16.7 Final Remarks

Optimization models can help researchers and managers in their business decision-making process.

Among the existing optimization models, we can mention linear programming, network programming, integer programming, nonlinear programming, goal or multiobjective programming, and dynamic programming. Linear programming is one of the most widely used tools.

This chapter introduced and presented the main concepts of optimization models, especially, the modeling of linear programming problems (general formulation in the standard and canonical forms and business modeling problems).

The use of optimization models, mainly linear programming, is being more and more disseminated in academia and in the business world. It may be applied to several areas (strategy, marketing, finance, operations and logistics, human resources, among others) and to several sectors (transportation, automobile, aviation, naval, trade, services, banking, food, beverages, agribusiness, health, real estate, metallurgy, paper and cellulose, electrical energy, oil, gas and fuels, computers, telecommunications, mining, among others). The greatest motivation is the huge saving that may happen, around millions or even billions of dollars, for the industries that use them.

Several real problems can be formulated through a linear programming model, including: a production mix problem, a mixture problem, a capital budget problem, an investment portfolio selection, production and inventory, an aggregated planning, among others.

The methods to solve a linear programming problem (graphical, analytical, by using the Simplex algorithm or by using computerized solutions) will be discussed in the next chapter.

16.8 Exercises

1. (1) Describe the main characteristics present in a linear programming model.

1. (2) Give examples of the main fields and sectors in which the linear programming technique can be applied.

1. (3) Transform the problems into the standard form:
   1. (a) 

      $\begin{array}{l}max\sum _{j=1}^2{x}_j\\ \mathrm{s}.\mathrm{t}.\\ 2x_1-5x_2=10\left(1\right)\\ x_1+2x_2\le 50\left(2\right)\\ x_1,x_2\ge 0\left(3\right)\end{array}$

      

   1. (b) 

      $\begin{array}{l}min24x_1+12x_2\\ \mathrm{s}.\mathrm{t}.\\ 3x_1+2x_2\ge 4\left(1\right)\\ 2x_1-4x_2\le 26\left(2\right)\\ x_2\ge 3\left(3\right)\\ x_1,x_2\ge 0\left(4\right)\end{array}$

      

   1. (c) 

      $\begin{array}{l}max10x_1-x_2\\ \mathrm{s}.\mathrm{t}.\\ 6x_1+x_2\le 10\left(1\right)\\ x_2\ge 6\left(2\right)\\ x_1,x_2\ge 0\left(3\right)\end{array}$

      

   1. (d) 

      $\begin{array}{l}max3x_1+3x_2-2x_3\\ \mathrm{s}.\mathrm{t}.\\ 6x_1+3x_2-x_3\le 10\left(1\right)\\ \frac{x_2}{4}+x_3\ge 20\left(2\right)\\ x_1,x_2,x_3\ge 0\left(3\right)\end{array}$

      

1. (4) Do the same here, but now into the canonical form.

1. (5) Transform the maximization problems into minimization problems:
   1. (a) 

      $maxz=10x_1-x_2$

      

   2. (b) 

      $maxz=3x_1+3x_2-2x_3$

      

1. (6) What are the hypotheses of a linear programming model? Describe each one of them.

1. (7) KMX is an American company in the automobile industry. It will launch three new car models next year: Arlington, Marilandy, and Lagoon. The production of each one of these models goes through the following processes: injection, foundry, machining, upholstery, and final assembly. The average operation times (minutes) of one unit of each component can be found in Table 16.1. Each one of these operations is 100% automated. The number of machines available for each sector can also be found in the same table. It is important to mention that each machine works 16 hours a day, from Monday to Friday. According to the commercial department, besides the minimum sales potential per week, the profit per unit of each automobile model can be seen in Table 16.2. Assuming that 100% of the models will be sold, formulate the linear programming problem that will try to determine the number of automobiles of each model to be manufactured, in order to maximize the company’s weekly net profit.

