Operations Research I

1. Costal States (15 points)

Costal States produces eight (8) chemical products that are sold to companies in various industries. Information about the products that Costal States provides on a daily basis is shown in the table below. This table shows profit margin for each product, capacity at each plant for producing the product and the consumption of natural gas required to produce each product. Additionally, the table shows the maximum production rate percentage that can be ascertained for each product on a daily basis. This maximum production rate for each product takes into account market demand requirements for each product and potential system losses in the production process. Note, that Costal States produces only one product (Ammonium Nitrate) to it’s maximum production rate, no other product is be produced at its full capacity rate.

An important ingredient to the production process is natural gas as this provides the heating for the furnaces in the production of each chemical product. The amount of natural gas required for each product is also detailed in the table

Costal States receives 36,000 cubic feet of natural gas per day from Cajun Pipelines. Recently, Cajun Pipelines has informed Costal States that it may see a reduction in natural gas supply by either 20% or 40% in the pursuing winter months. This curtailment is the result of government energy regulations. Use Linear Programming (not integer) to determine the following:

1. What is Costal States maximum profit with no curtailment in the natural gas resource
2. What is the impact to maximum profit and chemical production with a 20% curtailment of natural gas resource
3. What is the impact to maximum profit and chemical production with a 40% curtailment of natural gas resource
4. Suppose there is a 40% curtailment in natural gas resource. How much would Costal be willing to pay for additional gas resources (Assume Costal States is only looking to break even (no additional profits) and how much resources would they purchase at this price (answer this from the confines of sensitivity analysis for the model)

1. Shale Oil (20 points )

Shale Oil, located on the island of Aruba, has a capacity of 600,000 barrels of crude oil per day. The final products from the refinery include two types of unleaded gasoline; regular and premium. The refining process encompasses four stages:

1. The pure crude flows through a distillation tower that produces a feedstock
2. The feedstock output breaks up into two paths; the first path involves feedstock flowing into a cracker unit that refines the mixture into a gasoline stock. The second path has a portion of the feedstock flowing into the blender unit.
3. The gasoline stock (from the cracker unit) feeds into the blender.
4. The blender unit produces the final product, regular or premium gasoline. Both the regular and premium gasoline can be produced from either the feedstock or the gasoline stock during the blending process, although at different production costs. The company estimates that the net profit per barrel of regular gasoline is $5.20 from feedstock and $7.70 from gasoline stock. The corresponding profit values for the premium are $10.40 from the feedstock and $12.30 from the gasoline stock. According to design specifications, it takes five barrels of crude oil to produce one barrel of feedstock. The cracker units cannot use more than 40,000 barrels of feedstock per day. All remaining feedstock is used directly in the blender unit to produce the end product gasoline. The demand limits for regular and premium gasoline are 80,000 and 50,000 barrels per day.
   1. Using Linear Programming, determine the optimum production schedule for the refinery. Allow for the solution variables to be in fractions (non integer)
   2. Suppose the capacity of the distillation tower can be increased to 650,000 barrels per day at cost of $40,000. Is it worth making this investment. (answer this question using sensitivity analysis, stating shadow price, net profit or net loss and rationale. Extract all information from the lindo output used in part a).

1. Giant Motor Company II ( 25 points)
   This problem deals with strategic planning issues for a large company. The main issue is planning the company’s production capacity for the conning year. At issue is the overall level of capacity and the type of capacity - for example, the degree of flexibility in the manufacturing system. You are to use mixed integer programming and zero one programming to solve this problem
   Problem Statement :
   Giant Motor Company (GMC) produces three lines of cars for the domestic (U.S.) market: Lyras, Libras and Hydras. The Lyra is a relatively inexpensive subcompact car that appeals mainly to first-time car owners and to households using it as a second car for commuting. The Libra is a sporty compact car that is sleeker, faster, and roomier than the Lyra. Without any options, the Libra costs slightly more than the the Lyra. Additional options increase the price further. The Hydra is the luxury car of the GMC line. It is significantly more expensive than the Lyra and Libra, and it has the highest profit margin of the three cars.
   Retooling Options for Capacity Expansion
   Currently, GMC has three manufacturing plants in the US. Each plant is dedicated to producing a single line of cars. In its planning for the coming year, GMC is considering the retooling of its Lyra and/or Libra plants. Retooling either plant would represent a major expense for the company. The retooled plants would have significantly increased production capacities. Although having greater fixed costs , the retooled plants would be more efficient and have lower marginal production costs-that is, higher marginal profit contributions, In addition, the retooled plants would be flexible- they would have the capability of producing more than one line of cars.
   The characteristics of the current plants and the retooled plants are given in table 1. The retooled Lyra and Libra plants are prefaced by the word new . The fixed costs and capacities in Table 1 are given on an annual basis. A dash in the profit margin section indicates that the plant cannot manufacture that line of car. For example, the new Lyra plant would be capable of producing both Lyras and Libras but not Hydras. The new Libra plant would be capable of producing any of the three lines of cars. Note, however, that the new Libra plant has a slightly lower profit margin for producing Hydras than the Hydra plant. The flexible new Libra plant is capable of producing the luxury Hydra model but is not quite as efficient as the current Hydra plant that is dedicated to Hvdra production.
   The fixed costs are annual costs that are incurred by GMC independent of the number of cars that are produced by the plant. For the current plant configurations, the fixed costs include property taxes, insurance, payments on the loan that was taken out to construct the plant, and so on. If a plant is retooled, the fixed costs will include the previous fixed costs plus the additional cost of the renovation. The additional renovation cost will be an annual cost representing the cost of the renovation amortized over a long period.
   Table 1 – Plant Characteristics

