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1 Applications of Linear and Integer Programming Models - 2 Chapter 3.

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1 1 Applications of Linear and Integer Programming Models - 2 Chapter 3

2 2 3.5 Applications of Integer Linear Programming Models Many real life problems call for at least one integer decision variable. There are three types of Integer models: Pure integer (AILP) Mixed integer (MILP) Binary (BILP)

3 3 The use of binary variables in constraints AAny decision situation that can be modeled by “yes” / “no”, “good” / “bad” etc., falls into the binary category. To illustrate

4 4 Example A decision is to be made whether each of three plants should be built (Y i = 1) or not built (Y i = 0) RequirementBinary Representation At least 2 plants must be builtY 1 + Y 2 + Y 3  2 If plant 1 is built, plant 2 must not be builtY 1 + Y 2  1 If plant 1 is built, plant 2 must be builtY 1 – Y 2  One, but not both plants must be builtY 1 + Y 2 = 1 Both or neither plants must be builtY 1 – Y 2 =0 Plant construction cannot exceed $17 million given the costs to build plants are $5, $8, $10 million5Y1+8Y2+10Y3  17 The use of binary variables in constraints

5 5 Example - continued Two products can be produced at a plant. Product 1 requires 6 pounds of steel and product 2 requires 9 pounds. If a plant is built, it should have 2000 pounds of steel available. The production of each product should satisfy the steel availability if the plant is opened, or equal to zero if the plant is not opened. 6X 1 + 9X 2  2000Y 1 The use of binary variables in constraints If the plant is built Y 1 = 1. The constraint becomes 6x 1 + 9X 2  2000 If the plant is not built Y 1 = 0. The constraint becomes 6x 1 + 9X 2  0, and thus, X 1 = 0 and X 2 = 0

6 Personnel Scheduling Models Assignments of personnel to jobs under minimum required coverage is a typical integer problems. When resources are available over more than one period, linking constraint link the resources available in period t to the resources available in a period t+1.

7 7

8 8 You look so funny

9 9 The City of Sunset Beach staffs lifeguards 7 days a week. Regulations require that city employees work five days. Insurance requirements mandate 1 lifeguard per 8000 average daily attendance on any given day. The city wants to employ as few lifeguards as possible. Sunset Beach Lifeguard Assignments

10 10 Problem Summary Schedule lifeguard over 5 consecutive days. Minimize the total number of lifeguards. Meet the minimum daily lifeguard requirements Sun.Mon. Tue. Wed. Thr. Fri. Sat Sunset Beach Lifeguard Assignments

11 11 Decision Variables X i = the number of lifeguards scheduled to begin on day “ i ” for i=1, 2, …,7 (i=1 is Sunday) Objective Function Minimize the total number of lifeguard scheduled Constraints Ensure that enough lifeguards are scheduled each day. Sunset Beach Lifeguard Assignments

12 12 To ensure that enough lifeguards are scheduled for each day, identify which workers are on duty. For example: … Sunset Beach Lifeguard Assignments

13 13 X1 X1 Sunset Beach Lifeguard Assignments X6 X6 X5 X5 X4 X4 X3 X3 Tue. Wed. Thu. Fri. Sat Sun. Who works on Saturday ? Who works on Friday ? X2 Mon X3 X4 X5 X6 Repeat this procedure for each day of the week, and build the constraints accordingly.

14 14 Sunset Beach Lifeguard Assignments MinX 1 + X 2 +X 3 +X 4 + X 5 +X 6 + X 7 S.T.X 1 +X 4 + X 5 +X 6 + X 7  8 X 1 + X 2 + X 5 +X 6 + X 7  6 X 1 + X 2 + X 3 + X 6 + X 7  5 X 1 + X 2 + X 3 +X 4 +X 7  4 X 1 + X 2 + X 3 +X 4 +X 5  6 X 2 + X 3 +X 4 +X 5 +X 6  7 X 3 +X 4 +X 5 +X 6 +X 7  9 All the variables are non negative integers

15 15 Sunset Beach Lifeguard Assignments

16 16 Sunset Beach Lifeguard Assignments

17 17 TOTAL LIFEGUARDS OPTIMAL ASSIGNMENTS LIFEGUARDS DAYPRESENTREQUIREDBEGIN SHIFT SUNDAY 981 MONDAY 860 TUESDAY 651 WEDNESDAY 541 THURSDAY 663 FRIDAY 772 SATURDAY Note: An alternate optimal solution exists. Sunset Beach Lifeguard Assignments

