Does any variable have a slack value? If so, what does that mean?

Given the following linear programming formulation:
40x 1 + 25x 2 < 80,000(constraint 1)
20x 1 + 30x 2 < 60,000 (constraint 2)
x 1 + x 2 > 0 (non-negativity constraints
Solve the problem graphically
Solve the problem using excel solver
What is the total objective function?
Do both variables contribute to the solution? Why?
Does any variable have a slack value? If so, what does that mean?

10.2

The cost of providing public services at a local hospital has been scrutinized by management. Although these services are used as marketing tools for the hospital, the cost and availability of scarce resources require their optimal allocation while minimizing costs.
Two popular programs being assessed for this purpose are family planning (FP) and Health drive screenings (HDS) their costs to the hospital each offering $200 and $400 respectively. The health care manager in charge of operations found three common patterns of resource consumption for each of these services and the available resources, Shown Table EX 10.2

Resource type FP HDS Available resources per month
Staff Time 60 120 480 minutes
Materials 30 90 250 kits
Rent Space 1 3 occassions

Formulate this as a linear programming problem
Solve this problem graphically
Solve this problem using excel solver
In a given month how many FP and how many HDS should be offered?
With that proposed class offerings, how many kits will be left over (not distributed in classes?)
What is the yearly cost of these programs to the hospital?

10.4

A hospital is evaluating the feasibility of offerings among three technologies, on the basis of what would make the most profit. These new technologies are:
1. Closed chest by pass surgery with daVinici surgery robot
2. Gamma Knife
3. Positron emission tomography (PET) scanner

Table EX 10.4 gives the information on profit, the amount of common resources used by each of the three technologies per case, and their available resources per month.

Available
daVinici Gamma Knife PET Resources

Profit$ $2,000 $3,500 $2,000
Total Staff Time 15 12 1.5 2,000 hours
Maintenance 25 25 22 1,500 minutes
Computer Resources 20 25 10 3,000 minutes

Formulate this as a linear problem
Solve this problem using Excel solver
Based on optimal solution, which product(s) should be offered, and how many procedures can be offered in a month?
What is the expected contribution of new technology to the hospital’s monthly profits?

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