The Surest Concrete Company produces cement in a continuous process. Two ingredients in the cement are sand, which Surest purchases for $6 per ton, and gravel, which costs $8 per ton. The other ingredients are lumped together and referred to as “Other”. The cost of the other ingredients is $5 per ton. No more than 40 percent of the cement can be sand, and at least 30 percent of the cement must be gravel. The maximum amount of gravel available is 100 tons. The maximum amount available of the “Other” ingredients is 125 tons. The cement sells for $60 per ton. Fractional values of the inputs and outputs are allowed.

The company wants to maximize its profit.

a) Formulate the liner programming problem to maximize the profit.

b) Using linear programming, solve for the optimal solution, identifying the value of the objective function and the variables at optimality. Provide your linear programming solution.

c) Using the sensitivity analysis output from part b, provide any two sensitivity analysis interpretations. One of the interpretations must relate to the objective function and the second one must relate to one of the constraints.

d) How would your answers to a) and b) change, if you wanted to minimize cost and in addition to the other constraints you had to produce at least 325 tons of cement? Using linear programming, solve for the optional solution. Provide your linear programming solution.

You must submit your linear programming formulation and show the linear programming software solution to this problem to receive credit.

#2 Pik-Fast, a chain of convenience stores, is planning to open a new store. The activities required, their immediate predecessors, and the optimistic, most likely, and pessimistic estimates of their times (in days) are provided below.

a) Provide the project network.
b) Identify the critical path and projected project completion time using the expected completion times.
c) Identify the slack times for all activities.
d) Related to the critical path, identify the slack time for each of the other paths.
e) Find the probability that the critical path will be completed within 80 working days.
f) Illustrate or explain why the probability that the critical path will be finished in 80 days or less is not necessarily the probability that the project will be finished in 80 days or less. Assume that times for each activity will not change from what is provided below and no new activities are added nor are the predecessor relationships changed.

Activity Description Immediate Predecessors Optimistic Time (days) Most Likely Time (days) Pessimistic Time (days)
A Select manager — 10 12 20
B Select site — 8 15 22
C Hire workers A 17 21 31
D Basic training C 3 4 5
E Secure permits B 12 18 30
F Construct building E 26 30 40
G Purchase equipment — 30 45 60
H Install equipment F,G 4 6 8
I Advertise opening B 15 55 89
J Final training D,H 2 3 4
K Stock store H,I 4 6 8
L Open store J,K 1 1 1

#3 The Austin, Texas plant of Computer Products produces external hard drives for personal and small business computers. Gerald Knox, the plant’s production planning director, is looking over next year’s sales forecasts for these products and will be developing an aggregate capacity plan for the plant. The quarterly sales forecasts for the external hard drives are as follows and we will assume that these are accurate:

1st Quarter 2nd Quarter 3rd Quarter 4th Quarter
2,550 2,610 2,520 2,700

Ample machine capacity exists to produce the forecast. Each external hard drive takes an average of 20 labor-hours. In addition, you have collected the following information:

a. Inventory carrying cost is $100 per external hard drive per quarter. The inventory to use in this calculation is the inventory at the end of the quarter.
b. The plant works the same number of days in each quarter, 12 five-day weeks, 6 hours per day.
c. There are 300 units in inventory at the start of this year. They will be used to fill the immediate 1st Quarter demand.
d. The cost of hiring a worker is $800 while the cost of laying off a worker is $950.
e. The straight time labor rate is $22 per hour during the entire year.
f. Overtime work is paid at time and a half (150%) of the straight time work.
g. Outsourcing (contract work) is paid at the rate of $475 per disk unit for the labor and you provide the material
h. The number of workers the company had at the end of last year could produce 2,430 units per quarter. The demand for the first quarter of the next year (year following the year you are analyzing) is projected to be at the 2,700 unit level.
i. In a backlog situation, the customer will wait for his order to be filled but will expect a price reduction each quarter he waits. The backlog costs are $300 per external hard drive for the first quarter the customer waits, $700 for the second quarter the customer waits, and $900 for the third quarter the customer waits. In filling orders, backlog items will always be filled before current quarter items,

Which of the following plans would provide the lowest production cost? Assume that the material and overhead costs are the same in each plan.

i) This plan will continue to use the number of workers employed at the end of last year through this entire year. If the workforce can produce more units during a quarter then there is demand, the workers will still produce all of the units. These units will be used to meet demand in a future quarter. Any additional units required in a quarter that cannot be produced with the given workforce will be obtained through outsourcing.

ii) Using a matching demand plan that would ensure there is no inventory or shortage each quarter. The number of workers used will match the demand so there will be no underutilization and no overtime.

iii) A form of level production where the company successfully shifted the demand so that it was uniform (constant) throughout the entire year. To do this, the company had to reduce the price of each unit shifted to another quarter by $500. The number of workers used will match the demand so there will be no underutilization and no overtime.