   	Lyra Plant	Libra Plant	Hydra Plant	New Lyra Plant	New Libra Plant
   Capacity (in 1000s)	900	800	900	1600	1800
   Fixed Cost (in $millions)	2000	2000	2600	3400	3700

   
   Profit Margin by Car line (in $1000s)

   	Lyra Plant	Libra Plant	Hydra Plant	New Lyra Plant	New Libra Plant
   Lyra	2	-	-	2.5	2.3
   Libra	-	3	-	3.0	3.5
   Hydra	-	-	5	-	4.8

   
   Demand for GMC Cars
   Short-term demand forecasts have been very reliable in the past and are expected to be reliable in the future. Longer-term forecasts are not so accurate. The, demand for GMC cars for the coming year is given in Table 2.
   Table 2 Demand for GMC Cars

   	Demand (in 1000s)
   Lyra	1400
   Libra	2100
   Hydra	800

   
   A quick comparison of plant capacities and demands in Tables 1 and 2 indicates that GMC is faced with insufficient capacity. Partially offsetting the lack of capacity is the phenomenon of demand diversion. If a potential car buyer walks into a GMC dealer showroom wanting to buy a Lyra but the dealer is out of stock, frequently the salesperson can convince the customer to purchase the better Libra car, which is in stock. Unsatisfied demand for the Lyra is said to be diverted to the Libra. Only rarely in this situation can the salesperson convince the customer to switch to the luxury Hydra model.
   From past experience GMC estimates that 30% of unsatisfied demand for Lyras is diverted to demand for Libras and 5% to demand for Hydras. Similarly, 10% of unsatisfied demand for' Libras is diverted to demand for Hydras. For example, if the demand for Lyras is 1,400,000 cars, then the unsatisfied demand will be 500,000 if no capacity is added. Out of this unsatisfied demand, 150,000 (=500,000 x .3) will materialize as demand for Libras, and 25,000 (= 500,000 x .05) will materialize as demand for Hydras. Similarly, if the demand for Libras is 1,220,000 cars (1,100,000 original demand plus 120,000 demand diverted from Lyras), then the unsatisfied demand for Lyras would be 420,000 if no capacity is added. Out of this unsatisfied demand, 42,000 (= 420, 000 x . 1) will materialize as demand for Hydras. All other unsatisfied demand is lost to competitors. The pattern of demand diversion is summarized in Table 3
   Table 3 – Demand Diversion Matrix

   	Lyra	Libra	Hydra
   Lyra	N/A	0.3	.05
   Libra	0	N/A	.10
   Hydra	0	0	N/A

   
   Question:
   GMC wants to decide whether or not to retool the Lyra and Libra plants. In addition, GMC wants to determine its production plan at each plant in the coming year. Based on the previous data, formulate a mixed integer programming model for solving GMC's production planning-capacity expansion problem for the coming year.
2. JuiceCo (20 Points) Use Linear program to solve the following problem. Create a lindo model that allows the decision variables to be in fractions: JuiceCo manufactures two products: premium orange juice and regular orange juice. Both products are made by combining two types of oranges: grade 6 and grade 3. The oranges in premium juice must have an average grade of at least 5, those in regular juice, at least 4. During each of the next two months, Juiceco can sell up to 1000 gallons of premium juice and up to 2000 gallons of regular juice. Premium juice sells for $1 per gallon and regular juice sells for $.80 per gallon. At the beginning of month 1 Juiceco has 3000 gallons of grade 6 oranges and 2000 gallons of grade 3 oranges. At the beginning of month 2 Juiceco may purchase additional grade 3 oranges for $.40 per gallon and additional grade 6 oranges for $.60 per gallon. Juice spoils at the end of the month, so it makes no sense to make extra juice during month 1 in hopes of using it to meet month 2 demand. Oranges left at the end of month 1 may be used to produce juice for month 2. At the end of the month a holding cost of $.05 is assessed against each gallon of leftover grade 3 oranges and $.10 against each gallon of leftover grade 6 oranges. In addition to the cost of oranges, it cost $.10 to produce each gallon of (regular or premium) juice. Determine the mixture of oranges that maximizes profit (revenue - cost)
3. First West Chemical ( 20 Points )

First West Chemical Company produces two chemical ingredients for pharmaceutical firms; formula X and formula Y. Production of each ingredient requires two processes. A unit of Formula X requires 4 hours in process 1 and 3 hours in process 2. A unit in formula Y requires 2 hours in process 1 and 5 hours in process 2. The normal operating production times for the two processes are as follows: process 1, 70 hours and process 2, 60 hours. The production process of formula X results in one unit of a by-product called XZ. Four (4) units of X produce 1 unit of XZ. The production process for formula Y yields 5 units of a by-product, YK for each unit of formula Y. The unit profits for formulas X and Yare $10,000 and $15,000 respectively. By products XZ yields $6000 unit profit. By product YK yields a $3000 unit profit for up to 15 units. Because of the limited market and the danger involved in handling the material, any by product YK in excess of 15 units must be destroyed at a unit cost of $4,000. Also due to storage regulations, no more than 15 units of Formula X can be produced.

The management at this chemical company has established the following goals in order of importance:

1. Avoid any underutilization of normal operation hours of each of the two processes. Each of equal importance.
2. Meet the outstanding orders for 8 units of formula X and 7 units of Formula Y. Each of equal importance.
3. Limit any overtime operation of each of the two production process to 10 hours. Each of equal importance.
4. Achieve a profit goal of $220,000
5. Limit the production of by-product YK to 15 units.
6. Minimize the overtime operation of the production processes. Each of equal importance.

Formulate this problem into a mathematical model and determine the solution that best satisfies these goals.

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