   Table 16.1

   Average Operation Time (Minutes) of 1 Unit of Each Component and Total Number of Machines Available

   Sector	Operation Average Time (Minutes)			Machines Available
   	Arlington	Marilandy	Lagoon
   Injection	3	4	3	6
   Foundry	5	5	4	8
   Machining	2	4	4	5
   Upholstery	4	5	5	8
   Final assembly	2	3	3	5

   

   Table 16.2

   Profit Per Unit and Weekly Minimum Sales Potential Per Product

   Model	Profit Per Unit (U$)	Minimum Sales Potential (Units/Week)
   Arlington	2500	50
   Marilandy	3000	30
   Gristedes	2800	30

1. (8) Refresh is a company in the beverage industry that is rethinking its production mix of beers and soft drinks. The production of beer goes through the following processes: the extraction of malt (which can be manufactured internally or not), processing the wort, which produces the alcohol, fermenting (the main phase), processing the beer, and filling the bottles up (packaging). The production of soft drinks goes through the following processes: preparation of the simple syrup, preparation of the compound syrup, dilution, carbonation, and packaging. Each one of the beer and soft drink processing phases is 100% automated. Besides the total number of machines available for each activity, the average operation times (in minutes) of each beer component can be found in Table 16.3. The same data regarding the processing of soft drinks can be found in Table 16.4. It is important to mention that each machine works 8 hours a day, 20 business days a month. Due to the existing competition, we can state that the total demand for beer and soft drinks is not greater than 42,000 L a month. The net profit is US$ 0.50 per liter of beer produced and US$ 0.40 per liter of soft drink. Formulate the linear programming problem that maximizes the total monthly profit margin.

   Table 16.3

   Average Beer Operation Time and Number of Machines Available

   Sector	Operation Time (Minutes)	Number of Machines
   Extraction of malt	2	6
   Processing the wort	4	12
   Fermenting	3	10
   Processing the beer	4	12
   Packaging the beer	5	13

   Table 16.4

   Average Soft Drink Operation Time and Number of Machines Available

   Sector	Operation Time (Minutes)	Number of Machines
   Simple syrup	1	6
   Compound syrup	3	7
   Dilution	4	8
   Carbonation	5	10
   Packaging the soft drink	2	5

1. (9) Golmobilec is a company in the electrical appliance industry that is reviewing its production mix regarding the main household equipment used in the kitchen: refrigerators, freezers, stoves, dishwashers, and microwave ovens. The manufacturing of each one of these devices starts in the pressing process that molds, perforates, adjusts, and cuts each component. The next phase consists in the painting, followed by the molding process that gives the product its final shape. The last two phases consist in the assembly and packaging of the product final. Table 16.5 shows the time required (in hours/machine) to manufacture one unit of each component in each manufacturing process, besides the total time available for each sector.

   Table 16.5

   Time Necessary (in Hours/Machine) to Manufacture 1 Unit of Each Component in Each Sector

   Sector	Time Necessary (Hours/Machine) to Manufacture 1 Unit					Time Available (Hours/Machine/Week)
   	Refrigerator	Freezer	Stove	Dishwasher	Microwave oven
   Pressing	0.2	0.2	0.4	0.4	0.3	400
   Painting	0.2	0.3	0.3	0.3	0.2	350
   Molding	0.4	0.3	0.3	0.3	0.2	250
   Assembly	0.2	0.4	0.4	0.4	0.4	200
   Packaging	0.1	0.2	0.2	0.2	0.3	200

   

   
   Table 16.6 shows the total number of labor hours (hours/employee) necessary to manufacture one unit of each component in each manufacturing process, in addition to the total number of employees available who work in each sector. It is important to highlight that each employee works 8 hours a day, from Monday to Friday.

