Help with Optimization and Decision Support Modeling for Business HW2

Module 3-2 Practice Problems OSCM 471/571 Optimization and Decision Support Modeling for Business

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Study Materials: Lecture 3-2: LP What-if Analysis

1. Consider the Big M Co. problem presented in Section 3.5, including the spreadsheet in Figure 3.10 showing its formulation and optimal solution.

There is some uncertainty about what the unit costs will be for shipping through the various shipping lanes. Therefore, before adopting the optimal solution in Figure 3.10, management wants additional information about the effect of inaccuracies in estimating these unit costs.

Use Solver to generate the sensitivity report preparatory to addressing the following questions.

a. Which of the unit shipping costs given in Table 3.9 has the smallest margin for error without invalidating the optimal solution given in Figure 3.10? Where should the greatest effort be placed in estimating the unit shipping costs? Answer:

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All of the unit costs have a margin of error of 100 in at least one direction (increase or decrease). Factory 1 to Customer 2 and Factory 2 to Customer 2 have the smallest margins for error since it is 100 in both directions.

b. What is the allowable range for each of the unit shipping costs? Answer: The allowable range for Factory 1 to Customer 1 is Unit Cost≤ $800. The allowable range for Factory 1 to Customer 2 is $800 ≤ Unit Cost ≤ $1,000. The allowable range for Factory 1 to Customer 3 is Unit Cost ≥ $700. The allowable range for Factory 2 to Customer 1 is Unit Cost ≥ $700 The allowable range for Factory 2 to Customer 2 is $800 ≤ Unit Cost ≤ $900. The allowable range for Factory 2 to Customer 3 is Unit Cost ≤ $800.

c. How should the allowable range be interpreted to management? Answer: The allowable range for each unit shipping cost indicates how much that shipping cost can change before you would want to change the shipping quantities used in the optimal solution.

d. If the estimates change for more than one of the unit shipping costs, how can you use the sensitivity report to determine whether the optimal solution might change? Answer:

Use the 100% rule for simultaneous changes in objective function coefficients. If the sum of the percentage changes does not exceed 100%, the optimal solution definitely will still be optimal. If the sum does exceed 100%, then we cannot be sure.

2. University Ceramics manufactures plates, mugs, and steins that include the campus name

and logo for sale in campus bookstores. The time required for each item to go through the two stages of production (molding and finishing), the material required (clay), and the corresponding unit profits are given in the following table, along with the amount of each resource available in the upcoming production period.

Module 3-2 Practice Problems OSCM 471/571 Optimization and Decision Support Modeling for Business

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A linear programming model has been formulated in a spreadsheet to determine the production levels that would maximize profit. The solved spreadsheet model and corresponding sensitivity report are shown below.

For each of the following parts, answer the question as specifically and completely as is

possible without re-solving the problem with Solver. Note: Each part is independent (i.e., any change made in one part does not apply to any other parts).

a. Suppose the profit per plate decreases from $3.10 to $2.80. Will this change the optimal production quantities? What can be said about the change in total profit? Answer: The decrease is within the allowable decrease, so the optimal production quantities stay the same. Total profit will decrease by ($0.30)(300) = $90 to $2440.

b. Suppose the profit per stein increases by $0.30 and the profit per plate decreases by $0.25. Will this change the optimal production quantities? What can be said about the change in total profit? Answer: $0.30 is 0.30/0.65 = 46.2% of the allowable increase for steins. $0.25 is 0.25/0.37 = 67.5% of the allowable decrease for plates. 46.2% + 67.5% > 100%, so the optimal production quantities may or may not change.