18 18 These models involve a “go/no-go” situations, that can be modeled using binary variables. Typical elements in such models are: Budget Space Priority conditions Project selection Models

19 19 The Salem City Council needs to decide how to allocate funds to nine projects such that public support is maximized. Data reflect costs, resource availabilities, concerns and priorities the city council has. Salem City Council – Project Selection

20 20 Survey results X1X2X3X4X5X6X7X8X9X1X2X3X4X5X6X7X8X9 Salem City Council – Project Selection

21 21 Decision Variables : X j - a set of binary variables indicating if a project j is selected (X j =1) or not (X j =0) for j=1,2,..,9. Objective function: Maximize the overall point score of the funded projects Constraints: See the mathematical model. Salem City Council – Project Selection

22 22 Either police car or fire truck be purchased Sports funds and music funds must be restored before computer equipment is purchased Sports funds and music funds must be restored / not restored together The maximum amounts of funds to be allocated is $900,000 The number of police-related activities selected is at most 3 (out of 4) The number of new jobs created must be at least 10 Salem City Council – Project Selection The Mathematical Model Max 4176X X X X X X X X X 9 S.T.400X X X X X X X X X 9  900 7X 1 +X 3 + 2X 5 +X 6 + 8X 7 +3X 8 + 2x 9  10 X 1 +X 2 +X 3 +X 4  3 X 3 +X 5 =1 X 7 -X 8 =0 X 7 -X 9  0 x 8 -x 9  (X i = 0,1 for i=1, 2…, 9)

23 23 Salem City Council – Project selection =SUMPRODUCT(B4:B12,E4:E1 2) =SUMPRODUCT(B4:B12,C4:C12) =SUMPRODUCT(B4:B12,D4:D12) =SUM(B4:B7) =B6+B8 =B10-B11 =B10-B12 =B11-B12

24 Supply Chain Management Supply chain management models integrate the manufacturing process and the distribution of goods to customers. The overall objective of these models is to minimize total system costs The requirements concern (among others) Appropriate production levels Maintaining a transportation system to satisfy demand in timely manner.

25 25 Globe Electronics, Inc. manufactures two styles of remote control cable boxes, G50 and H90. Globe runs four production facilities and three distribution centers. Each plant operates under unique conditions, thus has a different fixed operating cost, production costs, production rate, and production time available. Globe Electronics, Inc.

26 26 Demand has decreased, and management is contemplating closing one or more of its facilities. Management wishes to: – Develop an optimal distribution policy. – Determine which plant to close (if any). Globe Electronics, Inc.

27 27 Data Production costs, Times, Availability Monthly Demand Projection Globe Electronics, Inc.

28 28 Transportation Costs per 100 units At least 70% of the demand in each distribution center must be satisfied. Unit selling price G50 = $22; H90 = $28. CityFrancisco CincinnatiKansasSan Philadelphia $ St.Louis New Orleans Denver Globe Electronics, Inc.

29 29 Ordering raw material Scheduling personnel Production 1.Production level for each product in each plant. 2.Distribution plan. Distribution centers 1. Storage 2. Sale and Dissemination to retail establishments The Globe problem Globe Electronics, Inc.

30 Philadelphia St. Louis New Orleans Denver Cincinnati Kansas City San Francisco G 11, H 11 G 12, H 12 G 22, H 22 G 31, H 31 G 13, H 13 Transportation variables Globe Electronics, Inc. - Variables G 41, H 41

31 Philadelphia St. Louis New Orleans Denver Cincinnati Kansas City San Francisco G 11, H 11 G 12, H 12 G 22, H 22 G 31, H 31 G 13, H 13 Total production of G50 in Philadelphia = G P = G 11 G 12 G 11 G 13 G 12 G 13 G 12 G 13 G 12 Production variables in each plant G 13 G 12 G 11 Globe Electronics, Inc. - Variables

32 Philadelphia St. Louis New Orleans Denver Cincinnati Kansas City San Francisco G 11, H 11 G 12, H 12 G 22, H 22 G 31, H 31 G 13, H 13 Shipment variables to each distribution center Total shipment of H90 to Cincinnati = H C = H 11 + H 21 + H 31 +H 41 Globe Electronics, Inc. - Variables

33 33 Globe Electronics Model No. 1: All The Plants Remain Operational

34 34 Objective function Max Gross Profit = 22(Total G50)+28(Total H90) – Total Production Cost – Total transportation Cost = Max 22G + 28H G = total number of G50 produced H = total number of H90 produced Globe Electronics – all plants opened

