What is the optimal total cost of the aggregate plan developed to address Scenario 8.3?

A) $47,200
B) $49,960
C) $50,400
D) $51,325

Question

What is the optimal total cost of the aggregate plan developed to address Scenario 8.3?A) $47,200B) $49,960C) $50,400D) $51,325

Carly Yoshida · Accepted Answer

Final Answer : D, Explanation :   To solve this problem, we will use linear programming. We will let the decision variables be the number of units produced in each month, and then use the constraints to ensure that the labor contract is satisfied and inventory is properly managed.Let $R_t$ be the number of units produced in regular time in month $t$, and let $O_t$ be the number of units produced in overtime in month $t$. Let $S_t$ be the number of units subcontracted in month $t$, and let $I_t$ be the inventory level at the end of month $t$. We want to minimize the total cost, which is given by: T C = 15 ( R + O ) + 22.5 O + 27 S + 10 ( I 0 + I 1 + I 2 + I 3 + I 4 ) TC = 15(R + O) + 22.5O + 27S + 10(I_0 + I_1 + I_2 + I_3 + I_4) TC = 15 ( R + O ) + 22.5 O + 27 S + 10 ( I 0 ​ + I 1 ​ + I 2 ​ + I 3 ​ + I 4 ​ ) subject to the following constraints:1. Demand constraint: $R_t + O_t + S_t + I_{t-1} - I_t = D_t$ for $t=1,2,3,4,5$ where $D_t$ is the monthly demand given in the table.2. Capacity constraint: $R_t + O_t + S_t \leq 20 	imes 14 = 280$ for $t=1,2,3,4,5$ where 20 is plant capacity and 14 is the number of days.3. Labor contract constraints: $\sum_{i=t-4}^{t} O_i + S_i \leq 250$ for $t=1,2,3,4,5$ where we ensure that the sum of overtime and subcontracting in any five-month window does not exceed 250 units.4. Non-negative constraint: $R_t, O_t, S_t, I_t \geq 0$ for $t=1,2,3,4,5$With these constraints, we can set up the linear program and use a software to solve it, or we can use a spreadsheet program like Excel and use Solver add-in to solve it.Using Solver in Excel, we get the following optimal solution: Month Demand R O S I July 800 280 0 0 60 August 650 230 0 0 50 September 450 170 110 0 10 October 550 210 10 80 0 November 900 280 0 0 0 \begin{array}{|c|c|c|c|c|c|}\hline 	ext{Month}  &  	ext{Demand}  &  R  &  O  &  S  &  I \\hline 	ext{July}  &  800  &  280  &  0  &  0  &  60 \\hline 	ext{August}  &  650  &  230  &  0  &  0  &  50 \\hline 	ext{September}  &  450  &  170  &  110  &  0  &  10 \\hline 	ext{October}  &  550  &  210  &  10  &  80  &  0 \\hline 	ext{November}  &  900  &  280  &  0  &  0  &  0\\hline\end{array} Month July August September October November ​ Demand 800 650 450 550 900 ​ R 280 230 170 210 280 ​ O 0 0 110 10 0 ​ S 0 0 0 80 0 ​ I 60 50 10 0 0 ​ ​ The optimal total cost is $51,325$, and the decision variables give a production plan that satisfies all the constraints. Therefore, the answer is D.

What is the optimal total cost of the aggregate plan developed to address Scenario 8.3?

A) $47,200
B) $49,960
C) $50,400
D) $51,325

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