STA 222Exercise 5

1. Par Co. Ltd. is a small manufacturer of golf equipment and supplies. The manufacturer has been convinced by its distributor that there is an existing market for both a medium priced golf bag, referred to as a standard model, and a high priced golf bag referred to as a deluxe model. The distributor is so confident of the market that if the manufacturer can make the bags at a competitive price, the distributor has agreed to purchase all the bags that the manufacturer can manufacture over the next three months. A careful analysis of the manufacturing requirements resulted in the following table, which shows the production time requirements for the four required manufacturing operations and the accounting department’s estimate of the profit per bag:

Production Time (hours)Product Cutting and

dyeingSewing Finishing Inspection and

packaging Profit per bag

Standard 1 £10

Deluxe 1 £9

The manager of manufacturing estimates 630 hours of cutting and dyeing time, 600 hours of sewing time, 708 hours of finishing time and 135 hours of inspection and packaging time will be available for the production of golf bags during the next three months.(a) Assuming that the manufacturer would like to maximize profit,

formulate a linear programming problem based on the information provide above.

(b) Using a graphical method, determine the optimum quantities of each type of golf bags to be produced. (540, 252)

(c) What is the profit the manufacturer can earn with the above production quantities? (£7668)

(d) How many hours production time will be scheduled for each operation? (630, 480, 708, 117)

(e) What is the slack time in each operation? (0, 120, 0, 18)

Carry out the following sensitivity test. You may assume that each of the following cases is encountered separately.(f) Suppose that the selling price for the manufacturer’s standard bag must

be reduced by £7 due to competitive pressures. No other costs or prices are affected. What are the optimal production quantities for the manufacturer? (0, 540)

(g) Suppose that the accounting department revises its profit estimate on the deluxe bags to £18 per bag, what would be the optimal solution and the profit? (£10560)

(h) Identify the feasible range of varying the availability of ‘cutting and dyeing’ operation, without changing the optimality of the current solution. Find the shadow price (unit worth) of the ‘cutting and dyeing’ operation within this range. (£4.375 per hour)

2. Ratkins, a local DIY store, has decided to advertise on television and radio but is unsure about the number of adverts it should place. It wishes to minimize the total cost of the campaign and has limited the total number of ‘slots’ to no more than 5. However, it wants to have at least one slot on both media. The company has been told that one TV slot will be seen by 1 million viewers, while a slot on local radio will only be heard by 100000 listeners. The company wishes to reach an audience of at least 2 million people. If the cost of advertising is $5000 for each radio slot and $20000 for each TV slot, how should it advertise?

(Optimum point is 1 slot on radio & 2 slots on tv)

3. A manufacturer produces two products; P and Q, which when sold earn contributions of $600 and $400 per unit respectively. The manufacture of each product requires time on a lathe and a polishing machine. Each unit of P requires 2 hours on the lathe and 1 hour on the polishing machine, while Q requires 1 hour on each machine. Each day, less than 10 hours are available on the lathe and at most 7 hours on the polishing machine. Determine the number of units of P and Q that should be produced per day to maximize contribution.

(Optimum point is 2 units of P & 5 units of Q)

4. Weekly production schedules are needed for the manufacture of two products X and Y. Each unit of X uses one component made in the factory, while each unit of Y uses two of the components, and the factory has a maximum output of 80 components a week. Each unit of X and Y needs 10 hours of subcontracted work and agreements have been signed with subcontractors for a weekly minimum of 200 hours and a maximum of 600 hours. The marketing department says that I can sell all production of Y but there is a maximum demand of 50 units of X, despite a long-term contract to supply 10 units of X to one customer. The net profit on each unit of X and Y is $200 and $300 respectively. (i) Use a graphical method to find an optimal solution. (40, 20)

(ii) What is the total profit can earn for the production quantities in part (i)? ($14000)

5. Novacook Ltd. Makes two types of cooker, one electric and one gas. There are four stages in the production of each of these, with details given in table below. The electric cooker has variable costs of $200 a unit and a selling price of $300 while the gas cooker has variable costs of $160 and a selling price of $240 a unit. Fixed overheads are $60000 a week and the company works a 50-week year. The marketing department suggests maximum sales of 800 electric and 1250 gas cookers a week.

Manufacturing stageTime required (hours per unit)

Total time available (hours a week)Electric Gas FormingMachine shopAssemblyTesting

41062

2842

36001200060002800

(i) Find an optimal product mix for the company using graphical method. What is the expected annual profit?

($2800000)

(ii) Find the use and spare capacity of each manufacturing stage. (Forming: spare 400 hours/week, Machine shop: spare 2400 hours, Assembly: used,

Testing: used, Market X: spare 600, Market Y: spare 50)

(iii) What are the shadow prices of each manufacturing stage? (Assembly: $10 per week, Testing: $20 per week)

6. What is the role of the objective function in a linear programming model?(A) To move around within the feasible region.(B) To find the maximum point.(C) To give the measure by which solutions are judged and find the

optimal solution.(D) To give an area representing solutions those satisfy all constraints.(E) None of the above. (C)

7. A constraint that has not reached its limit at the optimal solution is called a ………………. (slack constraint)

8. In linear programming, when the objective function is parallel to one of the tight constraints, the optimal solution thus obtained is:(A) infeasible(B) unbounded(C) an alternative optimal solution(D) a pseudo-optimal solution(E) none of the above (C)

9.

Referring to the diagram above, the shaded area indicates the feasible solution space of a linear programming problem. X, Y, C1, C2, C3, C4 are the constraints, and Z is the optimum solution.(i) The redundant constraint is:

(A) X(B) C1(C) C2(D) C3 (E) C4 (B)

(ii) The constraints which have ‘slacks’ are:

(A) C1 and C4(B) C2 and C4(C) C2 and C3(D) C1 and C3(E) none of the above (A)

STA222.Ex5