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a) Formulate an LP that the production department can use to minimize the total cost instead of the number of employees. (Let c;: the cost of workers who start work on day i) b) Suppose that the production department can meet its daily requirements using both full-time and part-time employees (one day at a time). Let p;: the cost of a part-time worker on day i, solve part a.

a) Formulate an LP that the production department can use to minimize the total cost instead of the number of employees. (Let c;: the cost of workers who start work on day i) b) Suppose that the production department can meet its daily requirements using both full-time and part-time employees (one day at a time). Let p;: the cost of a part-time worker on day i, solve part a.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Sunday. Suppose that the production department wants to meet its daily requirements using only full-time employees. Formulate an LP that the production department can use to minimize the number of full time employees who must be hired. # of Full-Time Employees Required 17 Day Monday Tuesday Wednesday 13 15 Thursday 19 Friday Saturday Sunday 14 16 11 Incorrect Formulation: X; : the number of employees working on day i Minimize Z- XI + X2 + X3 + X4 + X5 + X6 + X7 s.t. X1217 X2 2 13 X3 2 15 X4 2 19 Xs 2 14 X6 2 16 X72 11 Xi 20, vi = 1,2,.7 In this formulation, objective function is not the number of full-time employees. The current objective function counts each employee five times, not once. For example, each employee who starts work on Monday works Monday to Friday and is included in x1, x2, x3, X4 and xs. The key to correctly formulating this problem is to realize that the production department's primary decision is not how many people are working each day but rather how many people begin work on each day of the week. Correct Formulation: Xị : the number of employees beginning work on day i Minimize Z=x, +x,+x,+x,+x,+x, + X, ++x,217 +X, + X, + x,2 13 +x, +x,2 15 +X,2 19 2 14 st. X, + X, + 2 16 X, +x, +x, + x,+x, 2 11 X20, Vi = ,2.7 a) Formulate an LP that the production department can use to minimize the total cost instead of the number of employees. (Let c: the cost of workers who start work on day i) b) Suppose that the production department can meet its daily requirements using both full-time and part-time employees (one day at a time). Let p; the cost of a part-time worker on day i. solve part a.
Transcribed Image Text:Sunday. Suppose that the production department wants to meet its daily requirements using only full-time employees. Formulate an LP that the production department can use to minimize the number of full time employees who must be hired. # of Full-Time Employees Required 17 Day Monday Tuesday Wednesday 13 15 Thursday 19 Friday Saturday Sunday 14 16 11 Incorrect Formulation: X; : the number of employees working on day i Minimize Z- XI + X2 + X3 + X4 + X5 + X6 + X7 s.t. X1217 X2 2 13 X3 2 15 X4 2 19 Xs 2 14 X6 2 16 X72 11 Xi 20, vi = 1,2,.7 In this formulation, objective function is not the number of full-time employees. The current objective function counts each employee five times, not once. For example, each employee who starts work on Monday works Monday to Friday and is included in x1, x2, x3, X4 and xs. The key to correctly formulating this problem is to realize that the production department's primary decision is not how many people are working each day but rather how many people begin work on each day of the week. Correct Formulation: Xị : the number of employees beginning work on day i Minimize Z=x, +x,+x,+x,+x,+x, + X, ++x,217 +X, + X, + x,2 13 +x, +x,2 15 +X,2 19 2 14 st. X, + X, + 2 16 X, +x, +x, + x,+x, 2 11 X20, Vi = ,2.7 a) Formulate an LP that the production department can use to minimize the total cost instead of the number of employees. (Let c: the cost of workers who start work on day i) b) Suppose that the production department can meet its daily requirements using both full-time and part-time employees (one day at a time). Let p; the cost of a part-time worker on day i. solve part a.
Sunday. Suppose that the production department wants to meet its daily requirements using only full-time employees. Formulate an LP that the production department can use to minimize the number of full time employees who must be hired. Day Monday Tuesday Wednesday Thursday Friday Saturday # of Full-Time Employees Required 17 13 15 19 14 16 Sunday 11 Incorrect Formulation: X4 : the number of employees working on day i Minimize Z- xi + X2 + X3 + X4 + Xs + Xs + X7 s.t. XI2 17 X:2 13 X3 2 15 X42 19 Xs 2 14 X6 2 16 X) 211 X 2 0, vi-1,2.7 In this formulation, objective function is not the number of full-time employees. The current objective function counts each employee five times, not once. For example, each employee who starts work on Monday works Monday to Friday and is included in x1, x. X3, X4 and xs. The key to correctly formulating this problem is to realize that the production department's primary decision is not how many people are working each day but rather bow many people begin work on each day of the week. Correct Formulation: x; : the number of employees beginning work on day i Minimize Z-x, +x, +x, **, +x,217 +X, + X, + X,2 13 **, +x,2 15 *X,2 19 s.t. X, + X, + X, +X, +x, + X, +x, +X, + x, + X, +X, + X, + x, + x; X, +*, +*, +x, +x, X, +x, + x, +x, + x, 2 11 x20, Vi - 2.7 214 216 c) Suppose that the number of workers needed on day i is d.. Let w; be the actual number of workers on day i. Formulate an LP where the "cost" of having too many workers on day i is f; (w; – d.). d) Suppose that the production department wants to minimize the maximum of the surpluses on each day (max (w1 – dı, wz – d2,…..,w7– d;)). Formulate an LP. e) Suppose that the production department wants to ensure that at least 30% of the workers have Sunday off. Formulate a constraint for this case.
Transcribed Image Text:Sunday. Suppose that the production department wants to meet its daily requirements using only full-time employees. Formulate an LP that the production department can use to minimize the number of full time employees who must be hired. Day Monday Tuesday Wednesday Thursday Friday Saturday # of Full-Time Employees Required 17 13 15 19 14 16 Sunday 11 Incorrect Formulation: X4 : the number of employees working on day i Minimize Z- xi + X2 + X3 + X4 + Xs + Xs + X7 s.t. XI2 17 X:2 13 X3 2 15 X42 19 Xs 2 14 X6 2 16 X) 211 X 2 0, vi-1,2.7 In this formulation, objective function is not the number of full-time employees. The current objective function counts each employee five times, not once. For example, each employee who starts work on Monday works Monday to Friday and is included in x1, x. X3, X4 and xs. The key to correctly formulating this problem is to realize that the production department's primary decision is not how many people are working each day but rather bow many people begin work on each day of the week. Correct Formulation: x; : the number of employees beginning work on day i Minimize Z-x, +x, +x, **, +x,217 +X, + X, + X,2 13 **, +x,2 15 *X,2 19 s.t. X, + X, + X, +X, +x, + X, +x, +X, + x, + X, +X, + X, + x, + x; X, +*, +*, +x, +x, X, +x, + x, +x, + x, 2 11 x20, Vi - 2.7 214 216 c) Suppose that the number of workers needed on day i is d.. Let w; be the actual number of workers on day i. Formulate an LP where the "cost" of having too many workers on day i is f; (w; – d.). d) Suppose that the production department wants to minimize the maximum of the surpluses on each day (max (w1 – dı, wz – d2,…..,w7– d;)). Formulate an LP. e) Suppose that the production department wants to ensure that at least 30% of the workers have Sunday off. Formulate a constraint for this case.
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