A $3$PL $($Third$-$Party Logistics$)$ service provider company operates on a three$-$shift schedule, six days a week and aims to optimize labor shift arrangements to maximize outbound sales efficiency while maintaining minimum staffing requirements, meeting operational deadlines, and ensuring legal compliance.

Problem Details:

The company employs $45$ workers and operates three shifts:

Shift Time Minimum Workers Required

Shift A $9$:$00$ AM $-5$:$00$ PM $12$

Shift B $5$:$00$ PM $-1$:$00$ AM $10$

Shift C $1$:$00$ AM $-9$:$00$ AM $10$

During peak hours from $9$:$00$ AM to $5$:$00$ PM$,$ a total of $84$ man$-$hours are required for inbound operations. Shift B & C requires a minimum of $70$ man$-$hours.

The company aims to maximize its monthly outbound target, with a daily maximum threshold of $40,000$ cartons considering available docks, material handling machines & other resources. Distribution orders are received daily at $6$:$30$ PM and must be completed by the next day's $9$:$00$ AM deadline. Additionally, sales orders must be closed at $7$:$00$ AM four times a month to facilitate long$-$haul export vehicle trips per contract.

During outbound sales operations, each worker can pick $1,200$ cartons on average per shift. However, if the picking rate exceeds $1,300$ cartons per shift, observed average damage is $400$ cartons per month. The maximum allowable damage is $0\mathrm{.}0003\%$ of the monthly carton handling.

To comply with regulations, $24$ Working hours in total are available per day as overtime that can be distributed to workers among Shift B & C only with each worker availing maximum of $3$ hours per day depending on workload & to achieve high targets. The workers should be given $1-$hour break in each shift to maintain the energy levels.

The optimization challenge is to develop an optimal labor shift schedule that maximizes outbound sales while adhering to staffing requirements, meeting deadlines, and ensuring legal compliance. Need solution as per goal progarmming technique.