35 35 Objective function Max Gross Profit = 22(Total G50)+28(Total H90) – Total Production Cost – Total transportation Cost = Max 22G + 28H – 2H 11 – 3H 12 – 5H 13 – 1H 21 – 1H 22 – 4H 23 – 2H 31 – 2H 32 – 3H 33 – 3H 41 – 1H 42 – 1H 43 – 2G 11 – 3G 12 – 5G 13 – 1G 21 – 1G 22 – 4G 23 – 2G 31 – 2G 32 – 3G 33 – 3G 41 – 1G 42 – 1G 43 Transportation costs – 10G P – 12G SL – 8G NO – 13G D – 14H P – 12H SL – 10H NO – 15H D Production costs Globe Electronics – all plants opened

36 36 Constraints: Ensure that the amount shipped from a plant equals the amount produced in a plant (summation constraints). For G50 G 11 + G 12 + G 13 = G P G 21 + G 22 + G 23 = G SL G 31 + G 32 + G 33 = G NO G 41 + G 42 + G 43 = G D For H90 H 11 + H 12 + H 13 = H P H 21 + H 22 + H 23 = H SL H 31 + H 32 + H 33 = H NO H 41 + H 42 + H 43 = H D The amount received by a distribution center is equal to all the shipments made to this center (summation constraints). For G50 G 11 + G 21 + G 31 + G 41 = G C G 12 + G 22 + G 32 + G 42 = G KC G 13 + G 23 + G 33 + G 43 = G SF For H90 H 11 + H 21 + H 31 + H 41 = H C H 12 + H 22 + H 32 + H 42 = H KC H 13 + H 23 + H 33 + H 43 = H SF Globe Electronics – all plants opened

37 37 Constraints The amount shipped to each distribution center is at least 70% of its projected demand. The amount shipped to each distribution center does not exceed its demand. Cincinnati:G C  1400G C  2000 H C  3500H C  5000 Kansas CityG KC  2100G KC  3000 H KC  4200H KC  6000 San FranciscoG SF  3500G SF  5000 H SF  4900H SF  7000 Globe Electronics – all plants opened

38 38 Constraints: Production time used at each plant cannot exceed the time available :.06G P +.0 6H P  G SL +.08H SL  G NO +.07H NO  G D +.09H D  640 All the variables are non negative Globe Electronics – all plants opened

39 39 Globe Electronics – all plants opened spreadsheet =F10*F9+F19*F18- SUMPRODUCT(G23:G26,F5:F8) SUMPRODUCT(H23:H26,F14:F17 )- SUMPRODUCT(C5:E8,C23:E26)- SUMPRODUCT(C14:E17,C23:E26 )-SUM(F23:F26) =$I23*$F5+$J23*$F14 Drag to L24:L26

40 40 Globe Electronics 1 - Summary The optimal value of the objective function is $356, Note that the fixed cost of operating the plants was not included in the objective function because all the plants remain operational. Subtracting the fixed cost of $125,000 results in a net monthly profit of $231, Rounding down several non-integer solution values results in an integral solution with total profit of $231,550. This solution may not be optimal, but it is very close to it.

41 41 Globe Electronics Model No. 2: The number of plants that remain operational in each city is a decision variable.

42 42 High set up costs raise the question: Is it optimal to leave all the plants operational? Using binary variables the optimal solution provides suggestions for: Production levels for each product in each plant, Transportation pattern from each plant to distribution center, Which plant remains operational. Globe Electronics – which plant remains opened?

43 43 Binary Decision Variables Y i = a binary variable that describes the number of operational plants in city i. Objective function Subtract the following conditional set up costs from the previous objective function: 40,000Y P + 35,000Y SL + 20,000Y ND + 30,000Y D Constraints Change the production constraints.06G P +.0 6H P  640Y P.07G SL +.08H SL  960Y SL.09G NO +.07H NO  480Y NO.05G D +.09H D  640Y D Globe Electronics – which plant remains opened?

44 44 =F10*F9+F19*F18- SUMPRODUCT(G23:G26,F5:F8) - SUMPRODUCT(H23:H26,F14:F17) -SUMPRODUCT(C5:E8,C23:E26)- SUMPRODUCT(C14:E17,C23:E26) -SUMPRODUCT(F23:F26,A5:A8) Globe Electronics – which plant remains opened?

