Final Assignment
The Tastee Pie Company has entered a consortium with Equus Riding Stables and
the Ponderosa Lumber Company. Under the terms of the agreement, the Tastee
Pie Company can receive up to 1000 kg of assorted horse meat per week from Equus,
at a cost of $2.00 per kg, and up to 3000 kg of fine sawdust from Ponderosa, at
$0.10 per kg. In return, the Pie company will supply pies to the employee
cafeterias of Equus and Ponderosa.
The Tastee Pie company has two products: Hearty Steak and Kidney Pie, containing
225 gms of horsemeat, 500 gms of fine sawdust, and 50 mg of DuPont Steak-and-Kidney
Artificial Flavour; and Traditional Country Pie, containing 400 gms of horsemeat,
1000 gms of fine sawdust and 25 mg of monosodium glutamate.
(The cost of the artificial flavour and monosodium glutamate is negligible.)
Both pies are encased in
attractive pre-formed pie shells, obtained from Acme Pulp and Paper Products at $0.50
per shell.
Labour costs, depreciation on capital equipment and incidental running costs add up
to an additional total cost of $1.00 per pie.
The Hearty Steak and Kidney pie sells for $3.00, and the Traditional Country
Pie sells for $5.00. Market research has determined that the public demand for pies
does not exceed 4000 pies per week, and the demand for Country pies in particular
does not exceed 2000 pies per week.
The Tastee Pie Company is interested in finding the product mix that will maximize its
profits.
- What is the net profit per pie on each of the types of pie in the
product line?
- Choose suitable decision variables, define them, and write down the
objective function for the problem.
- What are the constraints on the problem? (After writing down the
constraints, you may find it useful to multiply one or two of them by
conversion factors, to reduce the left-hand-side coefficients to manageable size.)
- Solve the problem graphically. (Let your axes go from 0 to 6000 for both
variables; this should allow you to show all the details at a reasonable scale.)
- Give a general definition of a corner-point solution.
(Start your definition with ``In a linear programming problem with n decision variables
and m constraints ... ") Mark the corner point solutions on your graph with dots.
How many are there? (Several of the corner-point solutions may be beyond the edges of
the graph paper. Don't attempt to mark these with dots or to calculate their coordinates,
but include them in your count.)
- Among the solutions you've marked, pick out one example of each of
the following: a feasible solution; an infeasible solution; the optimal
solution.
- Define slack and surplus variables. Introduce slack variables
and construct the initial tableau for solving the pie problem by the simplex
method. Write down the initial augmented corner-point solution.
Solve the problem, using the tableau form of the simplex method.
Write down the final augmented corner-point solution.
What are the shadow prices for the resources used in achieving the
optimal solution?
What is the economic significance of these prices?
- One day, the Chairman of the Tastee Pie Company finds four proposals on his
desk. The first is from Equus, offering to make additional quantities of horsemeat
available for a premium of $0.10 per kg for any amount in addition to the basic 1000 kg.
The second is from
Ponderosa, offering additional quantities of fine sawdust for a premium of $0.01 per kg
for any amount beyond the basic 3000 kg. The third and fourth proposals both come from
Ananias & Belial, the advertising agency retained by Tastee; the agency estimates that
additional demand can be generated for pies by a new advertising campaign. The demand for
Hearty Steak and Kidney can be increased at a cost of $0.50 per additional pie sold;
it has proved harder to motivate consumers to eat Traditional Country Pies, but Ananias &
Belial believe that advertising costing $1.00 per additional pie will do the trick.
Which, if any, of these proposals should he accept?
John Jones
Tues March 13 2000