Question

High Point Custom Cabinetry (HPCC) specializes in two- and three-drawer custom made file cabinets, which generate $10 and $15 worth of profit, respectively, for each unit sold. It takes 2 hours of labor to create a two-drawer cabinet, and 3 hours of labor to generate a three-drawer cabinet, and the company has a weekly operating capacity of 1300 hours of labor. The demand for the two-drawer model is up to 600 cabinets per week, and the demand for the three-drawer model is up to 400 per week, but this includes a standing order for 100 three-drawer units/week from a major national retail chain.

HPCC has the following goals in order of importance:

1. Utilize the firm's full production capacity as closely as possible.

2. Attain a profit of at least $11,000 each week.

3. Sell at least as many two-drawer cabinets and three-drawer cabinets as the demand indicates.

Please complete the following:

(a) Set this up as a preemptive goal programming problem.

(b) Reformulate the problem as a general non-preemptive goal programming problem where the goals have different levels of importance.

(c) Reformulate part (b) to optimize the problem using the minimax criteria.

Answer 1

A) Problem formulations only

Decision Variables:

X1= number of 2 Drawer File Cabinet to produce
X2= number of 3 Drawer File Cabinet to produce

Functional Constraints:

X2>=100 (standing order)
X1,X2>=0 (Nonnegativity)
  
Goal equations:

The Goal equation has 3 priority levels :

Priority1:(Goal 1) Utilize firms full production capacity as closely as possible

2X1 + 3X2 +U1 = 1300
  
Priority2:(Goal 2) Attain profit of $11,000 each week

10X1 + 15X2 +U2 = 1100
  
Priority3:(Goal 3 & 4) Sell at least as many 2-drawer cabinets and 3-drawer cabinets as the demand indicates.

X1 +U3 =600
X2 +U4 =400   
  
Priority level objectives :

derimental derivatives are :

Priority1: Underachieving 1300 production capacity (U1)
Priority2: Underachieving 1100 profit (U2)
Priority3: Underachieving 600 and 400 units of production (U3,U4)

Thus corresponding priority objectives are :
  
Priority1: Minimize V1=U1
Priority2: Minimize V2=U2
Priority2: Minimize V3=U3+U4

Solution : Solving has to be done iteratively first

• (X2>=100, X1,X2>=0 ,2X1 + 3X2 +U1 = 1300) ->solve this in graph/excel to get Min(U1)=V1
• (X2>=100, X1,X2>=0 ,10X1 + 15X2 +U2 = 1100) ->solve this in graph/excel to get Min(U2)=V2 [provided V1 constraint is also satisfied]

(X2>=100, X1,X2>=0 , X1 +U3 =600,X2 +U4 =400 ) ->solve this in graph/excel to get Min(U3+U4)=V3 [provided above V1,V2 is also satisfied]

B &C) Non premtive goal programming with each goal is different level of importance WJ (Problem formulations only)  

Minimize W1*U1+W2*U2+W3*U3+W4*U4

ST

1:2X1 + 3X2 +U1= 1300
2: 10X1 + 15X2 +U2= 1100
3: X1 +U3=600
4: X2 +U4=400     
X2>=100

X1,X2,UJ>0 (for all J)

where WJ is the weight assigned for one unit of loss of Jth goal

The above is a linear programming poblem easily solvable in Excel (after putting the required weights)