Optimization

Question 1

Ed owns a small business called Kingston Anchors, which produces four types of anchor for pleasure craft:

Danforth: a lightweight, temporary anchor;

Small Plow: an all purpose anchor for boats less than 24';

Large Plow: same thing for boats longer than 24';

Hereshoff: a fancy, show-off anchor for larger boats.

Each anchor requires metal to be cut, ground, tumbled and finally welded and assembled. Because of earlier foresight, Ed has stockpiled plenty of raw material in the form of steel bars, sheets, and tubes. The production data for each anchor of each type in the next cycle is as follows:

Ed needs to know how many of each type of anchor he should plan to produce in order to maximize his contribution to profit. We will define D, SP, LP and H to be the number of Danforths, small plows, large plows and Herreshoffs that Ed decides to produce. Converting all the production times to minutes, the complete formulation will be as follows:

MAXIMIZE \[50D+110SP+130LP+200H\]

Subject to:

Cutting: & 5D & +10SP & +15LP & +30H & \le 7200 \\

Grinding: & 8D & +20SP & +20LP & +20H & \le 14400 \\

Tumbling: & 2D & +5SP & +7.5LP & +10H & \le 6000 \\

Weld/Ass'y: & 4D & +6SP & +8LP & +10H & \le 4800 \\

Danforths: & D & {} & {} & {} & \le 800 \\

Small\begin{matrix}

{} \\

\end{matrix}Plows: & {} & SP & {} & {} & \le 300 \\

L\arg e\begin{matrix}

{} \\

\end{matrix}Plows: & {} & {} & LP & {} & \le 200 \\

Herreshoffs: & {} & {} & {} & H & \le 100 \\

\end{matrix}\]

We must also constrain the number of each type of anchor to be non- negative.

We can enter this formulation into Excel using trial values for the decisions, the number of anchors of each type to be produced. (Using a trial value of 1 lets you quickly ensure that your formulas are correct; that is, that the value of the left hand side, or LHS, of each inequality is correct.) The constraint limits, or Right Hand Sides (RHS) are entered as numbers. The model is shown below.

After calling up the Solver and solving the problem the sheet appears:

The answer and sensitivity reports are as follows:

1. Ed notices that your plan does not call for any Hereshoffs to be produced. He then says that he has promised 2 of these to special customers. What can you tell Ed about the cost of his promise? (3 marks)
2. The strong demand for Danforths has led Ed to think of raising the price, which would result in a $6 increase per anchor in contribution. Ed has evidence that such a small increase in price will not affect the demand for Danforths. What will be the effect on the optimal solution of such a change? (Note: Don’t re-solve the problem. Use your output from a) to indicate what you can say about Ed’s change in price.) (3 marks)
3. Ed has the chance to hire some part-time welding support from another business, but it is expensive. The "standard" welding hour charge which he used to compute the contribution margins was $25/hr , but he will have to pay $75/hr to buy additional welding support. How many hours should Ed buy at this price? (Note that the units of measurement in the solution are minutes which must be converted to hours to complete this part of the question.) (3 marks)
4. Ed's son has agreed to help out by grinding for 30 hours. How will Ed's plan change by having more grinding hours available? (3 marks)
5. Recent competition in the small plow end of the market is causing Ed to consider cutting back on the price, and hence the profit contribution of this anchor type. How far can the contribution margin be cut before the production plan will change? (3 marks)
6. Suppose the metal cutter breaks down and 8 hours of cutting time are lost. Without resolving, what can you say about the effect on profit contribution? (5 marks)

Question 2

Linear programming models are used by many Wall Street and Bay Street firms to select a desirable bond portfolio for their clients. The following is a simplified version of such a model. TAL Private Investments has $1 million to invest for a client and is considering an investment in four bonds. The expected annual return, the worst-case annual return and the duration of each bond are given below. (The duration of a bond is a measure of its sensitivity to interest rate changes.)

TAL wants to maximize the expected return from the investment subject to three constraints that have been placed on it by the client.

1. The worst-case annual return of the portfolio should be at least 8%
2. The average duration of the portfolio must be no more than 6. (For example, a portfolio that was made up of $600,000 in Bond 1 and $400,000 in Bond 4 has an average duration of \[\frac{(600000\times 3)+(400000\times 9)}{600000+400000}=5.4\] It may also help you to recognize that \[\frac{10x}{10x+15y}\le 5\] can be rewritten as \[10x\le 50x+75y\] or \[40x+75y\ge 0\] .)
3. At most, 40% of the portfolio can be in invested in a single bond.

1. Formulate a linear model that will help TAL make the portfolio composition decision. Be sure to define your decision variables clearly including the units of each. (9 marks)
2. Solve the problem using Excel and Solver and submit a printout of your model as is done in Question 1 above. (6 marks)

Question 3

The Pigskin company will produce footballs over the next six months.  Forecasted demand and production costs over this time period are shown in the table below. Pigskin has a monthly production capacity of 300 cases.  The company currently has 50 cases of footballs in inventory, and has enough capacity to store up to 100 cases.  The holding cost of keeping a football in inventory for a month is estimated to be 5% of production cost. Pigskin has decided that they want to meet the entire demand for footballs over the 6-month period.

The selling price of the footballs is not relevant to the production decision since it is company policy to meet demand exactly when it occurs, regardless of selling price. Pigskin wants to find a production schedule that minimizes the total production and holding costs. You may assume that the company wants to have the ending inventory (at the end of month six) to be zero.

Formulate this as a transportation problem using Excel as shown in slide 50 of the PowerPoint presentation for Class 7. Solve it and submit printouts showing your Excel model. (Note that some "routes" are not feasible. For example, production in month two cannot be sold in month one, so a high cost penalty should be assigned to that cell to ensure that the "route" is not used.)

Question 4

The Ferguson Paper Company produces rolls of paper for use in adding machines, desk calculators, and cash registers. The rolls, which are 200 feet long, are produced in widths of 1 ½, 2 ½, and 3 ½ inches. The production process provides 200-foot rolls in 10-inch widths only. The firm must therefore cut the rolls to the desired final product sizes. The seven cutting alternatives and the amount of waste generated by each are as follows:

Cutting Number of Rolls Waste

Alternative 1½ in. 2½ in. 3½ in. (inches)

1 6 0 0 1

2 0 4 0 0

3 2 0 2 0

4 0 1 2 ½

5 1 3 0 1

6 1 2 1 0

7 4 0 1 ½

The minimum production requirements call for 1200 feet of 1½ in. paper, 3000 feet of 2½ in. rolls, and 6000 feet of 3½ in. rolls.

1. If the company wants to minimize the number of units of the 10-inch rolls that must be manufactured, how many 10-inch rolls will be processed on each cutting alternative? How many rolls are required, and what is the total waste (inches)? Assume that fractional rolls can be cut in this problem.
2. If the company wants to minimize the waste generated, how many 10-inch units will be processed on each cutting alternative? How many rolls are required, and what is the total waste (inches)? Comment on the comparison of the solutions of a) and b).

Hint: Think carefully about how you define the decision variables. It might help to think about the kind of instruction that you would have to give to the person in charge on the shop floor.

Price: $29.44

Solution: The downloadable solution consists of 12 pages, 1744 words and 5 charts.
Deliverable: Word Document