Reference no: EM132181492

Part A - Homework

Directions - The focus of this homework is on integer programming and multi-objective programming. You may talk to your classmates about the problems, but must turn in your own unique solution set. Please provide printouts of all spreadsheets/LINGO utilized to complete the homework.

Q1. Management of the Albert Franko Co. has established goals for the market share it wants each of the company's two new products to capture in their respective markets. Specifically, management wants Product 1 to capture at least 15 percent of its market and Product 2 to capture at least 10 percent of its market. Three advertising campaigns are being planned to try to achieve these market shares. One is targeted directly on the first product. The second targets the second product. The third is intended to enhance the general reputation of the company and all its products. Let x₁, x₂, and x₃ be the amount of money allocated (in million of dollars) to these respective campaigns, the resulting market share (expressed as a percentage) for the two products are estimated to be: Market share for Product 1 = 0.5x₁ + 0.2x₃ and Market share for Product 2 = 0.3x₂ + 0.2x₃. A total of $55 million is available for the three advertising campaigns, but management wants at least $10 million devoted to the third campaign.

(a) Formulate a goal programming model of this problem. (i.e., In place of an objective function you should have one or more goals.) Be sure to provide any constraints placed on your decisions.

(b) Formulate a model, and use this model to find a solution for this problem given that there is a clear priority placed on Product 1.

(c) Formulate a model, and use this model to find a solution for this problem given that there is a clear priority placed on Product 2.

(d) Suppose that management believes that the goals for each product have roughly the same priority. More precisely, suppose that if both market share goals cannot be achieved, management considers each i percent decrease in the market share from the goal to be equally serious for the two products (in other words, Product 1's market share being 14 percent is the same as Product 2's market share being 9 percent). Formulate a linear programming model for this situation and solve it using Excel Solver or LINGO. Provide the intuition behind allocating the marketing in this manner.

Q2. A developing country has 15,000,000 acres of publicly controlled agricultural land in its use. The government is planning a way to divide this land among three basic crops (labeled 1, 2, and 3) next year. A certain percentage of each of these crops is exported to obtain badly needed foreign capital (dollars) and the rest of each of these crops will be used to feed the population. Raising these crops provides employment for a proportion of the population. Therefore, the main factors that are considered in allocating the land to these crops are (1) the amount of foreign capital generated, (2) the number of citizens fed, and (3) the number of citizens employed in raising these crops. The following table shows how much each 1,000 acres of the various crops contribute towards these factors and the last column gives the goal established by the government for each of these factors.

(a) Formulate a goal programming model for this problem. (i.e In place of an objective function you should have one or more goals.) Be sure to provide any constraints placed on your decisions.

(b) The government has concluded that the following deviations from the goals are equally undesirable: each $100 under the foreign-capital goal, each person under the citizen-fed goal, and each deviation of one from the citizens employed goal. Formulate a linear programming model to solve the goal programming under these deviations. Use Excel Solver or LINGO to determine how to allocate the land across crops.

(c) The government has indicated that they are not too confident about the scale of the weight placed on the citizen-fed goal. In particular, they tend to believe that it is more important than the other two listed goals. Provide an appropriate analysis that will determine the level of importance that needs to be placed on the citizen-fed goal before the government should deviate from the plan from Part (b). Hint: Consider the role that the weights for this goal play into the solution.

Q3. The advertising division of a major company is planning how to allocate their advertising budget for TV programming during the spring season. There are ten options for shows to advertise during, each option has its own cost and 'reach' for two targeted sections of the population. The table below provides the reach (in thousands) of advertising on show i for each of the targeted sections and the cost (in hundreds of thousands) to advertise on the show.

The advertising budget is $3,000,000.

(a) You have been asked to provide an analysis in order for the advertising division to understand the trade-offs between targeting the two different sections of the population. They would like you to propose four different efficient solutions (where the objectives focus on either targeting Section 1 or targeting Section 2) in allocating their advertising budget. Provide this set of solutions along with the weights you placed on each objective in obtaining each of these four solutions.

(b) For the fall season, the advertising division has already chosen to sacrifice 25% of the best possible reach for Section i and 5°/0 of the best possible reach for Section 2. You have been hire to determine their advertising allocation for the Spring season. Management wants you to 'balance' the total sacrifices (Spring + Fall) across the two sections as best as possible. State what allocation achieves this balance, and provide supporting analysis to back up the selection.

Q4. Consider a restaurant that wants to decrease the time customers must wait for their food or equivalently speed up service. The cost to speed up service changes as there are multiple ways to improve service including hiring more workers, buying improved kitchen equipment, and buying improved computer systems, corresponding to 3 levels of improvement. Taking these into account, we have generalized this problem such that I can purchase units of speed up at different costs provided in the table below. To interpret this table consider the following example. If I desire my service to be 6 minutes faster I must first purchase 5 units from level 1 at a cost of $3/unit and then 1 unit from level 2 at a cost of $7/unit for a total of $22. Formulate an integer program of this problem determining how many units of speed up I should purchase such that I minimize cost.

Part B -

Directions - Formulate an integer program for the following description. Be sure to include definitions of decision variables, objective function, and constraints. Use this integer program to determine an optimal solution to the following program. Interpret your solution.

The five residents of Hometown live in houses represented by the letters "A" through "E" as shown on the left side of Figure 1. The offices where they will be working are represented by their matching letters on the island of Worktown.

Because a river lies between Hometown and Worktown, the residents are unable to get to work. They have in their budget enough funds to build two bridges that could connect Hometown to Worktown. The locations where these bridges could be built are indicated by the brown 1x3 hashed tiles. The two bridges can only be built in these approved areas.

Once the bridges are built, the residents would then be able to commute to work. A commuter will always take the shortest path from home to work and can only travel in up, down, left or right directions (no diagonals). Each tile represents a 1-km-by-1-km distance. As an example, if bridge four were built, resident "E" would have to travel lo km to reach his workplace.

Question: Which two bridges should be built in order to minimize the total commuting distance of all residents?