   Table 16.6

   Total Number of Labor Hours Necessary to Produce 1 Unit of Each Product in Each Sector, Besides the Total Number of Employees Available

   Sector	Total Number of Labor Hours to Manufacture 1 Unit					Employees Available
   	Refrigerator	Freezer	Stove	Dishwasher	Microwave Oven
   Pressing	0.5	0.4	0.5	0.4	0.2	12
   Painting	0.3	0.4	0.4	0.4	0.3	10
   Molding	0.5	0.5	0.3	0.4	0.3	8
   Assembly	0.6	0.5	0.4	0.5	0.6	10
   Packaging	0.4	0.4	0.4	0.3	0.2	8

   

   
   Due to storage capacity limitations, there is a maximum production capacity per product, as specified in Table 16.7. The same table also shows the minimum demand for each product that must be met, besides the net profit per unit sold.

   Table 16.7

   Maximum Capacity, Minimum Demand, and Unit Profit Per Product

   Product	Maximum Capacity (Units/Week)	Minimum Demand (Units/Week)	Profit Per Unit (US$/Unit)
   Refrigerator	1000	200	52
   Freezer	800	50	37
   Stove	500	50	35
   Dishwasher	500	50	40
   Microwave oven	200	40	29

   

   
   Formulate the linear programming problem that maximizes the total net profit.

1. (10) A refinery produces three types of gasoline: regular, green, and yellow. Each type of gasoline can be produced from the mixture of four types of petroleum: petroleum 1, petroleum 2, petroleum 3, and petroleum 4.
   Each type of gasoline requires certain specifications of octane and benzene:
   • – A liter of regular gasoline requires, at least, 0.20 L of octane and 0.18 L of benzene
   • – A liter of green gasoline requires, at least, 0.25 L of octane and 0.20 L of benzene
   • – A liter of yellow gasoline requires, at least, 0.30 L of octane and 0.22 L of benzene
   
   The octane and benzene compositions of each type of petroleum are:
   • – A liter of petroleum 1 contains 0.20 of octane and 0.25 of benzene
   • – A liter of petroleum 2 contains 0.30 of octane and 0.20 of benzene
   • – A liter of petroleum 3 contains 0.15 of octane and 0.30 of benzene
   • – A liter of petroleum 4 contains 0.40 of octane and 0.15 of benzene
   
   Due to contracts that have already been signed, the refinery needs to produce 12,000 L of regular gasoline, 10,000 L of green gasoline, and 8000 L of yellow gasoline daily. The refinery has a maximum production capacity of 60,000 L of gasoline a day, and can purchase up to 15,000 L of each type of petroleum daily.
   Each liter of regular, green, and yellow gasoline has a net profit of $ 0.40, $ 0.45 and $ 0.50, respectively. The purchase prices per liter of petroleum 1, petroleum 2, petroleum 3, and petroleum 4 are $ 0.20, $ 0.25, $ 0.30, and $ 0.30, respectively. Formulate the linear programming problem aiming at maximizing the daily net profit.

1. (11) Model Adrianne Medici Torres is upset about some localized fat and would like to lose a few kilos in a few weeks. Her nutritionist recommended a diet that is rich in carbs, moderate in fruit, vegetables, protein, legumes, milk and dairy products, and low in fats and sugar. Table 16.8 shows the food options that can be part of Adrianne’s diet and their respective compositions and characteristics. The data in Table 16.8 can also be found in the file AdrianneTorres’Diet.xls. According to her nutritionist, a balanced diet should contain between 4 and 9 portions of carbs, 3 to 5 portions of fruit, 4 to 5 portions of vegetables, 1 portion of legumes, 2 portions of protein, 2 to 3 portions of milk and dairy products, 1 to 2 portions of sugar and sweets, and 1 to 2 portions of fat. We tried to determine how many portions of each food must be ingested daily, at each meal, in order to minimize the total number of calories consumed, meeting the following requisites:Note: The soup contains 1 portion of carbs, 1 of protein, 1 of vegetables and 1 of fat. Now, a natural sandwich contains 2 portions of carbs, 1 of protein, 1 of milk and dairy product, 1 of vegetables, and 1 of fat.

