+ 2(amount over the level of labor hours per day goal)

+ 1(amount under the level of labor hours per day goal)

+ 3(amount over the material goal).

Table 1

Factor	Bowls	Mugs		Goal	Penalty
Profit	40	50		?1,600	0(+), 5(-)
Labor Hrs	1	2		=40	2(+), 1(-)
Material	4	3		?120	3(+), 0(-)
Unknown*:	Deviations (or di+(OverGoal i), di- (UnderGoal i)) (i=no. of goals)

*auxiliary variables (extra variables that are helpful for formulating the Linear Goal Programing model)

Question: Formulate a linear goal programming model for this problem to determine how many bowls and how many mugs should be produced in order to achieve these goals. How much of each product should be produced, and what is the revised total contribution to profit?