Ballco manufactures large softballs, regular softballs, and hardballs. Each type of ball requires time in three departments: cutting, sewing, and packaging, as shown in Table 65 (in minutes). Because of marketing considerations, at least 1,000 regular softballs must be produced. Each regular softball can be sold for $3, each large softball, for$5; and each hardball, for $4. A total of 18,000 minutes of cutting time, 18,000 minutes of sewing time, and 9,000 minutes of packaging time are available. Ballco wants to maximize sales revenue. If we define RS = number of regular softballs produced LS = number of large softballs produced HB = number of hardballs produced then the appropriate LP is

$\begin{aligned} \max z &=3 R S+5 L S+4 H B \\ \text { s.t. } & 15 R S+10 L S+8 H B \leq 18,000 \quad \text{(Cutting constraint)} \\ & 15 R S+15 L S+4 H B \leq 18,000 \quad \text{(Sewing constraint)}\\ 3 R S+4 L S+2 H B & \leq 9,000 \quad \text{(Packaging constraint)}\\ R S & \geq 1,000 \quad \text{(Demand constraint)}\\ R S, L S, H B & \geq 0 \end{aligned}$

The optimal tableau for this LP is shown in Table 66. a. Find the dual of the Ballco problem and its optimal solution. b. Show that the Ballco problem has an alternative optimal solution. Find it. How many minutes of sewing time are used by the alternative optimal solution? c. By how much would an increase of 1 minute in the amount of available sewing time increase Ballco’s revenue? How can this answer be reconciled with the fact that the sewing constraint is binding? d. Assuming the current basis remains optimal, how would an increase of 100 in the regular softball requirement affect Ballco’s revenue? TABLE 65:

$\begin{matrix} \text{Bills} & \text{Cutting Time} & \text{Sewing Time} & \text{Packaging Time}\\ \text{Regular softballs} & \text{15} & \text{15} & \text{3}\\ \text{Large softballs} & \text{10} & \text{15} & \text{4}\\ \text{Hardballs} & \text{8} & \text{4} & \text{2}\\ \end{matrix}$

TABLE 66:

$\begin{matrix} \text{z} & \text{RS} & \text{LS} & \text{HB} & \text{s1} & \text{s2} & \text{s3} & \text{e4} & \text{a4} & \text{rhs}\\ \text{1} & \text{0} & \text{0} & \text{0} & \text{0.5} & \text{0} & \text{0} & \text{4.5} & \text{M − 4.5} & \text{4,500}\\ \text{0} & \text{0} & \text{0} & \text{1} & \text{0.19} & \text{−0.125} & \text{0} & \text{0.94} & \text{−0.94} & \text{187.5}\\ \text{0} & \text{0} & \text{1} & \text{0} & \text{−0.05} & \text{0.10} & \text{0} & \text{0.75} & \text{−0.75} & \text{150}\\ \text{0} & \text{0} & \text{0} & \text{0} & \text{−0.17} & \text{−0.15} & \text{1} & \text{1.88} & \text{1.88} & \text{5,025}\\ \text{0} & \text{1} & \text{0} & \text{0} & \text{0} & \text{0} & \text{0} & \text{−1} & \text{1} & \text{1,000}\\ \end{matrix}$