   Table 16.8

   Composition and Characteristics of Each Food That Can Be Part of Adrianne’s Diet (File AdrianneTorres’Diet.xls)

   Food	Energy (cal/Portion)	Fibers (g/Portion)	% Vitamins and Minerals	Type of Food	Meals
   Lettuce	1	1	9	V	3, 5
   Plums/prunes	30	2.4	4	F	1, 2, 4
   Rice	130	1.2	0.5	C	3
   Brown rice	110	1.6	1	C	3
   Olive oil	90	0	0	TF	3, 5
   Banana	80	2.6	13	F	1, 2, 4
   Cereal bar	90	0.9	11	C	1, 2, 4
   Crackers	90	0.4	0.4	C	1, 2, 4
   Broccoli	10	2.7	15	V	3, 5
   Meat	132	0	1	P	3
   Carrots	31	2	19	V	3, 5
   Cereal	120	1.3	20	C	1
   Chocolate	150	0.2	0.5	SS	3, 5
   Spinach	18	2	28	V	3, 5
   Beans	95	7.9	6	L	3
   Chicken	112	0	1.5	P	3
   Jello	30	0.2	0	SS	3, 5
   Chickpeas	92	3.5	4	L	3
   Yoghurt	70	1.1	0.7	MD	1, 2, 4
   Apples	60	3	0.9	F	1, 2, 4
   Papayas	56	2.4	3.1	F	1, 2, 4
   Eggs	60	0.6	8.5	P	3
   Butter	100	0	0	TF	1, 5
   Bread	140	0.5	3.3	C	1, 5
   Wholewheat bread	142	0.8	12	C	1, 5
   Turkey ham	75	0.4	0.4	P	1, 5
   Fish	104	0.7	11	P	3
   Pears	88	4	1.2	F	1, 2, 4
   Cottage cheese	80	0.4	0.6	MD	1, 5
   Arugula	4	1	9.5	V	3, 5
   Natural sandwiches	240	1.4	19	Mixed	5
   Soya	85	3.9	8	L	3
   Soup	120	3.5	16	Mixed	5
   Tomatoes	26	1.5	5	V	3, 5

   

   C, carbs; V, vegetables; F, fruit; P, protein; L, legumes; MD, milk and dairy products; TF, total fat; SS, sugar and sweets; 1, food that can be eaten at breakfast; 2 food that can be eaten as a morning snack; 3, food that can be eaten at lunch; 4, food that can be eaten as an afternoon snack; 5, food that can be eaten at dinner.

   1. (a) The ideal number of portions ingested, of each type of food, must be respected.
   2. (b) Each food can only be ingested at the meal specified in Table 16.8. For example, in the case of cereal, we tried to determine how many portions must be ingested daily at breakfast. Now, in the case of cereal bars, we tried to specify how many portions can be ingested daily at breakfast and as part of the morning and afternoon snacks.
   3. (c) The total number of calories ingested at breakfast cannot be higher than 300 calories.
   4. (d) The total number of calories ingested as a morning snack cannot be higher than 200 calories.
   5. (e) The total number of calories ingested at lunch cannot be higher than 550 calories.
   6. (f) The total number of calories ingested as an afternoon snack cannot be higher than 200 calories.
   7. (g) The total number of calories ingested at dinner cannot be higher than 350 calories.
   8. (h) At breakfast, she must ingest, at least, 1 portion of carbs, 2 of fruit and 1 of milk and/or dairy products.
   9. (i) Lunch should contain, at least, 1 portion of the following types of food: carbs, protein, legumes, and vegetables.
   10. (j) The morning and afternoon snacks should contain, at least, 1 fruit each.
   11. (k) Dinner should contain, at least, 1 portion of carbs, 1 of protein, 1 of milk and dairy products, and 1 of vegetables.
   12. (l) A balanced diet should contain, at least, 25 g of fibers a day.
   13. (m) 100% of our daily needs of the main vitamins and minerals (iron, zinc, vitamins A, C, B1, B2, B6, B12, niacine, folic acid, etc.) must be met in order for our bodies to work properly. Table 16.8 shows the percentage guaranteed by each portion of food with regard to our daily needs of vitamins and minerals.
       Formulate the linear programming model for Adrianne's diet problem.

