Tutorial 2 Exercises

This tutorial trains you to quantify the elements of an optimisation model:

Question 1

With precisely 2700 sq. cm. of cardboard, you wish to construct a box (width: \(x\), depth: \(y\), height: \(z\)) containing a volume \(V\). You require the width to double its depth. You would like to maximise the volume of the box. Which values of \(x, y, z\) will maximise the volume?

Question 2

A manufacturing facility produces four types of wood panelling: Tahoe, Pacific, Savannah, and Aspen. Each product is manufactured using pine chips and oak chips and requires both gluing and pressing processes.

The profit earned per pallet is as follows:

  • $450 for Tahoe,
  • $1140 for Pacific,
  • $800 for Savannah, and
  • $400 for Aspen.

The factory operates under limited resource availability. During the planning period, there are 5,800 quarts of glue available, 730 pressing machine hours, and 29,200 pounds of pine chips. In addition, the supply of oak chips is limited to a fixed total amount.

Tahoe Pacific Savannah Aspen Capacity
Glueing (quarts) 50 50 100 50 5800 quarts
Pressing (hours) 5 15 10 5 730 hours
Pine chips (pounds) 500 400 300 200 29200 pounds
Oak chips (pounds) 500 750 250 500 60500 pounds
Profit per pallet ($) 450 1150 800 400 -

Let decision variables be \(X_1,X_2,X_3,X_4\) (pallets of Tahoe, Pacific, Savannah, Aspen). Write down the optimisation model to maximise profit, and solve for the optimal production plan. What is the maximum profit?

Question 3

Consider the problem of diet optimisation. There are four different types of food: Brownies, Ice Cream, Cola, and Cheese Cake. The nutrition values and cost per unit are as follows:

Brownies Ice Cream Cola Cheese Cake
Calories 400 200 150 500
Chocolate 3 2 0 0
Sugar 2 2 4 4
Fat 2 4 1 5
Cost ($) $0.50 $0.20 $0.30 $0.80

The objective is to minimise the cost of the diet while satisfying the following nutritional requirements:

  • At least 500 calories,
  • At least 5 units of chocolate,
  • At least 10 units of sugar, and
  • At least 8 units of fat.

Question 4

TowAlong makes trailers at Kansas City, Denver, and Raleigh plants and ships these units to Birmingham, Milwaukee, Los Angeles, and Seattle distribution centres. In planning production for the next year, TowAlong estimates Kansas City, Denver, and Raleigh plant’s unit shipping cost between any plant and distribution centre, plant capacities, and distribution centre demands. These numbers are given in the table.

Plant/DC Birmingham Milwaukee LA Seattle Capacity
Kansas City $35 $40 $60 $120 12,000
Denver $30 $30 $45 $130 8,000
Raleigh $60 $65 $50 $100 5,000
Demand 9,000 3,000 9,500 1,500 -

Question 5

Sunchem, a manufacturer of printing inks, has five manufacturing plants worldwide. Their locations and capacities are shown in Table 4 along with the cost of producing 1 ton of ink at each facility. The production costs are in the local currency of the country where the plant is located. The major markets for the inks are North America, South America, Europe, Japan, and the rest of Asia. Demand at each market is shown in Table 4. Transportation costs from each plant to each market in U.S. dollars are shown in Table 4. Management must come up with a production plan for the next year.

  1. If exchange rates are expected as in Table 2, and no plant can run below 50 percent of capacity, how much should each plant produce and which markets should each plant supply?
  2. If there are no limits on the amount produced in a plant, how much should each plant produce?
  3. Can adding 10 percent of capacity in any plant reduce costs?
  4. How should Sunchem account for the fact that exchange rates fluctuate over time?
Table 1. Transportation costs, production capacity, and cost by region
Supply Region N. America S. America Europe Japan Asia Capacity (tons/year) Production cost/ton
United States 600 1,200 1,300 2,000 1,700 185 $10,000
Germany 1,300 1,400 600 1,400 1,300 475 €15,000
Japan 2,000 2,100 1,400 300 900 50
Brazil 1,200 800 1,400 2,100 2,100 210 BRL13,000
India 2,200 2,300 1,300 1,000 800 80 INR400,000
Table 2. Exchange Rates as of April 2025
USD EUR JPY BRL INR
USD 1.0000 0.8794 141.94 5.6336 85.282
EUR 1.1374 1.0000 161.45 6.408 97.0
JPY 0.0070 0.0062 1.0000 0.045 0.60
BRL 0.1775 0.1560 22.22 1.0000
INR 0.0117 0.0103 1.67 0.066 1.0000