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DETAILS MY NOTES TANFIN12 3.4.002. PRACTICE ANOTHER
A company wishes to produce two types of souvenirs: Type $A$ and Type $B$ . Each Type $A$ souvenir will result in a profit of $\$ 1.40$ , and each Type $B$ souvenir will result in a profit of $\$ 1.60$ . To manufacture a Type $A$ souvenir requires 2 minutes on Machine I and 1 minute on Machine II. A Type B souvenir requires 1 minute on Machine I and 3 minutes on Machine II. There are 2 hours available on Machine I and 5 hours available on Machine II. (a) For a meaningful solution, the time available on Machine II must lie between $\square$ and $\square$ min. (Enter your answers from smallest to largest.) (b) If the time available on Machine II is changed from 300 min to $(300+k) \mathrm{min}$ , with no change in the maximum capacity for Machine I, then Ace Novelty's profit is maximized by producing $\square$ Type $A$ souvenirs and $\square$ Type $B$ souvenirs, where $\square$ $\leq k \leq$ $\square$ (c) Find the shadow price for Resource 2 (associated with constraint 2). (Round your answer to the nearest cent.) \\square$ Need Help? Read It Watch It

Studdy Solution

STEP 1

1. Each Type A souvenir results in a profit of $1.40.<br />2. Each Type B souvenir results in a profit of$ 1.60.
3. Type A requires 2 minutes on Machine I and 1 minute on Machine II.
4. Type B requires 1 minute on Machine I and 3 minutes on Machine II.
5. Machine I has 2 hours (120 minutes) available.
6. Machine II has 5 hours (300 minutes) available.
7. We need to determine the time range for Machine II, optimize production with a change in Machine II's time, and find the shadow price for Resource 2.

STEP 2

1. Determine the time range for Machine II for a meaningful solution.
2. Optimize production with a change in Machine II's time.
3. Find the shadow price for Resource 2.

STEP 3

Determine the time range for Machine II for a meaningful solution.
To find the meaningful range for Machine II, we need to consider the constraints of Machine I and the profit maximization. Machine I's constraint is 2 hours (120 minutes), and Machine II's constraint is 5 hours (300 minutes).
For a meaningful solution, the time on Machine II must allow for at least some production of Type A or Type B souvenirs. Therefore, the minimum time on Machine II should be enough to produce at least one Type A or Type B souvenir.
For Type A: Requires 1 minute on Machine II. For Type B: Requires 3 minutes on Machine II.
Thus, the minimum time on Machine II should be at least 1 minute.
The maximum time on Machine II is given as 300 minutes.
Therefore, the time range for Machine II is between 1 and 300 minutes.

STEP 4

Optimize production with a change in Machine II's time.
Given that the time available on Machine II changes from 300 minutes to $(300+k)$ minutes, we need to determine the optimal production quantities of Type A and Type B souvenirs.
The constraints are: - Machine I: $2A + 1B \leq 120$ - Machine II: $1A + 3B \leq 300 + k$
The objective function to maximize profit is: $\text{Profit} = 1.40A + 1.60B$
To solve this, we need to use linear programming techniques, such as the graphical method or the simplex method, to find the optimal values of $A$ and $B$ .
However, without specific values for $k$ , we can only determine the general approach. The range for $k$ will depend on the feasible region defined by the constraints and the intersection points of the lines representing the constraints.

STEP 5

Determine the range for $k$ .
The range for $k$ is determined by the point at which the constraint for Machine II becomes non-binding. This occurs when the additional time $k$ does not increase the feasible region for production.
To find this, we need to calculate the intersection points of the constraints and analyze the impact of increasing $k$ on the feasible region.
Without specific calculations, we can state that $k$ must be such that the constraint $1A + 3B \leq 300 + k$ remains binding or non-redundant.

STEP 6

Find the shadow price for Resource 2.
The shadow price for Resource 2 (Machine II) is the rate at which the objective function (profit) would increase with a one-unit increase in the availability of Machine II time, assuming all other constraints remain constant.
To find the shadow price, we need to calculate the change in the optimal value of the objective function per unit increase in the constraint for Machine II.
This requires solving the linear programming problem and analyzing the sensitivity report or using duality theory.

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