101 OPIM - Spring 2012
Recitation 8 Exercise
Section # (201,202,,214): _____________
Instructor (circle): Balas
Fishman
Horowitz
Saunders
Shah
Zhang
PennCard Last Name:
PennCard First Name:
Place your answers in the boxes provided.
Problem 1.
Veggie Soup company produces canned soup (hereafter referred to as product) for distribution to
several restaurant suppliers. Currently Veggie Soup operates three plants in the U.S. and distributes to 4
restaurant suppliers (i.e., customers of Veggie Soup). A big component of Veggie Soups costs are the
costs of shipping from its plants to its customers. The unit shipping costs are summarized in the next
table. A unit of product is 1,000 cases of soup.
Table. Unit Shipping Costs from Plants to Customers
(in $ per unit shipped)
Plant l
Plant 2
Plant 3
Cust 1
2.5
3.5
4
Cust 2
4
2.5
5
Cust 3
5
4
2
Cust 4
6
5
3
In the coming year, Veggie Soup has contracted to supply the four customers with 9,000, 10,000,
12,000, and 8,000 units of product, respectively. Each plant can produce in essentially unlimited
quantities at these costs.
Veggie Soup would like to devise an optimal distribution plan making sure that all demand must be
met during the year. In formulating an appropriate linear model, consider the following set of 12 decision
variables:
X11 units shipped from plant 1 to customer 1,
X12 units shipped from plant 1 to customer 2,
X13 units shipped from plant 1 to customer 3,
X14 units shipped from plant 1 to customer 4,
X21 units shipped from plant 2 to customer 1,
X22 units shipped from plant 2 to customer 2,
X23 units shipped from plant 2 to customer 3,
X24 units shipped from plant 2 to customer 4,
X31 units shipped from plant 3 to customer 1,
X32 units shipped from plant 3 to customer 2,
X33 units shipped from plant 3 to customer 3,
X34 units shipped from plant 3 to customer 4
1
Question 1.
In algebraic form, what is the objective function in this problem?
2.5X11 + 4X12 + 5X13 + 6X14 + 3.5X21 + 2.5X22 + 4X23 + 5X24 + 4X31 + 5X32 + 2X33 + 3X34
Question 2.
Including non-negativity, how many constraints does this problem have?
16.
(There are constraints 4 expressing demand for each customer and 12 non negativity constraints.)
Question 3.
Use Excel Solver to determine the optimal product mix (use the HFL template available from the Canvas,
HFL.xls). What is the optimal objective function value (in $)?
$95,500
Question 4.
Suppose that, due to previously accepted commitments, plant 1 cannot produce more than 5,000 units
during the year. Using decision variables defined above, express this constraint in an algebraic form.
X11 + X12 + X13 + X14 ! 5,000
Question 5.
Incorporate this new constraint into your Excel model and resolve it. For the new optimal solution, does
the optimal objective function value change? (circle Yes/No) Does the optimal value of any decision
variable change? (circle Yes/No)
Objective Function: Yes/No
Decision Variables: Yes/No
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Problem 2.
A pharmaceutical company Aspiron is considering financing its leading R&D project for the next 4 years
by investing in high-grade municipal bonds now (in 2012). Aspiron anticipates that the cash
requirements for this project will be $10 million in 2013, and then will grow by $10 million a year for
each of the following 3 years. Companys CEO considers 5 bonds available now with the following prices
and cash flows:
2012
2013
2014
2015
2016
Bond 1
-1.02
0.04
0.04
0.04
1.04
Bond 2
-0.98
0.03
0.03
1.03
Bond 3
-0.96
1.00
Bond 4
-0.92
0.00
1.00
Bond 5
-0.90
0.00
0.00
1.00
In addition, Aspiron can invest in a CD which pays 2% annual interest in any of the years 2013-2016.
Aspiron is trying to decide how much to invest in each of the bonds now to make sure that the funding
requirements for the R&D project are met in the least expensive way. Let the number of units (in
millions) of each bond to be bought are X 1, X2, X3, X4, and X5.
Question 6.
In algebraic form, what is the objective function in this problem?
Cost in $ millions = 1.02X1 + 0.98X2 + 0.96X3 +0.92X4 +0.90 X5.
Question 7.
Use Excel Solver to determine the optimal investment portfolio (use the ToysCash.xls template available
from the Canvas). What is the optimal objective function value (in $ million)?
89.86
3