The Toledo Leather Company has been producing leather goods for more than $30$ years. It purchases prepared hides from tanners and produces leather clothing accessories such as wallets, belts, and handbags. The firm has just developed a new leather product and has prepared a $1-$year production and sales plan for it$.$ The new product is best described as a combination billfold, key case, and credit card carrier. As company president Peggy Lane has noted, $"$It is a super carryall for small this$-$and$-$that." Lane has placed her administrative assistant, Harold Hamilton, in charge of the project.

Hamilton has established that material and variable overhead for the carryall should be about $$1\mathrm{.}50$ per case over the next year, given a $5-$day week and no overtime. Unit labor and machining costs, however, depend on the choice of machine that will be used for production. Hamilton has narrowed the choice down to two specialized pieces of equipment. Machine $1$ is a semi$-$automated machine that will cut the material to the size needed for one unit and also will sew it$,$ install the rings and snaps, and emboss it with two types of designs. This machine costs $$250,000$ and will add $$2\mathrm{.}50$ per case to the average variable cost for labor and other machine$-$related costs. This piece of equipment has a production capacity of $640$ units per day. However, estimated downtime for maintenance and repairs is $12\mathrm{.}5\%(1/8$th of the total time$).$

Machine $2$ is fully automated. It cuts, sews, installs rings and snaps, and is capable of embossing the case with three types of designs. This machine costs $$350,000$ and will add $$1\mathrm{.}75$ per case to the average variable cost for labor and other machine$-$related costs. Machine $2$ has a higher production capacity $($estimated at $800$ units per day$)$ than the semi$-$automated machine. However, estimated downtime is $25\%(1/4$th of the total time$),$ consistent with its great complexity.

Marketing estimates for the next year have been more difficult to project than production costs and capacity estimates. However, $$6\mathrm{.}00$ seems the likeliest selling price for the carryall. The price brings it in line with somewhat comparable products on the market, but because the carryall offers more features than these other products, it has the potential to outsell them. Sales volume estimates center on $140,000$ units for the year, but analysis of the potential market has been difficult because this new product is so different from the products now being sold. Hamilton's best estimates of sales at $$6\mathrm{.}00$ per unit and the probabilities attached to these volumes are as follows:

Sales Volume Probability

$120,000$ units $0\mathrm{.}15$

$130,000$ units $0\mathrm{.}25$

$140,000$ units $0\mathrm{.}40$

$150,000$ units $0\mathrm{.}15$

$160,000$ units $0\mathrm{.}05$

Given these marketing estimates and the machine capacities, the company will have to decide either to modify the machines to increase capacity or work overtime if demand is at the higher levels. Management can make this decision based on the first week's worth of sales, which are expected to be a good indicator of the annual sales level. Overtime premiums would raise the costs by $$1\mathrm{.}20$ per case on the semi$-$automated machine and by $$0\mathrm{.}90$ on the fully automated machine. Modification of Machine $1,$ the semi$-$automated one, would cost $$15,000$ to meet the highest level of sales. Modifications of Machine $2$ would cost $$20,000.$

Lane has directed Hamilton to make a decision based on first$-$year sales, since demand for a product such as this is uncertain after its initial popularity passes. Toledo operates on a fifty$-$week year because the company usually closes down for the winter holidays.

$1.$ Using a decision tree based on maximizing expected profit decide which machine Toledo Leather should select. Should overtime be scheduled? Or$,$ should a machine be modified and if so$,$ under what circumstances? Please show step by step.

$2.$ Set up a payoff matrix for the sales volumes given $($assume the machines cannot be modified and overtime is used$),$ and assume that the probability of occurrence of the five levels of sales are not known. Then, decide which machine should be purchased using the maximax criterion, the maximin criterion, and the equally likely criterion. Please show step by step.