Question

3. Waupaca is a foundry plant that does both sand casting and die-casting. It produces two types of connecting rods: Type A and B both by pre-casting and then metal-cutting; the associated costs to produce connecting rod types A and B are $13 and $15, respectively. The plant has 120 hours of casting and 36 hours of metal-cutting capacity per week. In addition, two people work halftime and one person works fulltime for casting and metal- cutting making available number of labor hour capacity 80 hours per week. Table 3 summarizes casting, metal-cutting, and labor hours required to produce each type of connecting rods. Demand for type A and B is 4 and 20, respectively. Table 3: Data for # of hours required to produce connecting rods Activity of hours required for #of hours required for connecting rod of Type-A connecting rod of Type-B Casting 4 Metal-cutting Labor hour 2 Formulate an LP model in order to minimize the total manufacturing cost for the plant.

Answer 1

Given:

Two types of connecting rods, Type A and Type B.

Associate cost to produce Type A rod = $13

Associate cost to produce Type B rod = $15

Activity	Hours required for connecting rod Type A	Hours required for connecting rod Type B	Total hours the plant works for in a week
Casting	3	4	120
Metal-cutting	2	1	36
Labor hour	1	3	80

Also, demand for Type A rod = 4

Demand for Type B rod = 20

Aim:

Minimize the total manufacturing cost for the plant, i.e,

13x + 15y =minimum total cost

Assumptions:

x = Number of Type A connecting rods

y = Number of Type B connecting rods

Solution:

We form all the equations to carry out the Linear Programming model in order to minimise the total manufacturing cost for the plant.

1) 3x + 4y $\geq$ 120

2) 2x + 1y  $\geq$ 36

3) 1x + 3y $\geq$ 80

4) y $\leq$ 20

5) x $\leq$ 4

Note:  In equations 1, 2 and 3 "we use less than equal to" because the total hours that the plant works for in a week gives us the maximum hours the plant can work and the production can stop before fully utilizing the plant hours. Similarly, equations 4 and 5 give us the demand for Type A and B rods in the market, this is the minimum number of rods we must produce to fulfill the market demand.

Now, we make the LP model to solve our problem,

muro 26 2089202 99099 Os Sh Of se o scot 91 OF $ =100 4 08 = ng + 20 3x+ y = 20 ma ancia -->

The graph above gives us the 4 ordered pairs namely, A, B, C and D that are common in all the equations. One of these points will give us the minimum cost of production.

A(4,20)

B(8,20)

C(5.6,24.8)

D(4,25.3)

substituting the above values in our minimization problem we get,

ans- (13*4) + (15*20) = 352

(13*8) + (15*20) = 404

(13*5.6) +(15*24.8) = 444.8

(13*4) + (15*25.3) = 431.5

Thus we get the minimum cost of production when we produce 4 units of Type A rods and 20 units of Type B rods.