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The change in total profit can not be definitively determined since it is not certain whether or not the production quantities change.

c. Suppose a worker in the molding department calls in sick. Now eight fewer hours are available that day in the molding department. How much would this affect total profit? Would it change the optimal production quantities? Answer: 8 hours, or 480 minutes, is within the allowable decrease for molding, so the shadow price is valid. The change in total profit is therefore ∆Profit = ($0.22)(–480) = –$105.60. The optimal production quantities will change.

d. Suppose one of the workers in the molding department is also trained to do finishing. Would it be a good idea to have this worker shift some of her time from the molding department to the finishing department? Indicate the rate at which this would increase or decrease total profit per minute shifted. How many minutes can be shifted before this rate might change? Answer: The shadow price for finishing ($0.28) is higher than the shadow price for molding ($0.22), so shifting minutes from molding to finishing would be beneficial, and would add $0.06 to total profit per minute shifted. This rate will remain valid at least until the 100% rule is violated. If x is the number of minutes shifted, the 100% rule will be violated when x/600 + x/2400 > 100%, or when x > 480 minutes.

e. The allowable decrease for the mugs’ objective coefficient and for the available clay constraint are both missing from the sensitivity report. What numbers should be there? Explain how you were able to deduce each number. Answer: 300. The shadow price is 0 because there is slack in this constraint. The shadow price will remain 0 so long as there is slack. There will remain slack so long as the right-hand side decreases no more than 300 minutes.

3. Consider the Super Grain Corp. case study as presented in Section 3.4, including the spreadsheet in Figure 3.7 showing its formulation and optimal solution. Use Solver to generate the sensitivity report. Then use this report to independently address each of the following questions.

a. How much could the total expected number of exposures be increased for each additional $1,000 added to the advertising budget? Answer:

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The total number of expected exposures can not be increased by adding an additional $1,000 to the advertising budget.

b. Your answer in part a would remain valid for how large of an increase in the advertising budget? Answer: This remains valid for any increases.

c. How much could the total expected number of exposures be increased for each additional $1,000 added to the planning budget? Answer: The total number of expected exposures can be increased by 35,000 by adding an additional $1,000 to the advertising budget.

d. Your answer in part c would remain valid for how large of an increase in the planning budget? Answer: This remains valid for increases of up to $22,500.

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e. Would your answers in parts a and c definitely remain valid if both the advertising budget and planning budget were increased by $100,000 each? Answer: Percentage of allowable increase for ad budget = (4,100 – 4,000) / ∞ = 0% Percentage of allowable increase for planning budget = (1,100 – 1,000) / 22.5 = 444% The sum is 444% > 100%, so the shadow prices may or may not be valid.

f. If only $100,000 can be added to either the advertising budget or the planning budget, where should it be added to do the most good? Answer: $100,000 is beyond the allowable increase for the planning budget. Therefore, the total impact of adding $100,000 to the planning budget can not be determined without re-solving. However, it would certainly be more worthwhile adding to the planning budget (35,000 additional exposures for each $1,000 spent up to $22,500) than adding to the advertising budget which would not increase the expected number of exposures at all.

g. If $100,000 must be removed from either the advertising budget or the planning budget, from which budget should it be removed to do the least harm? Answer: The $100,000 should be removed from the advertising budget. Since the shadow price is zero for the advertising budget (and the allowable decrease is $225,000), this will have no impact on the total number of exposures.

4. Reconsider the example illustrating the use of robust optimization that was presented in the

lecture note. Wyndor management now feels that the analysis described in the example was overly conservative for three reasons: (1) it is unlikely that the true values of the parameters 𝐻𝐻𝐷𝐷3 and 𝐻𝐻𝑊𝑊3, will be as far as half an hour off of the original estimate, (2) it is even more unlikely that both estimates will turn out to simultaneously lean toward the undesirable end of their range of uncertainty, and (3) there is a bit of latitude in the constraint to compensate for violating it by a tiny bit. Therefore, Wyndor management has asked its staff (you) to slove the model again while using ranges of uncertainty that are half as wide as those used in the lecture note below:

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a. Apply the procedure for robust optimization with independent parameters. What is the resulting optimal solution and how much would this increase the total profit per week? Answer: They should produce 1.385 doors and 6 windows per week. This increases the total profit by $158 as compared to the more conservative solution.

b. If Wyndor would need to pay a penalty of $150 per week to the distributor if the production rates fall below these new guaranteed minimum amounts, should Wyndor use these new guarantees? Answer: Probably yes. The increase in profit that would be expected every week ($158) exceeds the penalty of $150 that would need to be paid on occasion.