45 45 Globe Electronics 2 - Summary The Philadelphia plant should be closed, while the other plants work at capacity. Schedule monthly production according to the quantities shown in the Excel output. The net monthly profit will be $266,083 (after rounding down the non-integer variable values), which is $34,544 per month greater than the optimal monthly profit obtained when all four plants are operational.

46 46 Appendix 3.4 (CD): Advertising Models

47 47 Many marketing situations can be modeled by linear programming models. Typically, such models consist of: Budget constraints, Deadlines constraints, Choice of media, Exposure to target population. The objective is to achieve the most effective advertising plan. Appendix 3.4 (CD): Advertising Models

48 48 Vertex Software, Inc. Vertex Software has developed a new software product, LUMBER A marketing plan for this product is to be developed for the next quarter. The product will be promoted using black and white and colored full page ads. Three publications are considered: Building Today Lumber Weekly Timber World

49 49 Requirements A maximum of one ad should be placed in any one issue of any of the publication during the quarter. At least 50 full-page ads should appear during the quarter. at least 8 color ads should appear during the quarter. One ad should appear in each issue of Timber World. At least 4 weeks of advertising should be placed in each of the Building Today and Lumber Weekly publications. No more than $ should be spent on advertising in any one of the trade publications. Vertex Software, Inc.

50 50 Circulation and advertising costs PublicationFrequency Circulat. Cost/Ad Building Today5 day/week400,000Full pg.: $800 Half pg.: $500 Only B&W Lumber Weekly Weekly 250,000B&W pg.: $1500 Color pg.: $4000 Timber WorldMonthly 200,000B&W pg.: $2000 Color pg.: $6000 Key reader attitudesPercentage of Readership AttributeRatingBldng. Lumbr Timber Computer data-base user.5060%8090 Large Firm (>2M sales) Location (city / suburb) Age of firm (>5 years) Vertex Software, Inc.

51 51 Solution The requirements are: Stay within a $90,000 budget for print advertising. Place no more than 65 ads(=5 x 13 weeks) and no less than 20 ads (=5 X 4 weeks) in Building Today. Place no more than 13 and no less than 4 ads in Lumber Weekly. Place exactly 3 ads in Timber World. Place at least 50 full-page ads. Place at least 8 color ads. Spend no more than $40,000 on advertisement in any one of the trade publications. Vertex Software, Inc.

52 52 Variables X 1 = number of full page B&W ads placed in Building Today X 2 = number of half page B&W ads placed in Building Today X 3 = number of full page B&W ads placed in Lumber Weekly X 4 = number of full page color ads placed in Lumber Weekly X 5 = number of full page B&W ads placed in Timber World X 6 = number of full page color ads placed in Timber World Vertex Software, Inc.

53 53 The Objective Function The objective function measures the effectiveness of the promotion operation (to be maximized). It depends on the number of ads in each publication, as well as on the relative effectiveness per ad. A special technique (external to this problem) is applied to evaluate this relative effectiveness. Vertex Software, Inc.

54 54 Vertex Software, Inc. =SUMPRODUCT($B$6:$B$9,C6:C9) Drag to cells D11 and E11 =C$11*C$13*$B17 Drag across to D17:E17 then down to C19:E19. Then delete formulas in cells C17,D19, and E19

55 55  Budget The Mathematical Model Max X X X X X X 6 S.T. 800X X X X X X 6  X 1 + X 2  65 X 1 + X 2  20 X 3 + X 4  13 X 3 + X 4  4 X 5 + X 6 =3 X 1 + X 3 + X 4 +X 5 +X 6  50 X 4 + X 6 ³8 800X X 2  X X 4  X X 6  All variables non-negative Vertex Software, Inc. # of Building Today ads # of Lumber Weekly ads Timber World ads Full Page Colored Maximum spent In each magazine

56 56 Vertex Software, Inc.

57 57 Copyright  2002 John Wiley & Sons, Inc. All rights reserved. Reproduction or translation of this work beyond that named in Section 117 of the United States Copyright Act without the express written consent of the copyright owner is unlawful. Requests for further information should be addressed to the Permissions Department, John Wiley & Sons, Inc. Adopters of the textbook are granted permission to make back-up copies for their own use only, to make copies for distribution to students of the course the textbook is used in, and to modify this material to best suit their instructional needs. Under no circumstances can copies be made for resale. The Publisher assumes no responsibility for errors, omissions, or damages, caused by the use of these programs or from the use of the information contained herein.


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