1. (12) Company GWX is trying to obtain a competitive differential in the market and, in order to do this, it is considering five new investment projects for the following 3 years: the development of new products, investment in IT, investment in training courses, factory expansion, and warehouse expansion. Each project requires an initial investment and generates an expected return in the following 3 years, as shown in Table 16.9. Currently, the company has a maximum budget of US$ 1,000,000 to invest. For each investment project, the interest rate is 10% per year. It is important to highlight that the investment project in IT depends on the investment project in training, that is, it will only happen if the investment project in training is accepted. Besides, the factory and warehouse expansion projects are mutually excluding, that is, only one of them can be selected. Formulate the problem that has as its main objective to determine in which projects the company should invest, in order to maximize the current wealth generated from the set of investment projects being analyzed.

   Table 16.9

   Initial Investment and Return Expected in the Following 3 Years for Each Project

   Year	Cash Flow Each Year (US$ Thousand)
   	Product Development	Investment in IT	Training	Factory Expansion	Warehouse Expansion
   0	− 360	− 240	− 180	− 480	− 320
   1	250	100	120	220	180
   2	300	150	180	350	200
   3	320	180	180	330	310

   

1. (13) A financial analyst from a major brokerage firm is selecting a certain portfolio for a group of clients. The analyst intends to invest in several sectors, having as an option five companies from the financial sector, including banks and insurance companies, two from the metallurgy sector, one from the mining sector, one from the paper and cellulose sector, and another one from the electrical energy sector. Table 16.10 shows the monthly return history of each one of these stocks in a period of 36 months. These data are available in the file Stocks.xls.

   Table 16.10

   Monthly Return History of 10 Stocks From Different Industries in a Period of 36 Months (File Stocks.xls)

