Linear Programming -Minimisation

Special cases

Simplex maximisation

1.Innis Investments manages funds for a number of companies and wealthy clients. The investment strategy is tailored to each client’s needs. For a new client, Innis has been authorised to invest up to $1.2 million in two investment funds: a stock fund and a money market fund. Each unit of the stock fund costs $50 and provides an annual rate of return of 10%; each unit of the money market fund costs $100 and provides an annual rate of return of 4%.

The client wants to minimise risk subject to the requirement that the annual income from the investment be at least $60,000. According to Innis’s risk measurement system, each unit invested in the stock fund has a risk index of 8, and each unit invested in the money market fund has a risk index of 3; the higher risk index associated with the stock fund simply indicates that it is the riskier investment. Innis’s client has also specified that at least $300,000 be invested in the money market fund.

a.Determine how many units of each fund Innis should purchase for the client to minimise the total risk index for the portfolio. b.How much annual income will this investment strategy generate? c.Suppose the client desires to maximise annual return. How should the funds be invested?

(ASW: Ch 2, Qn 37 – Innis - min)

2.Photo Chemicals produces two types of photographic developing fluids. Both products cost Photo Chemicals $1 per gallon to produce. Based on an analysis of current inventory levels and outstanding orders for the next month, Photo Chemicals’ management has specified that at least 30 gallons of product 1 and at least 20 gallons of product 2 must be produced during the next 2 weeks. Management has also stated that an existing inventory of highly perishable raw material required in the production of both fluids must be used within the next 2 weeks. The current inventory of the perishable raw material is 80 pounds. While more of this raw material can be ordered if necessary, any of the current inventory that is not used within the next 2 weeks will spoil - hence, the management requirement that at least 80 pounds be used in the next 2 weeks. Furthermore, it is known that product 1 requires 1 pound of this perishable raw material per gallon and product 2 requires 2 pounds of the raw material per gallon. Since Photo Chemicals’ objective is to keep its production costs at the minimum possible level, the firm’s management is looking for a minimum-cost production plan that uses all the 80 pounds of perishable raw material and provides at least 30 gallons of product 1 and at least 20 gallons of product 2. What is the minimum-cost solution?

(ASW : Ch 2, Qn 38 – Photo - min)

3.Does the following linear program involve infeasibility, unbounded, and/or alternative optimal solutions? Explain.

Max4x1 + 8x2

s.t.

2x1 + 2x2 < 10

-1x1 + 1x2 > 8

x1, x2 > 0

(ASW : Ch 2, Qn 40 - infeasible)

4.Does the following linear program involve infeasibility, unbounded, and/or alternative optimal solutions? Explain

Max 1x1 + 1x2

s.t.

8x1 + 6x2 > 24

4x1 + 6x2 > -12

2x2 > 4

x1, x2 > 0

(ASW : Ch 2, Qn 41 - unbounded)

5.Reconsider the RMC situation.

a.Identify all the extreme points of the feasible region. b.Suppose RMC discovers a way to increase the profit of solvent base to $60 per ton. Does this change the optimal solution? If so, how? c.Suppose the profit for the solvent base is $50 per ton. What is the optimal solution now? Comment on any special characteristics that may exist with this profit for the solvent base.

(ASW : Ch 2, Qn 47)

6.Reconsider the RMC situation. Suppose that management adds the requirements that at least 30 tons of fuel additive and at least 15 tons of solvent...