Question

Kelson Sporting Equipment, Inc., makes two different types of baseball gloves: a regular model and a catcher’s model. The firm has 900 hours of production time available in its cutting and sewing department, 300 hours available in its finishing department, and 100 hours available in its packaging and shipping department. The production time requirements and the profit contribution per glove are given in the following table:

Production Time (hours)
  Model   Cutting and Sewing Finishing   Packaging and Shipping Profit/Glove  

Regular model 1 1/2 1/8 $5  

Catcher's model 3/2 1/3 1/4 $8

Assuming that the company is interested in maximizing the total profit contribution, answer the following:

a. What is the linear programming model for this problem?

b. Find the optimal solution using the graphical solution procedure. How many gloves of each model should Kelson manufacture?

c. What is the total profit contribution Kelson can earn with the given production quantities?

d. How many hours of production time will be scheduled in each department?

e. What is the slack time in each department?

Answer 1

Solution

Let x be the number of regular gloves, and y be the number of cather's gloves.

1 unit of regular glove makes a profit of $5, and 1 unit of cather's makes a profit of $8

Maximimise Profit Z = 5x + 8y

Total hours available :

Cutting (& Sewing) = 900(Constraints)

Finishing = 300

Packaging = 100

Now 1 unit of x requires 1 hr of cutting, 1/2 hr of finishing, and 1/8 hr of Packaging time

And 1 unit of y requires 3/2 hr of cutting, 1/3 hr of finsihing, and 1/4 hr of packaging time

Calculating total hrs required for cutting x numbers of regular & y numbers of catcher's  gloves = 1x + (3/2)y

which must not exceed 900 hrs, the maximum time available for cutting, i.e.

1x + (3/2)y <= 900, Constraint 1, similarly we can write,

(1/2)x + (1/3)y <= 300, Constraint 2, and

(1/8) x+ (1/4)y <= 100 constraint 3

Now to make equalities out of above inequalities, we can add slack(idle) hours, s1, s2, and s3 for cutting, finishing, and packaging respectively. W e have following equations,

x + (3/2) y + s1 = 900 (Eq. 1), Constraint 1

(1/2)x + (1/3) y + s2 = 300 (Eq. 2) 2

(1/8) x + (1/4) y + s3 = 100 (Eq. 3) 3

Maximize Z = 5x + 8y Objective function

x >= 0, y >= 0 Non negativity condition.

a. The three cobstraints, objective function, and non negativity condition make the Linear Programming model

b. Optimal solution (i.e. values of x and y which satisfy above 3 constraints, and maximize profit) is obtained by first plotting the 3 constraints equations, identifying the feasible region, drawing a number  of parallel iso profit lines, and observing the coordinat (x, y) of point which accomodates the highest profit within the feasible region.

That point occures at x = 500 (regular gloves) and y = 150 (Cather's gloves), where z = $3700.

c. Maximum profit obtained = $3700

d. With number of gloves as above, Cutting utilizes only 725 hrs, and its slack (idle time) is 175 hrs.

Slack for both Finishing, and Packaging is 0 hrs (zero, i.e.No idle time).  

See Graph below.