   	Stock 1	Stock 2	Stock 3	Stock 4	Stock 5	Stock 6	Stock 7	Stock 8	Stock 9	Stock 10
   	Banking (%)	Banking (%)	Banking (%)	Banking (%)	Insurance (%)	Metallurgy (%)	Metallurgy (%)	Mining (%)	Paper-Cellulose (%)	Electrical Energy (%)
   1	2.57	4.47	1.08	4.78	4.19	2.54	0.57	0.60	4.07	2.78
   2	3.14	4.33	0.87	3.41	3.08	2.69	0.98	5.78	3.57	3.69
   3	6.00	2.67	4.87	2.81	6.47	1.98	5.69	3.25	2.69	− 2.14
   4	2.14	− 3.59	3.57	6.70	8.05	− 3.14	− 3.10	− 0.88	2.02	4.01
   5	− 5.44	3.34	− 2.78	2.08	5.04	− 7.58	− 3.28	− 4.52	− 1.57	− 1.33
   6	11.30	2.09	− 5.69	− 3.00	− 3.47	6.85	− 8.07	− 2.88	− 2.33	4.21
   7	8.07	− 7.80	6.44	− 3.54	− 2.09	4.70	2.67	0.58	− 2.87	0.74
   8	2.77	− 6.14	6.87	2.97	− 2.56	11.02	3.69	− 3.69	− 0.05	0.65
   9	2.37	5.77	10.07	5.90	4.44	− 5.99	6.47	− 1.44	1.69	2.47
   10	2.14	− 3.23	− 5.64	− 7.01	6.07	0.14	0.22	− 4.22	5.87	− 3.54
   11	− 4.40	− 1.04	− 3.30	− 2.04	− 5.30	− 2.36	− 3.11	0.47	2.14	− 2.58
   12	− 2.10	− 3.02	− 2.27	3.50	− 2.07	2.14	− 4.55	0.05	1.01	5.47
   13	2.14	2.01	− 5.47	− 9.33	4.44	1.34	0.24	− 6.95	3.99	3.54
   14	4.69	3.67	− 2.10	− 8.07	− 6.14	0.98	− 3.50	8.41	− 1.47	2.57
   15	11.32	− 5.69	2.07	2.77	− 3.07	0.66	− 2.78	− 5.41	2.58	− 4.78
   16	− 4.69	− 2.00	3.47	5.48	− 2.05	2.89	− 8.40	0.22	3.57	− 1.23
   17	2.01	6.75	3.78	− 3.50	2.67	− 13.47	− 7.55	9.54	0.88	0.27
   18	− 7.65	9.47	3.89	6.41	3.07	− 4.23	0.07	− 11.02	− 2.34	3.55
   19	− 2.36	− 5.33	− 5.68	3.04	4.08	− 0.28	9.56	− 2.55	− 1.09	2.67
   20	− 11.47	− 6.01	− 3.46	2.08	4.99	2.63	5.04	− 12.23	7.03	0.74
   21	3.39	− 2.01	− 3.09	3.64	− 3.70	− 3.63	− 3.66	− 2.00	4.33	3.69
   22	− 8.43	5.03	1.01	− 6.80	− 8.02	2.47	− 4.40	4.47	− 5.87	− 0.25
   23	− 4.16	5.33	− 5.61	− 5.47	− 7.35	0.50	2.57	− 6.58	2.67	− 0.98
   24	− 2.37	− 3.36	− 7.43	− 6.17	2.44	− 7.99	− 3.01	− 8.80	7.80	4.36
   25	7.00	11.04	6.40	5.55	11.07	6.01	9.77	5.96	2.22	1.66
   26	3.22	4.64	6.43	4.58	− 2.47	14.15	6.41	3.22	1.49	− 0.20
   27	4.67	2.07	2.98	− 2.07	− 2.60	5.47	− 2.60	4.74	1.42	1.59
   28	3.20	3.68	− 3.10	− 2.65	3.18	− 3.14	− 3.01	− 2.33	− 0.77	5.67
   29	− 0.74	− 0.58	− 2.73	6.47	3.08	− 3.25	7.78	4.01	0.59	4.90
   30	− 5.02	− 7.04	− 9.40	6.07	2.00	− 1.08	8.36	4.32	3.07	3.92
   31	− 4.30	2.99	6.81	5.88	− 6.47	5.47	2.04	− 6.77	− 2.55	2.14
   32	2.64	7.66	6.90	− 0.47	6.13	11.01	2.15	− 2.64	− 0.84	− 0.71
   33	6.77	7.16	5.87	8.09	2.47	5.71	3.19	5.74	5.98	2.04
   34	6.70	− 3.41	6.80	6.47	2.08	− 14.33	− 2.03	9.12	0.25	4.33
   35	2.98	− 2.01	5.32	− 5.00	4.43	− 5.44	6.07	8.40	− 0.50	− 2.36
   36	5.70	11.52	6.00	0.27	2.29	2.47	5.73	6.47	1.00	1.60

   

   
   In order to increase diversification, it was established that the portfolio could contain 50% of stocks from the financial sector (banks and insurance companies) and 40% from each asset, at the most. Besides, the portfolio should contain, at least, 20% of stocks from the banking sector, 20% from the metallurgy or mining sector, and 20% from the paper and cellulose or electrical energy sector. Investors expect the average return of the portfolio to reach a minimum value of 0.80% a.m. Furthermore, the portfolio’s risk, measured by using the standard deviation, cannot be more than 5%. Elaborate the linear programming model that minimizes the portfolio’s mean absolute deviation.

1. (14) Redo the previous exercise considering a period from 1 to 24 months. However, in this case, the model must be formulated for three distinct goals: (a) to minimize the MAD (mean absolute deviation) as in the previous case; (b) to minimize the square root of squared deviations from the mean; (c) min-max (to minimize the highest absolute deviation).

1. (15) CTA Investment Bank manages third parties’ financial resources, operating in several different investment modalities, ensuring to its clients the best return with the lowest risk. Robert Johnson, a client of CTA Investments, wishes to invest US$ 500,000.00 in investment funds. According to Robert’s profile, his bank account manager selected 11 types of investment funds that could be a part of his portfolio. Table 16.11 shows a description of each fund, their annual profitability, their risks, and the necessary initial investment. The annual return expected was calculated as the weighted moving mean of the last five years. The risk of each fund, measured from the standard deviation of the return history, is also specified in Table 16.11. The maximum risk allowed for Robert’s portfolio is 6%. Besides, due to his conservative profile, Robert would like to invest, at least, 50% of his capital in index-pegged funds and fixed income funds and 25% in each one of the other investments, at the most. Formulate the linear programming problem that tries to determine how much to invest in each fund, in order to maximize the expected annual return, respecting the portfolio’s maximum risk constraints, minimum investment in fixed income, and minimum initial investment in each fund.

   Table 16.11

   Characteristics of Each Fund

   Fund	Annual Yield/Return (%)	Risk (%)	Initial Investment (US$)
   Index-pegged fund A	11.74	1.07	30,000.00
   Index-pegged fund B	12.19	1.07	100,000.00
   Index-pegged fund C	12.66	1.07	250,000.00
   Fixed income fund A	12.22	1.62	30,000.00
   Fixed income fund B	12.87	1.62	100,000.00
   Fixed income fund C	12.96	1.62	250,000.00
   Commercial paper A	16.04	5.89	20,000.00
   Commercial paper B	17.13	5.89	100,000.00
   Multimarket fund	18.10	5.92	10,000.00
   Stock fund A	19.53	6.54	1000.00
   Stock fund B	22.16	7.23	1000.00

   

1. (16) Company Arts & Chemical, a leader in the chemical sector, manufactures m products, including plastic, rubber, paints, and polyurethane, among others. The company plans to integrate production, inventories, and transportation decisions. The merchandise can be produced in n different facilities that distribute these products to p different retailers located in the regions of Washington, Baltimore, Philadelphia, New York, and Pittsburgh. The period analyzed is T periods. In each period, we intend to determine which of the n facility alternatives should produce and deliver each one of the m products to each one of the different p retailers. Each facility can cater to more than one retailer. However, the total demand for each retailer must be met by a single facility. The production and storage capacity of each one of the facilities is limited and differs from one another depending on the product and period. Unit production, transportation, and inventory maintenance costs also differ per product, facility, and period. The main objective is to designate retailers to facilities, to determine how much to produce and the level of inventories of each product in each facility and period—in such a way that the sum of the total production, transportation, and inventory maintenance costs is minimized, the demand for each retailer is met, and capacity constraints are not rejected. From the general production and inventory model proposed in Section 16.6.6, elaborate a general model that integrates production, inventory, and distribution decisions.

Note: Since we have a binary decision variable (if product i is delivered by facility j to retailer k in period t, its value will be 1, otherwise, the value will be 0), we have a mixed programming problem.

1. (17) From the previous exercise, consider a case in which each retailer can receive supplies/products from more than one facility. Elaborate the general adapted model.

Note: In this case, we must define a new decision variable that consists in determining the amount of product i to be transported from facility j to retailer k in period t.

1. (18) Pharmabelz, a company in the cosmetics and cleaning products industry, would like to define the aggregate planning of the production of Leveza, a type of soap, for the first semester of the following year. In order to do that, the sales department provided the demand expected for the period being studied, as shown in Table 16.12.

   Table 16.12

   Soap Demand Expected (kg) for the First Semester of the Following Year

   Month	Demand
   January	9600
   February	10,600
   March	12,800
   April	10,650
   May	11,640
   June	10,430

The production data are:

Each employee usually works 6 business hours a day, 20 business days a month, and he/she is only allowed to work 20 hours overtime a month, at the most. Formulate the aggregate planning model (mixed integer program) that will minimize the total production, labor and storage costs for the period analyzed, respecting the system’s capacity constraints.